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$$+\binom43\left(\frac12\right)^3\left(1-\frac12\right)^{4-3} \cdot\binom22\left(\frac23\right)^2\left(1-\frac23\right)^{2-2}$$
$$=\frac1{16}\cdot\frac49+\frac14\cdot\frac49=\frac5{36}$$
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# Parallel Vectors: Missing a Solution
#### (A new question of the week)
We were recently asked to check work on an interesting little question about parallel vectors, and I was almost convinced that there was no solution … until I realized there was one! How was it missed? How can we avoid doing that? That’s our goal today.
## When will these vectors be parallel?
Here is the question, from Brine in mid-September:
Here is my work:
Brine is using column-vector form for the vectors; I’ll be using the more compact bracket notation for convenience.
He’s found the vectors $$\overrightarrow{PQ}=\left\langle3,x^2\right\rangle-\left\langle x,0\right\rangle=\left\langle3-x,x^2\right\rangle$$ and $$\overrightarrow{P_1Q_1}=\left\langle 6,x^2\right\rangle-\left\langle2x,1\right\rangle=\left\langle6-2x,x^2-1\right\rangle$$, and then wrote an equation to say that the components of the two vectors are proportional: $$\frac{6-2x}{3-x}=\frac{x^2-1}{x^2}$$ Solving this yields only an imaginary solution, so it appears to be impossible for the vectors to be parallel.
Hi, Brine.
So your answer is, that there is no such real number x?
Your work almost had me convinced, until I tried solving a different way, and saw that there is an answer.
Look carefully at your work, and think about whether there is any step in which you made an unstated assumption
### How to recognize unstated assumptions
I had graphed the situation in GeoGebra, in such a way that I could vary x and observe the two vectors, to see how the problem worked, expecting perhaps to see why there would be no solution. Here is the case $$x=1.5$$, for example: | {
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But I found that I could make them parallel (as we’ll see below). That gave me reason to look more closely at the work. I’d been fooled by following along with his work rather than starting from scratch to solve the problem myself. (That’s a risk even when you check over your own work, because your mind will follow the same paths again. I should have known better, and I soon did!)
Brine replied,
I tried again but unfortunately I keep ending up that they are not parallel. Can you please give me a hint?
I answered with a more detailed hint:
As I said before, “think about whether there is any step in which you made an unstated assumption“. Here are several points in your work that you should think about:
First, you wrote this equation to say that the vectors are parallel:
When can’t you write such an equation? What fact are you using to justify the equation?
Second, here you canceled to simplify a fraction. When can’t you cancel?
Or you could think of this more generally as a rational equation. What is the domain of this equation? What does simplifying change?
Third, here you multiplied both sides of an equation. When can’t you do that?
Not all of these represent actual errors in your work; but they represent situations where you are making unstated assumptions, or ignoring special cases.
Thinking about exactly what you are doing, and what justifies each step, can be a valuable exercise when you have a wrong (extraneous or missing) solution. The answers to my questions are … | {
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1. He’s justifying the proportion because parallel vectors have proportional components … but that isn’t true when any of those components are zero.
2. You can’t cancel when the factor you cancel can be zero.
3. Simplifying by canceling can change the domain, introducing extraneous roots that are valid in the new equation, but are not in the domain of the original.
4. You can’t multiply an equation by a factor that might be zero, because the new equation may be true when the original is not.
Which is the key issue? We’ll see.
### Examples
The same sorts of problems can arise in more familiar problems, so I pointed out a couple examples:
For a similar issue, when you want to check whether two lines are parallel, you have to consider two cases: Either their slopes are equal or … what?
For another similar issue, consider how to solve an equation like x2 = 2x. It seems obvious that you can divide by x and get x=2; but that misses a solution. Why? Because you might be dividing by 0, which isn’t legal. When you divide by x, you are implicitly assuming that x≠0, which you don’t know to be true. The better way is to rearrange and factor: x2 -2x = 0 factors as x(x-2) = 0, and the solutions are x=0 and x=2.
I’m intentionally trying not to give direct hints too soon, because this is a very valuable lesson to learn; as I indicated, I was almost fooled, so it is rather subtle. But it happens a lot, and I want you to have a big “aha” moment you won’t forget.
In checking for parallel lines, checking the slopes won’t catch a case where they don’t have slopes (vertical lines); that is the most direct hint I gave here. The second example illustrates how a solution can be missed because of an unstated assumption (that what you divide by is not zero).
### Finding the answer by an imperfect method
Brine wrote back:
Oh ok I understand you. Is this the answer? 3? | {
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This time Brine has cross-multiplied without first canceling: $$\frac{6-2x}{3-x}=\frac{x^2-1}{x^2}\$$6-2x)(x^2)=(x^2-1)(3-x)\\6x^2-2x^3=3x^2-x^3-3+x\\-x^3+3x^2-x+3=0 and then solved the cubic equation by factoring (by grouping). This gave two solutions: the imaginary solutions \(x=\pm i$$ we saw before, and also $$x=3$$. This work is better, but … I answered, Yes, that’s correct. What you’ve done here is, I think, to cross-multiply your equation (6-2x)/(3-x) = (x2-1)/x2, and solve the resulting cubic. This amounts to factoring instead of dividing, as I recommended, though you started with a division (in writing the equation in the first place). Presumably you then checked that x = 3 worked. Another way to think of it is to go ahead and do what you originally did, but then check whether canceling the x-3 factors eliminated a solution. That is, one could just recognize that it was assumed that $$x-3\ne0$$, and check to see if the opposite assumption produces a solution. Rather than avoiding the assumption, we would be making the assumption explicit, and taking the contrary assumption as a second case. ### The equation doesn’t fully represent the problem Unfortunately, that check would fail! He’s missed a more fundamental error: But did you observe that, when you put x=3 into your initial equation, you get 0/0 = 8/9? So 3 is not actually a solution of that equation! (It would be called extraneous.) And it isn’t obvious that it is a solution of the actual problem, until you check it in the original problem. The real problem arises earlier than that: The equation doesn’t really represent the problem. I generally tell students to check their answer, not by plugging it into their equation, but by checking if it works in the problem itself. When the equation is wrong, this will catch it. But here, the equation turns out to be wrong but our answer is correct! So how do you check the solution? You have go back to the problem: Let P(x,0), Q(3,x2), P1(2x,1), and Q1(6,x2). Find | {
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the solution? You have go back to the problem: Let P(x,0), Q(3,x2), P1(2x,1), and Q1(6,x2). Find all possible values of x ∈ R such that PQ || P1Q1. If x=3, the points are P(3,0), Q(3,9), P1(6,1), and Q1(6,9), and the vectors are PQ = <0,9>, P1Q1 = <0,8>. Are they parallel? Yes, because both are vertical — a case not covered by your equation! That is the ultimate cause of your difficulty. This is what I’d found graphically (for $$x=3$$), which made me look again: So, how can we solve the problem without making the unintentional assumption that the vectors are not vertical? ### Back to square one: What really makes parallel vectors parallel? What you never wrote, at the start, was what defines parallel vectors. I asked, “What fact are you using to justify the equation?”, and I’d still like an answer: What perspective were you taking on parallelism when you wrote that equation? What have you been taught about the meaning of “parallel”? Two vectors are parallel if one is a scalar multiple of the other. In two dimensions, <a,b> and <c,d> are parallel if there is a non-zero scalar k such that <c,d> = k<a,b>; i.e. c = ka and d = kb. You have translated this to c/a = d/b = k; but that assumes that a and b are both non-zero. If you use this form, you need to separately check the contrary case. Equivalently, one could take vectors to be parallel if their direction angles are the same, and then compare the tangents of those angles (their slopes): b/a = d/c. But those don’t exist for vertical vectors. If you do that, you need to separately consider the latter possibility. His proportion was$$\frac{6-2x}{3-x}=\frac{x^2-1}{x^2}.$$So, using his form, my “c/a = d/b”, you need to ask what happens when a denominator is zero, namely $$3-x=0$$ or $$x^2=0$$. The former leads to our missing solution. If you used the “b/a=d/c” form, representing equal slopes,$$\frac{x^2}{3-x}=\frac{x^2-1}{6-2x},$$you’d need to check when these denominators are zero, and in this case both are zero for | {
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need to check when these denominators are zero, and in this case both are zero for $$x=3$$ (that is, both vectors are vertical). So if I were doing this carefully, I would not have started with the equality of two fractions, but with the definition using scalar multiplication: PQ = <3-x,x2> P1Q1 = <6-2x,x2-1> These are parallel if P1Q1 = kPQ, so that 6-2x = k(3-x) x2-1 = kx2 From the first equation, either k = (6-2x)/(3-x) = 2, or x = 3. If k = 2, the second equation implies that x2 = -1, which is impossible. If x = 3, then PQ = <0,9> and P1Q1 = <0,8>, and these are indeed parallel, with k = 8/9. The solution of that first equation would look like this, done carefully:$$6-2x = k(3-x)\\2(3-x)-k(3-x)=0\$$2-k)(3-x)=0\\k=2\text{ or }x=3 Or I might eliminate k from the equations by writing your rational equation, while keeping in mind that in dividing by x2 and by x-3, I was assuming both are non-zero, and check those cases. If we check the case \(x^2=0$$, we find that $$\overrightarrow{PQ}=\left\langle3,0\right\rangle-\left\langle 0,0\right\rangle=\left\langle3,0\right\rangle$$ and $$\overrightarrow{P_1Q_1}=\left\langle 6,0\right\rangle-\left\langle0,1\right\rangle=\left\langle6,-1\right\rangle$$. These are not parallel. | {
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Merely recognizing that your equation does not allow x = 3, and checking that separately, would also be valid, though it’s better to understand why. Just solving without canceling, as you have now done, is less appropriate, but it does lead to the answer (as long as you check it).
This is a very interesting problem, isn’t it?
Brine closed it out:
Yes you are right. I was actually taught that parallel vectors are parallel if they have a k value such that a=kb (where a and b are vectors) I forgot to apply this rule because in grade 12 I was taught by equaling the two slopes if they are parallel. I did learn the other method in the previous grade too but I forgot and I do remember it. The way you showed is way better that the one I did. Thank you so much for teaching me Dr. Peterson!
Sometimes we learn a shortcut method, or transfer it from one topic to another, and neglect to pay attention to conditions that are attached to them. It’s a good lesson!
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# The total number of subarrays
I want to count the number of subarrays for a vector (not combinations of elements).
Ex.
A[1,2,3]
It has 6 subarrays :
{1}, {2}, {3}, {1,2}, {2,3}, {1,2,3}
I think that for a vector of N elements the total number of subarrays is N*(N+1)/2.
I am not able to prove it, can someone do it?
• Welcoming the Math.SE! Here we have a culture of showing what you've tried, as it helps both the responders to give better answers and you to get a better understanding! :) – frogeyedpeas Mar 17 '15 at 22:08
• Looks you forgot to list $\{3, 1\}$ – AgentS Mar 17 '15 at 22:13
• No! I didn't forget {3,1}. A subarray has to have contiguous elements – user72708 Mar 18 '15 at 13:56
Suppose that your vector is $\langle a_1,a_2,\ldots,a_n\rangle$. Imagine a virtual element $a_{n+1}$ at the end; it doesn’t matter what its value is. A subarray is completely determined by the index of its first element and the index of the element that immediately follows its last element. For example, the subarray $\langle a_3,\ldots,a_{n-2}\rangle$ is determined by the indices $3$ and $n-1$, the subarray $\langle a_k\rangle$ is determined by the indices $k$ and $k+1$, and the subarray $\langle a_2,\ldots,a_n\rangle$ is determined by the indices $2$ and $n+1$. Moreover, each pair of distinct indices from the set $\{1,2,\ldots,n+1\}$ uniquely determines a subarray. Thus, the number of subarrays is the number of pairs of distinct indices from the set $\{1,2,\ldots,n+1\}$, which is
$$\binom{n+1}2=\frac{n(n+1)}2\;.$$ | {
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$$\binom{n+1}2=\frac{n(n+1)}2\;.$$
• I am confused why you use the index of the element that immediately follows its last element instead of just the last element. I was thinking the answer should be n C 2, instead of n+ 1 C 2 that you mentioned. Please correct me wherever I am wrong. – Abhishek Bhatia Jul 7 '18 at 19:33
• @Abhishek , You are right, It is n C 2 where we did not calculate the single character substrings. There can be n single character substring . So the result is (n C 2 + n C 1) = (n+1) C 2 . – shuva Oct 31 '18 at 20:15
Consider an arbitrary array of N DISTINCT ELEMENTS (if the elements are the same then I am afraid the formula you are seeking to prove no longer works!).
Naturally there exists 1 array consisting of all the elements (indexed from 0 to N-1)
There exist 2 arrays consisting of N-1 consecutive elements (indexed from 0 to N-2)
and in general there are k arrays consisting of N-k+1 consecutive elements (indexed from 0 to N-k-1)
Proof:
We can access elements 0 ... N-k-1 as the first array, then 1 ... N-k+2 is the second array, and this goes on for all N-k+r until N-k+r = N-1 (ie until we have hit the end). The r that does us is can be solved for :
$$N-k+r = N-1 \rightarrow r -k = -1 \rightarrow r = k-1$$
And the list $$0 ... k-1$$ contains k elements within it
Thus we note that the total count of subarrays is
1 for N elements
2 for N-1 elements
3 for N-2 elements
.
.
.
N for 1 element
And the total sum must be:
$$1 + 2 + 3 ... N$$
Let us see if your formula works
if:
$$1 + 2 +3 ... N = \frac{1}{2}N(N+1)$$
then
$$1 + 2 + 3 ... N+1 = \frac{1}{2}(N+1)(N+2)$$
We verify:
$$\frac{1}{2}N(N+1) + N+1 = (N+1)(\frac{1}{2}N + 1) = (N+1)\frac{N+2}{2}$$
So you're formula does indeed work! Now we verify that for N = 1
$$\frac{1*(1+1)}{2} = 1$$
And therefore we can use the above logic to show that for any and ALL whole numbers N the formula works! | {
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This calculation can be seen as an arithmetic series (i.e. the sum of the terms of an arithmetic sequence).
Assuming the input sequence: $$(a_0, a_1, \ldots, a_n)$$, we can count all subarrays as follows:
\begin{align} \; 1 & \; \text{subarray from} \; a_0 \; \text{to} \; a_{n-1}\\ + \; 1 &\; \text{subarray from} \; a_1 \; \text{to} \; a_{n-1}\\ & \; \ldots \\ + \; 1 & \; \text{subarray from} \; a_{n-1}\; \text{to} \; a_{n-1}\\ = & \; n \end{align}
$$+$$
\begin{align} \; 1 & \; \text{subarray from} \; a_0 \; \text{to} \; a_{n-2}\\ + \; 1 &\; \text{subarray from} \; a_1 \; \text{to} \; a_{n-2}\\ & \; \ldots \\ + \; 1 & \; \text{subarray from} \; a_{n-2}\; \text{to} \; a_{n-2}\\ = & \; n-1\\ \end{align}
$$+ \; \ldots$$
\begin{align} \; \; \; 1 & \; \text{subarray only containing} \; a_0\\ = & \; 1\\ \end{align}
which results in the arithmetic series: $$n + n-1 + … + 1$$.
The above can also be represented as $$\sum_{i=1}^{n}i\;$$ and adds up to $$n (n+1)/2$$.
Elaborating on Brian's answer, lets assume we count all the single character sub-strings. There can be n such single character sub-string. By using the Binomial Coefficient, $$\binom{n}{r}$$ notation, we can say n = $$\binom{n}{1}$$ .
$$total_1 = n = \binom{n}{1}$$
Now let's assume we count all the sub-string that are not single character. Because we have to choose the beginning and end of the sub-string regardless of the order, the count should be $$\binom{n}{2}$$.
$$total_* = \binom{n}{2}$$
Now the total number of sub-string,
$$total = total_1 + total_* = \binom{n}{1} + \binom{n}{2}$$
Now recall Pascal triangle and recurrence relation $$\binom{n+1}{r} = \binom{n}{r} + \binom{n}{r-1}$$. So we can write,
$$total = total_1 + total_* = \binom{n}{1} + \binom{n}{2} = \binom{n+1}{2} = n(n+1)/2$$
There we have the mathematical deduction. (I kind of feel we do not need the fancy recurrence relation to get the final answer though).
• Use LaTeX please. – Michael Rozenberg Oct 31 '18 at 20:31 | {
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• Use LaTeX please. – Michael Rozenberg Oct 31 '18 at 20:31
Trying to explain in layman terms.
Let's say f(0) = 0
f(1) = 1
f(2) = 3
f(3) = 6
f(4) = 10
f(5) = 15
By observation, you can see that each result is just an addition of previous result and current number.
f(n) = n + f(n-1)
So with this formula let's expand f(5).
f(5)
=> 5 + f(4)
=> 5 + 4 + f(3)
=> 5 + 4 + 3 + f(2)
=> 5 + 4 + 3 + 2 + f(1)
=> 5 + 4 + 3 + 2 + 1 + f(0)
=> 5 + 4 + 3 + 2 + 1 + 0
===> This is equal to sum of n numbers = n(n+1)/2 | {
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Given the planar curve, find the equation of the plane
Tags:
1. Oct 15, 2016
dlacombe13
1. The problem statement, all variables and given/known data
$r(t) = < 2e^t - 5 , e^t +3t^2 , 4t^2 +1>$
Is a curve that lies within a plane. Find the equation of this plane.
3. The attempt at a solution
I am not sure if my approach is correct. These are my results:
$x=2e^t - 5$
$y = e^t +3t^2$
$z = 4t^2 + 1$
$z = 4t^2 + 1 ~~\Rightarrow~~ t = \sqrt{\frac{z-1}{4}}$
$x = 2e^\sqrt{\frac{z-1}{4}} - 5 ~~\Rightarrow~~ \frac{x+5}{2} =e^\sqrt{\frac{z-1}{4}}$
$y = e^\sqrt{\frac{z-1}{4}} + \frac{3z-3}{4}$
$y = \frac{x+5}{2} + \frac{3z-3}{4}$
$= \frac{x}{2} + \frac{5}{2} + \frac{3z}{4} - \frac{3}{4}$
$= \frac{1}{2}x - y + \frac{3}{4}z = -\frac{15}{8}$
2. Oct 15, 2016
pasmith
One way of specifying a plane is as $\{ \lambda \mathbf{a} + \mu\mathbf{b} + \mathbf{c} : (\lambda , \mu) \in \mathbb{R}^2\}$ for given vectors $\mathbf{a}$, $\mathbf{b}$ and $\mathbf{c}$ where $\mathbf{a}$ and $\mathbf{b}$ are linearly independent. Then for non-constant functions $f: \mathbb{R} \to \mathbb{R}$ and $g: \mathbb{R} \to \mathbb{R}$ we have that $$\mathbf{r}(t) = f(t)\mathbf{a} + g(t)\mathbf{b} + \mathbf{c}$$ is a curve which lies on this plane.
3. Oct 15, 2016
Ray Vickson
Easier: we have $2y- x = 6 t^2 + 5$, which has eliminated the $e^t$ terms. Now to eliminate the $t^2$ terms, just add or subtract a suitable multiple of $z$. That will leave you with a constant, having no $t$ in it anywhere.
4. Oct 15, 2016
Staff: Mentor | {
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4. Oct 15, 2016
Staff: Mentor
"I am not sure if my approach is correct."
You can (and really should) check your final equation. To do this choose three t values in your equation for r(t). That will give you three vectors, the endpoints of which lie in the plane. These three points should satisfy the plane equation you ended with. If all three points work, then you can be 99% sure that your work is correct. (Subtracting a tiny amount for arithmetic errors you might make). If the three points don't satisfy the plane equation, that's a sign that you have done something wrong.
5. Oct 15, 2016
LCKurtz
I didn't check all your work, but look at your last 3 lines. You start out with $y =$ and end up with $= \frac{1}{2}x - y + \frac{3}{4}z = -\frac{15}{8}$.
Does $y=$ that or is your final equation supposed to be $\frac{1}{2}x - y + \frac{3}{4}z = -\frac{15}{8}$? If that is your answer then notice that $\vec r(0) = \langle -3,1,1\rangle$, so that point must be on the plane. It doesn't seem to satisfy that last equation. It may be a simple arithmetic mistake.
But one reason I didn't check your work is that I don't think eliminating the parameter is a sensible way to do the problem. I would suggest a different approach. All you need for a plane is a point and a normal vector. It's easy to get a point on the plane. Then notice that given the curve is planar, that means the tangent vectors to the curve must be in the plane. So you could calculate $\vec r'(0)\times \vec r'(1)$ to get a normal.
6. Oct 15, 2016
dlacombe13 | {
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6. Oct 15, 2016
dlacombe13
Thank you all for your replies. I tried plugging in t=0,1,2 into each parameter, and plugged it into the equation for the plane. For all three points, I get: -1.75. -15/8 = -1.875.
Thank you all for you're replies, I tried checking my values and got close to -15/8, but I don't like the error, since I think my arithmetic is right. Also, my original thinking was exactly as LCKurtz said. However I ran into issues, but I think that is because I did it wrong. I used:
$T = \frac{r'(t)}{|r'(t)|}$
T x r(t)
But it didn't seem to work. So what you're saying is I just need to take find two random tangent points along the curve, which will give me two random vectors on the plane it is within, and cross them which will yield an orthogonal vector (n), and then just form the equation from it?
7. Oct 15, 2016
dlacombe13
Thanks everyone, I did get the equation finally, using r'(0) x r'(1) = <8,-16,12> and got the equation:
8x - 16y +12z +28 = 0
I verified it by plugging in points, and it works out. One last question before I go...
Why didn't my attempt at using T, the unit tangent vector work? I mean it is still a vector that is in the direction of the curve, and thus on the plane, right?
8. Oct 15, 2016
LCKurtz
Yes, it should have worked. But dividing by the magnitude is unnecessary, adds square roots, and makes the calculations more error prone, which apparently got you. | {
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# Convergence in distribution to a degenerate distribution
This question came up based on a disagreement I had with a TA. This was the specific example:
Let $$X_{1},…,X_{n}X_{1},...,X_{n}$$ be an iid random sample from a population with pdf $$f(x)=3(1-x)^2, 0. The $$nthnth$$ order statistic is represented as $$X_{(n)}X_{(n)}$$.
Question: Find a constant $$vv$$ such that $$n^v(1-X_{(n)})n^v(1-X_{(n)})$$ converges in distribution.
I believe that the intent of the question was to prompt the invocation of Slutsky's theorem (or maybe not); however, to which distribution the presented snippet was supposed to converge was not specified. So I presented a lazy alternative answer as follows:
As the sample size approaches infinity, $$X_{(n)}X_{(n)}$$ becomes arbitrarily close to 1. Therefore, we can simply set $$v=0v=0$$ and the statement will converge to a degenerate distribution. In this case, the order statistic converges to 1, so with $$v=0v=0$$, the statement converges to 0. I later realized that it actually converges to 1 after the transformation, which is reflected in the final degenerate distribution written toward the bottom of the question. I was surprised when the TA said that 0 is not a random variable or a distribution, and my answer made no sense. I second guessed myself and went further:
The cdf of the original distribution is $$F(x)=x^3-3x^2+3xF(x)=x^3-3x^2+3x$$ Using this, I derived the pdf of the order statistic,
$$f_{X_{(n)}}(x)=3n(x^2-2x+1)(x^3-3x^2+3x)^{n-1} f_{X_{(n)}}(x)=3n(x^2-2x+1)(x^3-3x^2+3x)^{n-1}$$
And the cdf of the order statistic, $$f_{X_{(n)}}(x)=(x^3-3x^2+3x)^nf_{X_{(n)}}(x)=(x^3-3x^2+3x)^n$$
I then used this to graph the cdf of the order statistic with $$n=1, n=100, n=1,000,000,000n=1, n=100, n=1,000,000,000$$. Predictably, the graph showed the cdf getting thinner and steeper, until with huge samples it visually looks like a vertical line at $$X_{(n)}=1X_{(n)}=1$$. | {
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I reiterated my argument, and showed the visualization: it is converging to a degenerate distribution. Again, I was rebuffed. After pestering the TA and asking the instructor for the course, I still have no answer for why my answer was wrong, but I can't keep bothering them about it.
Can someone here tell me if my argument is solid, and if not, tell me specifically what I've done that is in error?
EDIT: Here is the final formalism that I gave to tie my argument together. Following Casella and Berger's definition,
A sequence of random variables, $$X_{1}, X_{2},...,X_{1}, X_{2},...,$$ converges in distribution to a random variable $$XX$$ if $$\lim_{n \to \infty} F_{X_n}(x) = F_X(x)\lim_{n \to \infty} F_{X_n}(x) = F_X(x)$$ at all points $$xx$$ where $$F_X(x)F_X(x)$$ is continuous
Since $$\lim_{n \to \infty} F_{X_(n)}(x) = 1\lim_{n \to \infty} F_{X_(n)}(x) = 1$$
And we can define a degenerate variable $$YY$$ with the cdf
$$F_Y(x)=\begin{cases}1, & x\ge1\\ 0, & else \\ \end{cases}F_Y(x)=\begin{cases}1, & x\ge1\\ 0, & else \\ \end{cases}$$
We can say
$$\lim_{n \to \infty} F_{X_(n)}(x) = F_Y(x) \lim_{n \to \infty} F_{X_(n)}(x) = F_Y(x)$$
So by definition, $$X_{(n)}X_{(n)}$$ converges in distribution to the degenerate distribution of $$YY$$.
EDIT: Adding a bounty to this, as it has become important. I did not formally perform the transformation $$Y=1-X_(n)Y=1-X_(n)$$ above, but doing so results in $$YY$$ having a distribution that still converges to the same degenerate distribution above. I was now told that my answer does not prove convergence in distribution at all. Question is answered if someone can definitively prove either 1) my answer is correct, and show exactly how you would have gone about proving it formally and rigorously, or 2) my answer is wrong, exactly why it is wrong, and prove any correct answer that uses degenerate distributions or degenerate variables if such an answer exists. | {
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Your answer is correct (assuming that you have accurately transcribed the question). The proof:
Let $$F_n(c)F_n(c)$$ be the cdf of $$(1 - X_{(n)})(1 - X_{(n)})$$, where $$X_{(n)}X_{(n)}$$ is the greatest element in a sample of size $$nn$$. Let $$F(c)F(c)$$ be the cdf for the constant 0 distribution.
For $$c < 0c < 0$$, of course $$F_n(c) = 0 = F(c)F_n(c) = 0 = F(c)$$.
For $$c > 1c > 1$$, of course $$F_n(c) = 1 = F(c)F_n(c) = 1 = F(c)$$.
For $$0 < c \le 10 < c \le 1$$:
\begin{align} F_n(c) &= P(1 - X_{(n)} \le c) \\ &= P(X_{(n)} \ge 1-c) \\ &= 1-P(X_1 < 1-c, ..., X_n < 1-c) \\ &= 1 - P(X_1 < 1-c)^n \to 1 = F(c) \end{align} \begin{align} F_n(c) &= P(1 - X_{(n)} \le c) \\ &= P(X_{(n)} \ge 1-c) \\ &= 1-P(X_1 < 1-c, ..., X_n < 1-c) \\ &= 1 - P(X_1 < 1-c)^n \to 1 = F(c) \end{align}
And the case $$c = 0c = 0$$ doesn't matter, because $$FF$$ isn't continuous at $$00$$.
If the people you are arguing with don't realise that convergence in distribution to a constant is a thing, you could point them to e.g. Wikipedia's Proofs of convergence of random variables article. | {
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# Did Feynman mentally compute $\sqrt[3]{1729.03}$ by linear approximation?
In the biopic "infinity" about Richard Feynman. (12:54) He computes $\sqrt[3]{1729.03}$ by mental calculation. I guess that he uses linear approximation. That is, he observe that $1728=12^3$. Let $f(x)=\sqrt[3]{x}$. Then $f'(x)=\frac{1}{3\sqrt[3]{x^2}}$ and $f'(1728)=\frac{1}{3\sqrt[3]{1728^2}}=\frac{1}{3\cdot 12^2}$. Therefore, $$\sqrt[3]{1729.03}=f(1729.03)\approx f(1728)+f'(1728)(1729.03-1728)=12+\frac{1.03}{3\cdot 12^2}=12.002384\overline{259}.$$
Question 1. If he used the linear approximation, how did he compute $\frac{1.03}{3\cdot 12^2}=0.002384\overline{259}$ by a mental calculation?
Question 2. If he didn't use the linear approximation, what is another method he might have used?
• You're asking about how a mental computation was accomplished in a scene from a fictional movie...? (Just clarifying; I'm not willing to click a youtube link to get the content of the question.) – Andrew D. Hwang May 1 '16 at 18:04
• @AndrewD.Hwang Yes. Although it is a fictinal movie, I think that maybe the scenarist had a method to approximate $\sqrt[3]{1729.03}$. And I hope this scene can motivate my student to learn linear approximation. – bfhaha May 1 '16 at 18:12
• $1.03/3\approx.34333\approx.34332$ $.34332/144=.05722/24=.02861/12\approx.002384$. – Mark S. May 1 '16 at 18:30
• You should mention what value was computed so that the question is self-contained. – Bill Dubuque May 1 '16 at 18:41
• @Andrew D. Hwang It is mentioned in his book "Surely You're joking, Mr. Feynman!". Give it a shot. – user537100 Jul 1 '18 at 13:09
Feynman tells the story in one of his books of anecdotes.
http://www.ee.ryerson.ca/~elf/abacus/feynman.html
$12$ is a very good first approximation and the linear term of the series expansion suffices to get high precision.
$$\sqrt[3]{1728 + d} = 12\sqrt[3]{1+x} = 12 + 4x + O(x^2)$$ | {
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$$\sqrt[3]{1728 + d} = 12\sqrt[3]{1+x} = 12 + 4x + O(x^2)$$
where $d = 1.03$ and $x = \frac{d}{1728}$ is, in Feynman's words, about 1 part in 2000, so that the error term is of order $10^{-6}$.
Feynman says that he computed $12 + \frac{4d}{1728}$ as the approximate value.
The number was 1729.03. I happened to know that a cubic foot contains 1728 cubic inches, so the answer is a tiny bit more than 12. The excess, 1.03 is only one part in nearly 2000, and I had learned in calculus that for small fractions, the cube root's excess is one-third of the number's excess. So all I had to do is find the fraction 1/1728, and multiply by 4 (divide by 3 and multiply by 12). So I was able to pull out a whole lot of digits that way.
He describes that as though $d=1$ for this part of the calculation, so maybe $12 + \frac{1}{432}$ was what he actually computed. By "adding two more digits" (to 12.002) he seems to mean working out the division in the fraction. It could also mean adding (0.03)/432 as two more decimal digits of accuracy to $(12 + 432^{-1})$, which requires only a multiplication by 3 of an already computed quantity 1/432.
Feynman's method is the one that would have been immediate for anyone familiar with the binomial series and with $12^3 = 1728$. He said that knew the latter as ft^3/in^3 and other people might know it from the Ramanujan 1729 story. The other ingredient, as Feynman says in the story, was being good at integer division.
I upvoted the other answer as it comes from the other book of anecdotes, but one way to calculate this without Feynman's experience might be the following.
You can start with the linear approximation, and then you have to calculate $(\dfrac{1.03}{3})/12^2$. $\dfrac{1.03}{3}\approx .34333$, but that doesn't lend itself to division by $12^2$, so you can change the last digit to get $\dfrac{1.03}{3}\approx .34332$ | {
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(You know this will be helpful by the divisibility test for $3$, and can note it's also divisible by $2$ and $4$ if you're thinking ahead.)
Then $\dfrac{.34332}{12^2}=\dfrac{.05722}{12*2}=\dfrac{.02861}{12}\approx.002384$.
• "linear approximation" here means taking the linear part of $(12 + x)^3$ when solving $(12 + x)^3 = 12^3 + 1.03$. Then $3x(12^2) = 1.03$ and the rest is as in the answer. Thinking of it as 1 or 1.03 divided by 432, how to divide by 432 with minimum calculation? The error in approximating that 432 by 500 is (1/432 - 1/500) = (500- 432)/500*432 or about 68/200000. This gives 0.002 + 0.00034 = 0.00234 using only division by 2, 4*5=20, and 100-32=68. – zyx May 2 '16 at 2:49
• Thanks MarkS. You and zyx gave the best answer for me. But it can only has one answer. So please forgive me to use upvote instead of mark it as an answer. – bfhaha May 2 '16 at 5:12 | {
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Ask your own question, for FREE!
Mathematics
OpenStudy (waheguru):
is there a special rick or is it just trial and error to solve this porblem
OpenStudy (waheguru):
OpenStudy (waheguru):
it would take a really long time to find he asnwer by trial and answer
jimthompson5910 (jim_thompson5910):
Best way to do this is to break things down. Start with 100. Clearly 1+0 = 1 which is not 0, so that doesn't work. But 101 does since 1+0 = 1 which is the last digit. Now ask yourself: are there any more numbers in the range of 100 to 109 that work? The answer is no since 1+0 = 1 is unique to this set of numbers (ie there are no more numbers in this range that have a 1 as the last digit) So again, in the list of numbers from 100 to 109, the only upright number is 101 Similarly, for numbers in the range 110 to 119, the only upright number is 112 Keep going to find the third upright number to be 123 Etc, etc When you come upon numbers that start with 19, you won't find any upright numbers since 1+9 = 10, which is not a single digit. So , in the range of 100 to 199, there are 8 numbers that are upright numbers So from 200 to 299, there are 7 numbers that work. Why 7 and not 8? notice that for numbers that start with 29 add to 2+9 = 11, but that's not a single digit. Likewise, numbers starting with 28 add to 2+8 = 10, again not a single digit. However, the rest work. Keep going and you'll find that only 6 numbers work from 300 to 399 etc etc After all that, you'll have the following number counts: 8, 7, 6, 5, 4, 3, 2, 1, 0 Add them all up to get 36 So the answer is choice C
OpenStudy (anonymous): | {
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OpenStudy (anonymous):
Interesting question. Here's how I tackled it:Look at the first few 3 digit numbers.100101102103104105106107108109They all have the same first two digits, but different last digits. You'll notice that exactly one of these numbers is upright. You'll find the same thing for 110 through 119. I can keep going unless the sum of the first two digits is greater than 9. For example, any number with 67 as the first two digits can never be upright because 6+7 is a two digits number.So to start counting upright numbers, add up all the possible 2 digit combinations you can get. you can use 10 through 99, so that's 90 possible two digit pairs. Now you need to subtract off the pairs that don't work, like 67. 10, 11, 12, 13, ... , 18 all work, but 19 doesn't. That's 9 upright numbers. 20, 21, ... , 27 all work, but 28 and 29 don't. That's 8 more upright numbers. 90 works, but 91, 92, ... , 99 don't. That's just one more upright number. See the pattern? So find 1+2+3+4+5+6+7+8, and you have your answer.
jimthompson5910 (jim_thompson5910):
you have the right idea pmilano, but you fell into the trap that there's a number in the range of 190 to 199 when there isn't one
OpenStudy (anonymous):
By the way, the fast way to add the numbers 1 through 8 (or, more generally, 1 through n) is by using the formula (n)(n+1)/2 So 8(8+1)/2 = 36
OpenStudy (anonymous):
Thanks jim.
OpenStudy (anonymous):
my answer is D)45. There is actually a pattern if you try to list down the first numbers. Starting at 1__, there are 9 possible answers. As you move up to the next hundreds digit, you decrease that number by one, because you can't have a sum of more than 9. 9+8+7+...+2+1 = 45
OpenStudy (anonymous):
I actually realized that before I posted but didn't proof read my post well enough (hence my correct answer and incorrect counting).
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jimthompson5910 (jim_thompson5910):
My bad, I was ignoring zero Start with the range 100-199 and moving to the next 200-299, etc, the following upright numbers are found in the tens digits listed below 0, 1, 2, 3, 4, 5, 6, 7, 8 0, 1, 2, 3, 4, 5, 6, 7 0, 1, 2, 3, 4, 5, 6 0, 1, 2, 3, 4, 5 0, 1, 2, 3, 4 0, 1, 2, 3 0, 1, 2 0, 1 0 So there are 45 here
OpenStudy (waheguru):
i am getting 36?
OpenStudy (anonymous):
Here is another method lol (incase you didnt have enough) When the last digit is 'n', you want to ask yourself how many solutions are there to: x + y = n where x and y are non-negative integers? For example, if n was 3, then these are the possible solutions for x and y: x = 0, y = 3 x = 1, y = 2 x = 2, y = 1 x = 3, y = 0 These correspond to the three digits numbers: 033 123 213 303 Of course we would discard the '033' case. Anywhos, the equation that gives you the number of solutions to "x + y = n" is: $\left(\begin{matrix}n+2-1 \\ 2-1\end{matrix}\right) = \left(\begin{matrix}n+1 \\ 1\end{matrix}\right)$ We need to do this for n = 1, 2, 3,....,9, while remembering to not count the case of the first digit being 0. So we end up with: $\left(\begin{matrix}2 \\ 1\end{matrix}\right)+\left(\begin{matrix}3 \\ 1\end{matrix}\right)+\left(\begin{matrix}4 \\ 1\end{matrix}\right)+\ldots +\left(\begin{matrix}10 \\ 1\end{matrix}\right)-9$$= 2+3+4+\ldots+10-9 = 1+2+3+\ldots+9 = 45$
OpenStudy (anonymous):
First set of numbers: 101,112,123,134,145,156,167,178,189 ---> 9 numbers. For 2__, that is one less than the 1 hundreds because u can only use 0-7 for the 2nd digit. The last possible number is 909, the only number in the 9 hundreds.
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# Integrating $\int \frac{-\sin x}{1+\cos x}\, dx$, I get $\ln(1 + \cos x)$. WolframAlpha gives $2 \ln(\cos \frac x 2)$. Is WA wrong?
So, I'm watching a tutorial on differential equations, where I encountered this little trick:
$$\int \frac{y'}{y}\, dx = \ln(y)$$ It seems perfectly logical and easy to justify, but something fishy happens to this integral:
$$\int \frac{-\sin x}{1+\cos x}\, dx$$
The trick gives $$\int \frac{-\sin x}{1+\cos x}\, dx = \ln(1 + \cos x)$$
while WolframAlpha gives $$\int \frac{-\sin x}{1+\cos x}\, dx = 2 \ln(\cos \frac x 2)$$.
You guys who know this stuff - does WolframAlpha mess up here or is it something I've missed? Taking the derivative of $$2 \ln(\cos \frac x 2)$$ gives me $$-\tan \frac x 2$$, so I don't see how WA may be right.
• Remember that $\cos{2x}=2\cos^2x-1$ – Don Thousand Aug 12 at 11:57
• They are different in terms of their symbols, but they give you exactly the same values. What you have is a trigonometric identity. – Pixel Aug 12 at 12:01
• As others have mentioned, the key here is to use a trig identity to bridge the two solutions. ... Often, the trickiest part of Calculus is remembering your Pre-Calculus. :) – Blue Aug 12 at 12:03
• For future reference, you can always check your (or WA's) answer by differentiating it and checking it against the expression you're integrating (the integrand). If they match up then you know your answer is correct (up to a constant), even if you don't know any trig! – Ben Aug 12 at 12:05
## 5 Answers
$$1+\cos x = 2\cos^2\frac{x}{2}$$
$$\ln (1+\cos x )=\ln( 2\cos^2\frac{x}{2})=\ln2+2\ln\cos\frac{x}{2}$$
And this $$\ln2$$ adds together with the arbitrary constant $$c$$ in indefinite integral and gets cancelled in definite integral.
• Thank you so much! This makes perfect sense! – Seigemann Aug 12 at 11:58
• You're welcome! – Ak19 Aug 12 at 11:58
As Ben suggested in comments, it's always good to check an integral by taking the derivative: | {
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As Ben suggested in comments, it's always good to check an integral by taking the derivative:
$$\dfrac d {dx} \ln(1+\cos x)=\dfrac{-\sin x}{1+\cos x}=\dfrac{-2\sin\dfrac x2 \cos \dfrac x2}{2\cos^2\dfrac x2}=\dfrac{-\sin\dfrac x2}{\cos \dfrac x2}=\dfrac d {dx} 2 \ln \cos \dfrac x2$$
• N.B. The expressions are undefined when $x=\pi$ – J. W. Tanner Aug 12 at 13:11
$$1+\cos \, x=2\cos^{2}(\frac x 2)$$ so $$\ln (1+\cos \, x)=2 \ln (\cos (\frac x 2))+\ln 2$$ . Also, $$-\tan (\frac x 2)$$ is same as $$-\frac {\sin \, x} {1+\cos \, x}$$ because $$\sin \,x =2 \sin (\frac x 2)\cos (\frac x 2)$$ and $$1+\cos \, x=2\cos^{2}(\frac x 2)$$.
$$2\ln\cos\frac x2=\ln\cos^2\frac x2=\ln\frac{1+\cos x}2=\ln(1+\cos x)-\ln2$$ by the half-angle formula, so your answer and WA's are the same up to the integration constant.
When you do the one integral $$I(x)$$ by different methods you get different expressions $$I_1(x),I_2(x),I_3(x),.....$$, however the difference between any two of these is a constant independent of $$x$$. For instance $$I(x)=\int \sin \cos x dx =\frac{1}{2}\int \sin 2x~ dx =-\frac{1}{4} \cos 2x +C_1 =I_1(x).$$
Next if you do integration by parts you get $$I=\sin ^2x -\int \sin x \cos \Rightarrow I=\frac{1}{2} \sin^2 x +C_2=I_2(x).$$
Further I you use a substitution $$\cos x =-t$$, then $$I(x)=-\frac{1}{2}\cos^2 x +C_3=I_3(x).$$
Now check that the difference between any two of $$I_1,I_2,I_3$$ is just a constant.
As pointed in other solutions the thing stated above is happening in your case as well. | {
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Thus, the complex numbers of t… The subsets of the real numbers can be r… However, real numbers have multiplication, and the complex numbers extend the reals by adding i. Intro to complex numbers. The set of complex numbers is closed under addition and multiplication. The set of complex numbers is denoted by C R is a subset of C 118 When adding from MAT 1341 at University of Ottawa Lv 7. hace 5 años. The values a and b can be zero, so the set of real numbers and the set of imaginary numbers are subsets of the set of complex numbers. We define the complex number i = (0,1).With that definition we can write every complex number interchangebly as A complex number such as $5-2 i$ then corresponds to 5 on the real axis and $-2$ on the imaginary axis. Yes. In situations where one is dealing only with real numbers, as in everyday life, there is of course no need to insist on each real number to be put in the form a+bi, eg. Click to see full answer. The table below describes important subsets of the real numbers. Bundle: Elementary Algebra, 9th + Student Workbook (9th Edition) Edit edition. In the complex number a + bi, a is called the real part and b is called the imaginary part. Real numbers can be considered a subset of the complex numbers … In general, all the arithmetic operations can be performed on these numbers and they can be represented in the number line, also. This is because the Real Numbers are a subset of the Complex Numbers (). The set of real numbers is a proper subset of the set of complex numbers. To which subsets of the real numbers does -7 belong? Similarly, since a can be zero, the imaginary numbers are a subset of the complex numbers. Why does it make sense to talk about the 'set of complex numbers'? Notational conventions. The set of complex numbersis, therefore; This construction allows to consider the real numbers as a subset of the complex numbers, being realthat complex number whiose imaginary part is null. Remember that under the set of rational | {
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being realthat complex number whiose imaginary part is null. Remember that under the set of rational numbers, we have the subcategories or subsets of integers, whole numbers, and natural numbers. The real numbers include both rational and irrational numbers. A and B may be equal; if they are unequal, then A is a proper subset of B. 2 I. Complex does not mean complicated; it means that the two types of numbers combine to form a complex, like a housing complex — a group of buildings joined together. However, $\mathbb{C}$ comes with a canonical embedding of $\mathbb{R}$ and in this sense, you can treat $\mathbb{R}$ as a subset of $\mathbb{C}$. In other words, i 2 = –1. Solved Example on Real Numbers Ques: Name the subset(s) of the real numbers to which '- 25' belongs. Find the real part of a complex number: Find the real part of a complex number expressed in polar form: Plot over a subset of the complex plane: Use Re to specify regions of the complex plane: The axiom of mathematical induction is for our purposes frequently The real numbers are complex numbers … Bundle: Elementary and Intermediate Algebra: A Combined Approach + Student Solutions Manual (6th Edition) Edit edition. What are rational and irrational numbers. In 1882, Ferdinand von Lindemann proved that π is not just irrational, but transcendental as well. Since b can be equal to 0, you see that the real numbers are a subset of the complex numbers. The set of real numbers is a subset of the set of complex numbers? they are of a different nature. Addition and multiplication of real numbers are defined in such a way that expressions of this type satisfy all field axioms and thus hold for C. For example, the distributive law enforces Therefore, a set of real numbers is bounded if it is contained in a … A complex numberis defined as an expression of the form: The type of expression z = x + iy is called the binomial form where the real part is the real number x, that is denoted Re(z), and the imaginary partis the | {
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form where the real part is the real number x, that is denoted Re(z), and the imaginary partis the real number y, which is denoted by Im(z). Natural Number (N) Subset N is the set of Natural Number or Counting Numbers given N = {1, 2, 3, ..… Set of Real Numbers Set of Real Numbers is a universal set. Real numbers are just complex numbers with no imaginary part. A mathematical operation of subtracting a complex number from another complex number is called the subtraction of complex numbers.. Introduction. There is a thin line difference between both, complex number and an imaginary number. Let Sbe a subset of the set Nof natural numbers. While the real numbers are a subset of the complex numbers, there are very many complex numbers that are not real numbers. Intro to complex numbers. In mathematics, a set A is a subset of a set B, or equivalently B is a superset of A, if A is contained in B. @HagenvonEitzen All the different constructions of $\mathbb{R}$ rely on the fact that we have already constructed $\mathbb{N}$ before (?). What Number Set Contains The Subset of Complex Numbers? Examples: 1 + i, 2 - 6i, -5.2i, 4. It is important to note that if z is a complex number, then its real and imaginary parts are both real numbers. Notational conventions. Milestone leveling for a party of players who drop in and out? Real numbers $$\mathbb{R}$$ The set formed by rational numbers and irrational numbers is called the set of real numbers and is denoted as $$\mathbb{R}$$. generating lists of integers with constraint. Read More -> The number {3} is a subset of the reals. For example, the set $\mathbf{C}^{2}$ is also a real vector space under the same addition as before, but with multiplication only by real scalars, an operation we might denote $\cdot_{\mathbf{R}}$. Can you put laminate flooring in a mobile home? Bundle: Elementary Algebra + Math Study Skills Workbook (4th Edition) Edit edition. Complex. Complex numbers can be represented as points on a “complex plane”: the | {
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Edit edition. Complex. Complex numbers can be represented as points on a “complex plane”: the rectangular x-y plane, in which the x-axis corresponds to the real numbers, and the y-axis corresponds to the imaginary numbers. Since $\mathbb{Q}\subset \mathbb{R}$ it is again logical that the introduced arithmetical operations and relations should expand onto the new set. The real numbers can be "said to be" a subset of the complex numbers. (examples: -7, 2/3, 3.75) Irrational numbers are numbers that cannot be expressed as a fraction or ratio of two integers. Set Theoretic Definition of Complex Numbers: How to Distinguish $\mathbb{C}$ from $\mathbb{R}^2$? iota.) ): Includes real numbers, imaginary numbers, and sums and differences of real and imaginary numbers. [1] [2] Such a number w is denoted by log z . An imaginary number is a complex number that can be written as a real number multiplied by the imaginary unit i, which is defined by its property i2 = −1. If a jet engine is bolted to the equator, does the Earth speed up? On the same footing, $\mathbb{N} \not \subset \mathbb{Z} \not \subset \mathbb{Q} \not \subset \mathbb{R}$. The relationship between the real and complex numbers from a set theoretic perspective. To make notation a little bit easier, we call a complex number z. A complex number is a number of the form a + bi, where a and b are real numbers, and i is an indeterminate satisfying i 2 = −1.For example, 2 + 3i is a complex number. Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. p S S S II) i.W 2 lIT ~and ir are two of very many real numbers that are not rational numbers. site design / logo © 2021 Stack Exchange Inc; user contributions licensed under cc by-sa. 9 is a real number because it belongs to the set of whole numbers and the set of whole numbers is a subset of real numbers. Real numbers are simply the combination of rational and irrational numbers, in the number | {
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numbers. Real numbers are simply the combination of rational and irrational numbers, in the number system. A real number is a number that can take any value on the number line. definition. D. Irrational Use MathJax to format equations. Be sure to account for ALL sets. To learn more, see our tips on writing great answers. What is internal and external criticism of historical sources? For example, 5i is an imaginary number, and its square is −25. 10, as 10 + 0i - that would be too pedantic, to say the … We call x +yi the Cartesian form for a complex number. Strictly speaking (from a set-theoretic view point), $\mathbb{R} \not \subset \mathbb{C}$. It solves x²+1=0. MathJax reference. 1 See answer AnshulDavid3143 is waiting for your help. 2/5 A. Slideshare uses cookies to improve functionality and performance, and to provide you with relevant advertising. But since the set of complex numbers is by definition $$\mathbb{C}=\{a+bi:a,b\in\mathbb{R}\},$$ doesn't this mean $\mathbb{R}\subseteq\mathbb{C}$, since for each $x \in \mathbb{R}$ taking $z = x + 0i$ we have a complex number which equals $x$? Is Delilah from NCIS paralyzed in real life? To which subset of real numbers does the following number belong? What is the "Ultimate Book of The Master". By now you should be relatively familiar with the set of real numbers denoted $\mathbb{R}$ which includes numbers such as $2$, $-4$, $\displaystyle{\frac{6}{13}}$, $\pi$, $\sqrt{3}$, …. Real numbers are a subset of complex numbers. Two complex numbers a + bi and c + di are defined to be equal if and only if a = c and b = d. If the imaginary part of a complex number is 0, as in 5 + 0i, then the number corresponds to a real number. Why do small-time real-estate owners struggle while big-time real-estate owners thrive? Complex numbers are distinguished from real numbers by the presence of the value i, which is defined as . Complex numbers are an important part of algebra, and they do have relevance to such things as solutions to | {
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numbers are an important part of algebra, and they do have relevance to such things as solutions to polynomial equations. Complex Numbers $\mathbb{C}$ Examples of complex numbers: $(1, 2), (4, 5), (-9, 7), (-3, -20), (5, 19),...$ $1 + 5i, 2 - 4i, -7 + 6i...$ where $i = \sqrt{-1}$ or $i^2 = -1$ The real numbers are a subset of the complex numbers. THE REAL AND COMPLEX NUMBERS AXIOM OF MATHEMATICAL INDUCTION. Any time you deal both with complex vector spaces and real vector spaces, you have to be certain of what "scalar multiplication" means. Dedekind cuts or Cauchy sequences for $\mathbb R$) these ZFC. What is the difference between simple distillation and steam distillation? So, I was taught that $\mathbb{Z}\subseteq\mathbb{Q}\subseteq\mathbb{R}$. The real axis is the line in the complex plane consisting of the numbers that have a zero imaginary part: a + 0i. Complex numbers, say … square root of 30 . The complex numbers C consist of expressions a + bi, with a, b real, where i is the imaginary unit, i.e., a (non-real) number satisfying i 2 = −1. Asking for help, clarification, or responding to other answers. Why did the design of the Boeing 247's cockpit windows change for some models? The irrational numbers are a subset of the real numbers. Imaginary number is expressed as any real number multiplied to a imaginary unit (generally 'i' i.e. Two complex numbers a + bi and c + di are defined to be equal if and only if a = c and b = d. If the imaginary part of a complex number is 0, as in 5 + 0i, then the number corresponds to a real number. Example 1. Pi is an irrational number, which means that it is a real number that cannot be expressed by a simple fraction. Complex numbers can be visualized geometrically as points in the complex (Argand) plane. How do I provide exposition on a magic system when no character has an objective or complete understanding of it? The set of complex numbers includes all the other sets of numbers. rev 2021.1.18.38333, The best answers are | {
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of complex numbers includes all the other sets of numbers. rev 2021.1.18.38333, The best answers are voted up and rise to the top, Mathematics Stack Exchange works best with JavaScript enabled, Start here for a quick overview of the site, Detailed answers to any questions you might have, Discuss the workings and policies of this site, Learn more about Stack Overflow the company, Learn more about hiring developers or posting ads with us, You have $\not\subset$ if you construct them one after another. The imaginary numbers are also a subset of the complex: the complex numbers whose real part is zero. If I am blending parsley for soup, can I use the parsley whole or should I still remove the stems? Is it safe to keep uranium ore in my house? Complex numbers are numbers in the form a + b i a+bi a + b i where a, b ∈ R a,b\in \mathbb{R} a, b ∈ R. And real numbers are numbers where the imaginary part, b = 0 b=0 b = 0. Why did flying boats in the '30s and '40s have a longer range than land based aircraft? The set {0,1, 2+i, 2-i} is NOT a subset of the real numbers. © AskingLot.com LTD 2021 All Rights Reserved. But already the fact that there are several constructions possible (e.g. Every real number graphs to a unique point on the real axis. Why set of real numbers not a set of ordered pairs? The term is often used in preference to the simpler "imaginary" in situations where. mam is real numbers a subset of complex numbers - Mathematics - TopperLearning.com | 8v26wq66 3. Start studying Field of Quotients, the Rational Numbers, the Real Numbers, & Complex Numbers. The real numbers are a subset of the complex numbers, so zero is by definition a complex number ( and a real number, of course; just as a fraction is a rational number and a real number). One can represent complex numbers as an ordered pair of real numbers (a,b), so that real numbers are complex numbers whose second members b are zero. At the same time, the imaginary numbers are the un-real numbers, which cannot be | {
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members b are zero. At the same time, the imaginary numbers are the un-real numbers, which cannot be expressed in the number line and is commonly used to represent a complex number. The set of real numbers can be drawn as a line called “the number line”. Subset. Would coating a space ship in liquid nitrogen mask its thermal signature? Complex numbers, such as 2+3i, have the form z = x + iy, where x and y are real numbers. a complex logarithm of a nonzero complex number z, defined to be any complex number w for which e w = z. a real number is not a set. Real numbers, rational numbers. Expressing complex numbers in form $a+bi$. Each complex number corresponds to a point (a, b) in the complex plane. The square of an imaginary number bi is −b2. In mathematics, a real number is a value of a continuous quantity that can represent a distance along a line. can in general assume complex values with nonzero real parts, but in a particular case of interest, the real part is identically zero. Practice: Parts of complex numbers. In the last example (113) the imaginary part is zero and we actually have a real number. Rational numbers are numbers that can be expressed as a fraction or part of a whole number. It only takes a minute to sign up. There are three common forms of representing a complex number z: Cartesian: z = a + bi JR is the set of numbers that can be used to measure a distance, or the negative of a number used to measure a distance. 5.1.2 The Reals as a Subset of the Complex Numbers Since the complex numbers were seen as an extension of the set of real numbers, it is natural to believe that R is a subset of C. Of course, to prove this subset Similarly, it is asked, is every real number is a complex number? A complex number is a number that can be written in the form a + b i a + bi a+bi, where a and b are real numbers and i is the imaginary unit defined by i 2 = − 1 i^2 = -1 i2=−1. Classification of Real Numbers Examples. The real numbers have the following important | {
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= -1 i2=−1. Classification of Real Numbers Examples. The real numbers have the following important subsets: rational numbers, irrational numbers, integers, whole numbers, and natural numbers. (0,1) = (-1,0), which is purely real and equals to -1. Example 2 : Tell whether the given statement is true or false. Thus we can consider the complex number system as having embedded within it, as a subset the real number … Furthermore, each real number is in the set of complex numbers,, so that the real numbers are a … Choices: A. integers, rational numbers, real numbers B. whole numbers, integers, rational numbers, real numbers C. natural numbers, whole numbers, integer numbers, rational numbers, real numbers D. irrational numbers, real numbers Correct Answer: A Complex numbers are often graphed on a plane. What do you call a 'usury' ('bad deal') agreement that doesn't involve a loan? Some examples of irrational numbers are $$\sqrt{2},\pi,\sqrt[3]{5},$$ and for example $$\pi=3,1415926535\ldots$$ comes from the relationship between the length of a circle and its diameter. The horizontal axis is the real axis and the vertical axis is the imaginary axis. Oh I suppose Russel has a definition where the real number 3 is the set of all things there are 3 of. A composite number is a positive integer which is not prime (i.e., which has factors other than 1 and itself). How are Quaternions derived from Complex numbers or Real numbers? B. ¿Cuáles son los 10 mandamientos de la Biblia Reina Valera 1960? The real numbers are a subset of the complex numbers. So, $$i \times i = -1$$ $$\Rightarrow i = \sqrt{-1}$$. To subscribe to this RSS feed, copy and paste this URL into your RSS reader. Complex numbers introduction. We will now introduce the set of complex numbers. The complex numbers are a plane with an additional real axis to calculate square roots (and other even roots) of negative numbers. The set of complex numbers C with addition and multiplication as defined above is a field with | {
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The set of complex numbers C with addition and multiplication as defined above is a field with additive and multiplicative identities (0,0) and (1,0).It extends the real numbers R via the isomorphism (x,0) = x. Suppose that (1) 1 2S: (2) If a natural number kis in S;then the natural number k+ 1 also is in S: Then S= N:That is, every natural number nbelongs to S: REMARK. There are several types of subsets of real numbers—numbers that can be expressed as a decimal. Thus we can consider the complex number system as having embedded within it, as a subset the real number system. If you're seeing this message, it means we're having trouble loading external resources on our website. These numbers are called irrational numbers, and $\sqrt{2}$, $\sqrt{3}$, $\pi$... belong to this set. Is there even such a set? Proof that π is irrational. Email. But no real number, when squared, is ever equal to a negative number--hence, we call i an imaginary number. Better user experience while having a small amount of content to show. Explain your choice. Complex numbers in the form a + bi can be graphed on a complex coordinate plane. Complex Numbers. x is called the real part and y is called the imaginary part. (The counting numbers are 1,2,3,....) All of these types of numbers are real numbers. We will addres s complex (or imaginary) numbers in the Quadratic Functions chapter. That is the adjacent surface to our 3D! Learn vocabulary, terms, and more with flashcards, games, and other study tools. The real numbers are all the numbers on the number line, where you group rational numbers with a so called dedekind cut (you can form this cut so that it result is irrational). Popular Trending (In fact, the real numbers are a subset of the complex numbers-any real number r can be written as r + 0 i, which is a complex representation.) Real numbers 21.5 pi. Some fixed point O is chosen to represent the complex number … In the 1760s, Johann Heinrich Lambert proved that the number π (pi) is | {
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the complex number … In the 1760s, Johann Heinrich Lambert proved that the number π (pi) is irrational: that is, it cannot be expressed as a fraction a/b, where a is an integer and b is a non-zero integer. Therefore we have: z = Re(z) + iIm(z). Why or why not? In complex analysis, the complex numbers are customarily represented by the symbol z, which can be separated into its real (x) and imaginary (y) parts: = + for example: z = 4 + 5i, where x and y are real numbers, and i is the imaginary unit.In this customary notation the complex number z corresponds to the point (x, y) in the Cartesian plane. A set S of real numbers is called bounded from above if there exists some real number k (not necessarily in S) such that k ≥ s for all s in S.The number k is called an upper bound of S.The terms bounded from below and lower bound are similarly defined.. A set S is bounded if it has both upper and lower bounds. The complex numbers form a COMPLETE system of numbers of which the real numbers form a subset. The conjugate of a complex number z= a+ biis created by changing the sign on the imaginary part: z = a bi: Thus the conjugate of 2 + iis 2 + i= 2 i; the conjugate of p 3 ˇiis p 3 ˇi= p 3 + ˇi. | {
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# Is every odd order skew-symmetric matrix singular?
We call a square matrix $$A$$ a skew-symmetric matrix if $$A=-A^T$$. A matrix is said to be singular if its determinant is zero. Is every odd order skew-symmetric matrix with complex entries singular?
## 3 Answers
Yes, that holds, since: $$\det A=\det{(-A^T)}=(-1)^{odd}\det{A^T}=-\det A,$$ from where we get $\det{A}=0$.
This is actually the case :
Suppose, $A$ is an $n\times n$-matrix.
We have $$\det(A)=\det(-A^T)=(-1)^n\cdot \det(A^T)=(-1)^n\cdot \det(A)$$
Since $n$ is odd, we can conclude $\ \det(A)=-\det(A)\$ implying $\ \det(A)=0\$
• But the problem here is that the entries are complex. Is it true for complex entries as well? Nov 18 '16 at 12:05
• Yes, it is also true for complex entries. The proof does not assum real entries. Nov 18 '16 at 12:47
Consider the example $$\begin{bmatrix} 0 & i & -3\\ -i & 0 & 2i\\ 3 & -2i & 0\\ \end{bmatrix}$$
• I mean a 3 by 3 matrix with rows (0 i -3), ( -i 0 2i) and (3 -2i 0). Nov 18 '16 at 12:21
• If this was intended to be a counterexample, then it is not! The determinant of this matrix is zero. May 6 at 15:03 | {
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##### Cartesian to cylindrical coordinates | {
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8. As with spherical where we again observe that the basis vector in cylindrical coordinates is dependent on the two basis vectors e → x and e → y in Cartesian coordinates. Conversion between Cylindrical and Cartesian Coordinates The rectangular coordinates $$(x,y,z)$$ and the cylindrical coordinates $$(r,θ,z)$$ of a point are related as follows: These equations are used to convert from cylindrical coordinates to rectangular coordinates. Feb 24, 2010 · The cylindrical coordinate system is a 3-D version of the polar coordinate system in 2-D with an extra component for . Find the integrals that compute its volume, using cartesian, cylindrical, and spherical coordinates. Magnitude of a cartesian coordinates (a,b) is given bysqrt(a^2+b^2) and its angle is given by tan^-1(b/a) Let r be the magnitude of (-4,3) and theta be its angle. The old vvvv nodes Polar and Cartesian in 3d are similar to the geographic coordinates with the exception that the angular direction of the longitude is inverted. Jan 27, 2017 · We can write down the equation in Cylindrical Coordinates by making TWO simple modifications in the heat conduction equation for Cartesian coordinates. Layered (2,π2). To Convert from Cartesian to Polar. We will concentrate on cylindrical coordinates in this activity, but we will address spherical coordinates in a later activity. If called with a single matrix argument then each row of C represents the Cartesian coordinate (x, y (, z)). Use Question: A sphere, centered at the origin, has radius 3. A, then, has three vector components, each component corresponding to the projection of A onto the three axes. Cylindrical coordinates are chosen to take advantage of symmetry, so that a velocity component can disappear. 8. I know how to generate the strain tensor in a rotated coordinate system (also a Cartesian one), but just don't know how to apply the rules found in the second link to derive the strain components in the cylindrical coordinates, if I have strain tensor in | {
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link to derive the strain components in the cylindrical coordinates, if I have strain tensor in the corresponding Cartesian coordinates. i tried converting to cartesian coordinates, then plotting but at some points function jumps up or down (i think due to trigonometric functions) , so i wanted to see it in cylindrical coordinates. For the conversion between cylindrical and Cartesian coordinates, it is convenient to assume that the reference plane of the former is the Cartesian xy-plane (with equation z = 0), and the cylindrical axis is the Cartesian z-axis. Each point is uniquely identified by a distance to the origin, called r here, an angle, called ϕ {\displaystyle \phi } ( phi ), and a height above the plane of the coordinate system, called Z in the picture. We can slightly modify our arc length equation in polar to make it apply to the cylindrical coordinate system given that , . Mar 02, 2013 · Cartesian Coordinates vs Polar Coordinates In Geometry, a coordinate system is a reference system, where numbers (or coordinates) are used to uniquely determine the position of a point or other geometric element in space. Cartesian Coordinates Cylindrical coordinates consist of (1) a coordinate plane, plus (2) an axis perpendicular to the plane through the origin. In the coordinate plane, two coordinates describe position: (1) an angle, θ (azimuth angle, measured positive counterclockwise relative to a Richard Fitzpatrick 2016-01-22 a) x 2 - y = 25 to cylindrical coordinates. One of these is when the problem has cylindrical symmetry. 6 Volume element in Cartesian coordinates. 6. Figure 1. Replace (x, y, z) by (r, φ, θ) b. 8: Differential length, area, and volume. 1. Visit https://www. 2 s pherical Convert the three-dimensional Cartesian coordinates defined by corresponding entries in the matrices x, y, and z to cylindrical coordinates theta, rho, and z. Cylindrical Coordinate System: In cylindrical coordinate systems a point P(r 1, θ 1, z 1) is the intersection of the | {
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System: In cylindrical coordinate systems a point P(r 1, θ 1, z 1) is the intersection of the following three surfaces as shown in the following figure. Cylindrical coordinates. The Cylindrical to Cartesian calculator converts Cylindrical coordinates into Cartesian coordinates. Cartesian base vectors. Choose the source and destination coordinate systems from the drop down menus. which means that . Cylindrical coordinates in IR3. (1) The (orthogonal) base vectors in the two systems of coordinates are linked by er i and ˜xi could be two Cartesian coordinate systems, one moving at a con-stant velocity relative to the other, or xi could be Cartesian coordinates and ˜xi spherical polar coordinates whose origins are coincident and in relative rest. We shall see er and eθ in terms of their cartesian components along i and j. Define a spherical data set. The cylindrical coordinate system basically is a combination of the polar coordinate system xy ¡ plane with an additional z ¡ coordinate vertically. This system is a generalization of polar coordinates to three dimensions by superimposing a height axis. Conversion between cylindrical and Cartesian coordinates Cylindrical coordinates are obtained from Cartesian coordinates by replacing the x and y coordinates with polar coordinates r and theta and leaving the z coordinate unchanged. Expressed in Cartesian coordinates, a vector is defined in terms of Jul 22, 2020 · This blog will explain how to create a Stacked Contour Plot if user has data with Cylindrical Coordinates(ro, theta and Z), It will involve the following steps: interpolate data and convert data from cylindrical coordinates to Cartesian coordinates; Use XYZ gridding to convert xyz data into matrix; Clip data in matrix with circle. 2 Cylindrical Coordinates We first choose an origin and an axis we call the -axis with unit vector pointing in the increasing z-direction. 1 c oordinate systems a1. advanced. To use this calculator, a user just enters in the (r, φ, z) | {
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1 c oordinate systems a1. advanced. To use this calculator, a user just enters in the (r, φ, z) values of the cylindrical coordinates and then clicks 'Calculate', and the cartesian coordinates will be automatically computed and Jul 11, 2018 · Understand thoroughly about the Conversion between Cylindrical & Cartesian systems for Electromagnetism. We should bear in mind that the concepts covered in Chapter 1 and demonstrated in Cartesian coordinates are equally applicable to other systems of coordinates. A set of equations relating the Cartesian basis vectors to the basis vectors of the new coordinate system Another way of looking at it is that we take polar coordinates $$(r,\theta)$$ and slap on the rectangular coordinate z to the end to get $$(r,\theta,z)$$ and call this cylindrical coordinates. The cylindrical coordinates (r,θ,z) are related to the Cartesian coordinates (x1,x2,x3) by the following relations r = x2 1 +x 2 2 1/2, θ = tan−1 x2 x1, z = x3, and x1 = rcosθ, x2 = rsinθ, x3 = z. I know the material, just wanna get it over with. bjc a2. In a cylindrical coordinate system, the location of a three-dimensional point is decribed with the first two dimensions described by polar coordinates and the third dimension described in distance from the plane containing the other two axes. In the last two sections of this chapter we’ll be looking at some alternate coordinate systems for three dimensional space. And these coordinates are called Cartesian coordinates, named for Rene Descartes because he's the guy that came up with these. Related Calculators: You can always start in Cartesian because the kinetic energy is a scalar and thus independent of the coordinate system in which you choose to evaluate it, although scalar products are most easily computed in Cartesian coordinates. (1a): Triple integral in Cartesian coordinates x,y,z (1b): Triple integral in cylindrical coordinates r,theta,z (2a): Triple integral in cylindrical coordinates r,theta,z (2b): Triple integral | {
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r,theta,z (2a): Triple integral in cylindrical coordinates r,theta,z (2b): Triple integral in spherical coordinates rho,phi,theta Cylindrical coordinates are defined with respect to a set of Cartesian coordinates, and can be converted to and from these coordinates using the atan2 function as follows. Coordinates in the GIS are measured from the origin point. 25in}z = z\] In order to do the integral in cylindrical coordinates we will need to know 2 We can describe a point, P, in three different ways. To convert a point from Cartesian coordinates to cylindrical coordinates, use equations and In the spherical coordinate system, a point in space is represented by the ordered triple where is the distance between and the origin is the same angle used to describe the location in cylindrical coordinates, and is the angle formed by the positive z Express A using Cartesian coordinates and spherical base vectors. For the x and y components, the transormations are ; inversely, . I have to prove it by simple geometry & calculus, without using jacobian or linear algebra (basis). in 6. As shown in Figure 1-2a, any point in space is defined by the intersection of the three perpendicular surfaces of a circular It is easier to consider a cylindrical coordinate system than a Cartesian coordinate system with velocity vector V=(ur,u!,uz) when discussing point vortices in a local reference frame. It is simplest to get the ideas across with an example. It is good to begin with the simpler case, cylindrical coordinates. Under the formula of the stress tensor of the cylindrical wall under the polar coordinate system, the 30 Mar 2016 Cylindrical Coordinates. Although we have considered the Cartesian system in Chapter 1, we shall consider it in detail in this chapter. Convert the cylindrical coordinates defined by corresponding entries in the matrices theta, rho, and z to three-dimensional Cartesian coordinates x, y, and z. This is no longer the case in spherical! Cylindrical coordinates To | {
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coordinates x, y, and z. This is no longer the case in spherical! Cylindrical coordinates To get a third dimension, each point also has a height above the original coordinate system. Converting Polar Coordinates to Cartesian. Cartesian systems use linear distances while polar systems use radial and The cylindrical coordinate system extends polar coordinates into 3D by using the Cylindrical coordinates are defined with respect to a set of Cartesian In this lesson, we introduce two coordinate systems that are useful alternatives to Cartesian coordinates in three dimensions. any help ? Jul 07, 2009 · If you have any vector in Cartesian coordinates then to transform it to Cylindrical coordinates you use r = sqrt(x^2 + y^2) theta = atan(y/x) z = z That part is easy. edu. The following are the conversion formulas for cylindrical coordinates. Khan Academy is a 501(c)(3) nonprofit organization. May 11, 2019 · Approach 1 for deriving the Divergence in Cylindrical. If there’s a one to one mapping between coordinate systems, we can convert between them. For example, x, y and z are the parameters that define a vector r in Cartesian coordinates: r =ˆıx+ ˆy + ˆkz (1) Similarly a vector in cylindrical polar coordinates is described in terms of the parameters r, θ and z since a vector r can be written as r = rrˆ+ zˆk. system Homework This calculator allows you to convert between Cartesian, polar and cylindrical coordinates. Similarly, the angle that a line makes with the horizontal can be defined by the formula θ = tan-1(m), where m is the slope of the line. There are three commonly used coordinate systems: Cartesian, cylindrical and spherical. The scalar components can be expressed using Cartesian, cylindrical, or spherical coordinates, but we must always use Cartesian base vectors. Convert coordinates from Cartesian to spherical and back. Morrison, Michigan Technological University Cartesian Coordinates L ì̃ ë ë ì̃ ë ì ì̃ ë í ì̃ ì ë ì̃ ì ì ì̃ ì í ì̃ í ë ì í ì ì̃ í í M ë ì í L ä | {
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Cartesian Coordinates L ì̃ ë ë ì̃ ë ì ì̃ ë í ì̃ ì ë ì̃ ì ì ì̃ ì í ì̃ í ë ì í ì ì̃ í í M ë ì í L ä É È È È Ç 2 ò R ë ò T ò R ë ò U E ò R ì ò T Mar 04, 2017 · My question is: does it make a difference if I solve with 2-D cylindrical or 2-D cartesian coordinates and formulation of the Navier Stokes equation? If my mesh is 2-D in r and z, and the flow has no dependence, it seems that the cylindrical form should reduce to the cartesian form (because they can both equally describe my 2D mesh). Both cylindrical and Our page on Cartesian Coordinates introduces the simplest type of coordinate system, where the reference axes are orthogonal (at right angles) to each other. Syntax: set mapping {cartesian | spherical | cylindrical} A cartesian coordinate system is used by default. $\endgroup$ – paisanco Jun 14 '14 at 15:57 pol2cart. 1213 0 -5]' x = 4×1 1. To run this script: Download the attached ZIP folder containing the BAS script file and two SRF files: crv2xyz10. Spherical coordinates, also called spherical polar coordinates (Walton 1967, Arfken 1985), are a system of curvilinear coordinates that are natural for describing positions on a sphere or spheroid. 2 Cylindrical Coordinates The cylindrical system is used for problems involving cylindri-cal symmetry It is composed of: (1) the radial distance r2„0;1/, (2) the azimuthal angle, ˚2„0;2ˇ/, and z2. For example, you might be studying an object with cylindrical symmetry: uid ow in a pipe, heat ow in a metal rod, or light propagated through a cylindrical optical ber. The relationships between (x;y) and (r; ) are exactly the same as in polar coordinates, and the zcoordinate is unchanged. Using Cartesian coordinates on the plane, the distance between two points (x 1, y 1) and (x 2, y 2) is defined by the formula, which can be viewed as a version of the Pythagorean Theorem. Current Location > Math Formulas > Linear Algebra > Transform from Cartesian to Cylindrical Coordinate Transform from Cartesian to Cylindrical Coordinate | {
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from Cartesian to Cylindrical Coordinate Transform from Cartesian to Cylindrical Coordinate Don't forget to try our free app - Agile Log , which helps you track your time spent on various projects and tasks, :) The "magnitude" of a vector, whether in spherical/ cartesian or cylindrical coordinates, is the same. In cylindrical coordinates, (r; ;z), the continuity equation for an incompressible uid is 1 r @ @r (ru r) + 1 r @ @ (u ) + @u z @z = 0 In cylindrical coordinates, (r; ;z), the Navier-Stokes equations of motion for an incompress-ible uid of constant dynamic viscosity, , and density, ˆ, are ˆ Du r Dt u2 r = @p @r + f r+ 52u r u r r2 2 r2 @u Online calculator for definite and indefinite multiple integrals using Cartesian, polar, cylindrical, or spherical coordinates. The rotated Cartesian coordinate method to remove the axial singularity of cylindrical coordinates in finite‐difference schemes for elastic and viscoelastic waves Mingwei Zhuang Department of Electronic Science, Institute of Electromagnetics and Acoustics, Xiamen University, Xiamen, 361005 China Converts from Cartesian (x,y,z) to Cylindrical (ρ,θ,z) coordinates in 3-dimensions. The spherical coordinates of a point are related to its Cartesian coordinates as follows: Online calculator for definite and indefinite multiple integrals using Cartesian, polar, cylindrical, or spherical coordinates. 2. Nov 20, 2009 · Converting to Cylindrical Coordinates. 12 Compute $\ds \int_{-3}^3\int_0^{\sqrt{9-x^2}} \sin(x^2+y^2)\,dy\,dx$ by converting to cylindrical coordinates. ] Show that your equation in step 5 is equivalent to r = c in cylindrical coordinates. Express the values from Steps 1 and 2 as a Cylindrical and spherical coordinates Review of Polar coordinates in IR2. The calculator converts cylindrical coordinate to cartesian or spherical one. , the vector connecting the origin to a general point in space) onto the - plane and the -axis. For example, the circle of radius 2 may be described as the set of all | {
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the - plane and the -axis. For example, the circle of radius 2 may be described as the set of all points whose coordinates x and y satisfy the equation x 2 + y 2 = 2 2 . The n- and t-coordinates move along the path with the particle Tangential coordinate is parallel to the velocity The positive direction for the normal coordinate is toward the center of curvature ME 231: Dynamics Path variables along the tangent (t) and normal (n) In 3D Cartesian coordinates, Burkhart addressed the definition, existence, and uniqueness of the DGF and derived asymptotic expansion formulae, applicable at distances far from the source. Recall that $$x=r*cos Cylindrical coordinates are defined with respect to a set of Cartesian coordinates, and can be converted to and from these coordinates using the atan2 function as follows. Ex. In[1]:= Oct 26, 2005 · Spherical coordinates, also called spherical polar coordinates (Walton 1967, Arfken 1985), are a system of curvilinear coordinates that are natural for describing positions on a sphere or spheroid. Cylindrical Coordinates. When converted into cartesian coordinates, the new values will be depicted as (X, Y, Z). If all you want is the ability to specify velocities in cylindrical coordinates, just pick this option in the boundary condition GUI. 1 As the cylinder had a simple equation in cylindrical coordinates, so does the sphere in spherical coordinates: \rho=2 is the sphere of radius 2. In this section, we provide a working definition of the DGF and a numerical method to calculate Θ in Cartesian and cylindrical coordinates. www. We shall choose coordinates for a point P in the plane z=zP as follows. In Cartesian coordinates, the three unit vectors are denoted i x, i y, i z. e. The formula for it is as follows: It’s important to take into account that the definition of \(\rho$$ differs in spherical and cylindrical coordinates. A Cartesian coordinate system (UK: / k ɑː ˈ t iː zj ə n /, US: / k ɑːr ˈ t i ʒ ə n /) is a coordinate system that | {
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system (UK: / k ɑː ˈ t iː zj ə n /, US: / k ɑːr ˈ t i ʒ ə n /) is a coordinate system that specifies each point uniquely in a plane by a set of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in the same unit of length. Consider a differential element in Cartesian coordinates… Write a Cartesian equation of the cylindrical surface of radius c in the left-hand figure above. What we’ll need: 1. θr Cylindrical coordinates just adds a z-coordinate to the polar coordinates (r,θ). The partial derivatives with respect to x, y and z are converted into the ones with respect to ρ, φ and z. When we know a point in Cartesian Coordinates (x,y) and we want it in Polar Coordinates (r,θ) we solve a right triangle with After rectangular (aka Cartesian) coordinates, the two most common an useful coordinate systems in 3 dimensions are cylindrical coordinates (sometimes called 16 Jun 2018 PDF | On Jun 15, 2018, Audu Eliazar Elisha and others published Laplacian Equation: From Cartesian to Cylindrical Coordinate System | Find, Download scientific diagram | Transformation from Cartesian to cylindrical coordinates preserving the waveguide's width and length. Selecting Z outputs translation along the Z-axis in both Cartesian and cylindrical coordinate systems. The cylindrical coordinate system is a generalization of two-dimensional polar coordinates to three dimensions. r = square root of (x 2 + y 2) Θ = tangent inverse(y/x) z = z cylindrical,and spherical coordinates CM3110 Fall 2011Faith A. THETA is a counterclockwise angular displacement in radians from the positive x -axis, RHO is the distance from the origin to a point in the x-y plane, and Z is the height above cylindrical-coordinate wave equation 2 2 2 2 2 2 2 2 2 2 1 z q c t∂ ∂ + ∂ + ∂ = + ρ φ, (1) which allowed us to transform Eq. 2 Introduction Gradient of a scalar field Divergence of a vector field As shown below, the results for the scattering cross | {
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of a scalar field Divergence of a vector field As shown below, the results for the scattering cross section computed using cylindrical coordinates agree well with the 3d Cartesian simulation. We can see here that r=2 and Examples of orthogonal coordinate systems include the Cartesian (or rectangular ), the cir- cular cylindrical, the spherical, the elliptic cylindrical, the parabolic The colored area under the picture is the unit area in polar coordinates. In the Transform Sensor dialog box, coordinates that make up more than one coordinate system appear only once. The second set of coordinates is known as cylindrical coordinates. 3 Resolution of the gradient The derivatives with respect to the cylindrical coordinates are obtained by differentiation through the Cartesian coordinates, @ @r D @x @r @ @x DeO rr Dr r; @ @˚ D @x @˚ @ @x DreO ˚r Drr ˚: Nabla may now be resolved on the Convert the three-dimensional Cartesian coordinates defined by corresponding entries in the matrices x, y, and z to cylindrical coordinates theta, rho, and z. 5708 3. The polar coordinates of a point P = (x,y) in the first quadrant are given So I'll say that point has the coordinates, tells me where to find that point, negative 2, negative 5. . (The subject is covered in Appendix II of Malvern's textbook. Illustration of cylindrical coordinates illustrating the effect of changing each of the three cylindrical coordinates on the location of a point. 0000 The polar coordinates of a point P ∈ R2 is the ordered pair (r,θ) defined by the picture. The radial part of the solution of this equation is, unfortunately, not May 16, 2011 · Transformation of cartesian coordinates or rectangular coordinates to cylindrical coordinates: The cylindrical coordinates can be transformed to cartesian or rectangular coordinates and vice versa and the relations will be: x = rcos Θ. Continuing with our example, let's sketch the surface represented by z = x 2 Converts 3D rectangular cartesian coordinates to | {
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let's sketch the surface represented by z = x 2 Converts 3D rectangular cartesian coordinates to cylindrical polar coordinates. Draw solids bounded by quadric surfaces using Cartesian Coordinates; Polar Coordinates; Cylindrical Coordinates; Spherical Coordinates; Let us discuss all these types of coordinates are here in brief. Exercises. The vector field is already expressed with Cartesian base vectors, therefore we only need to change the Cartesian coordinates in each scalar component into spherical coordinates. The same is true of triple integrals. Referring to figure 2, it is clear that there is also no radial velocity. E 9. 3-D Cartesian coordinates will be indicated by $x, y, z$ and cylindrical coordinates with $r,\theta,z$. Select the appropriate separator: comma, semicolon, space or tab (use tab to paste data directly from/to spreadsheets). The painful details of calculating its form in cylindrical and spherical coordinates follow. The Cartesian coordinate system provides a straightforward way to describe the location of points in space. The inputs x, y (, and z) must be the same shape, or scalar. 1 Cartesian Coordinate System . Conversion from cartesian to spherical coordinates: Cartesian [x, y, z] Spherical [r, θ, φ] Conversion from spherical to cylindrical coordinates: Spherical [r, θ, φ] Cylindrical [ρ, φ', z'] ρ = r sin θ φ' = φ z' = r cos θ Conversion from cylindrical to spherical coordinates: Cylindrical [ρ, φ, z] Spherical [r, θ, φ'] θ = arctan Transform Cartesian coordinates to polar or cylindrical coordinates. 5–2) must be given. Ex 15. To find a1 requires a two step process: 1) Project xˆ onto the line formed by rˆ and its projection onto the xy plane Aug 28, 2012 · if i,j,k are unit vectors in cartesian system & e(r), e(θ), e(z) are unit vectors in cylindrical system, i have to show that- 'i'= 'e(r)' cosθ – 'e(θ)' sinθ 'j' = 'e(r)' sinθ+ 'e(r)' cosθ 'k'= 'e(r)' the quantities in '_ ' are vectors. r is the distance to the z-axis (0, 0, z). Faith | {
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'k'= 'e(r)' the quantities in '_ ' are vectors. r is the distance to the z-axis (0, 0, z). Faith A. The idea behind cylindrical and spherical coordinates is to use angles instead of Cartesian coordinates to specify points in three dimensions. In this coordinate system, a point P is represented by the triple (r; ;z) where r and are the polar coordinates of the projection of Ponto the xy-plane and zhas the same meaning as in Cartesian coordinates. therightgate. Since we assume (Δr)2 is negligable Three most common coordinate systems used in 3-dimensional representations are: a) Cartesian coordinates b) Cylindrical (polar) coordinates c) Spherical in two dimensions and cylindrical and spherical coordinates in three dimensions. xr x y zˆˆˆˆ ˆ ab c a11 1 1 Note: a1 is the projection of xˆ onto rˆ. Use the unit circle to get . cylindrical polar coordinates In cylindrical polar coordinates the element of volume is given by ddddvz=ρρϕ. So the cylindrical coordinates conversion equations are given in Table 1 and Figure 1 shows this relationship. d) x + y + z = 1 to spherical coordinates. Spherical Unit Vectors in relation to Cartesian Unit Vectors rˆˆ, , θφˆ can be rewritten in terms of xyzˆˆˆ, , using the following transformations: rx yzˆ sin cos sin sin cos ˆˆˆ Mar 28, 2019 · In 3D Cartesian coordinates, Burkhart addressed the definition, existence, and uniqueness of the DGF and derived asymptotic expansion formulae, applicable at distances far from the source. Matlab provides a simple utility for doing that. For a spherical coordinate system, the data occupy two or three columns (or using Code for converting Cartesian (x,y,z) to Cylindrical (ρ,θ,z) coordinates 2D/3D Hot Network Questions Do you really need to fire flashes regularly when not otherwise used? x, and y allow you to change (x, y) coordinates into polar . 3. Recall that the position of a point in the plane can be described using polar coordinates $(r,\theta)$. 7. (2) We now go through a separation-of-variable | {
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be described using polar coordinates $(r,\theta)$. 7. (2) We now go through a separation-of-variable procedure similar to that which we carried out using Cartesian coordinates in But so are cylindrical coordinates (an extension of two-dimensional polar coordinates to three-dimensional) and spherical polar coordinates. 1. Some surfaces and volumes are more easily (simply) described in cylindrical coordinates. plot(cartesian, We describe three different coordinate systems, known as Cartesian, cylindrical and spherical. Given the azimuthal sweep around the z axis theta as well as the radius of the cylinder r, the Cartesian co-ordinates within a cylinder is defined as: x = r*cos(theta) y = r*sin(theta) z = z If data are provided to splot in spherical or cylindrical coordinates, the set mapping command should be used to instruct gnuplot how to interpret them. Cylindrical Coordinates Orientation relative to the Cartesian standard system: The origins and z axes of the cylindrical system and of the Cartesian reference are coincident. 22 Jul 2014 This video explains how to convert rectangular coordinates to cylindrical coordinates. 9 Coordinate Systems in Space. f) ρsin θ = 1 to Cartesian coordiantes. theta describes the angle relative to the positive x-axis. In such cases, one has to first transform these coordinates into Cartesian coordinate system (X,Y, Z) and then only molecular graphics software can be used to visualize these molecules. Conversion from cylindrical to rectangular coordinates requires a simple application of the equations listed in Conversion between Cylindrical and Cartesian Coordinates: x = r cos θ = 4 cos 2 π 3 = −2 y = r sin θ = 4 sin 2 π 3 = 2 3 z = −2. $\begingroup$ The OP does need to compute the unit vectors in cylindrical coordinates and use the divergence, curl and Laplacian in cylindrical coordinates to solve Maxwell's equations, but the question was how to transform the tensor. Initializes a set of Cartesian coordinates from the provided set | {
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was how to transform the tensor. Initializes a set of Cartesian coordinates from the provided set of Cylindrical coordinates. A very common case is axisymmetric flow with the assumption of no tangential velocity ($$u_{\theta}=0$$), and the remaining quantities are independent of $$\theta$$. Using change-of-coordinate functions to change from a known parameterization to another Like polar coordinates, cylindrical coordinates will be useful for describing shapes in that are difficult to describe using Cartesian coordinates. Consider the case when a three dimensional region $$U$$ is a type I region , i. any straight line parallel to the $$z$$-axis intersects the boundary of the region Compute Areas and Volumes in Non-Cartesian Coordinates The "nut" defined by revolving the curve about the axis can be easily parameterized in cylindrical coordinates. If , , and are smooth scalar, vector and second-order tensor fields, then they can be chosen to be functions of either the Cartesian coordinates , , and , or the corresponding real numbers , , and . Cylindrical and spherical coordinates Recall that in the plane one can use polar coordinates rather than Cartesian coordinates. theta = [0 pi/4 pi/2 pi]' theta = 4×1 0 0. com. By using this website, you agree to our Cookie Policy. A circular cylindrical surface r = r 1; A half-plane containing the z-axis and making angle φ = φ 1 with the xz-plane; A plane parallel to the xy-plane at z = z 1 Sep 19, 2014 · Cylindrical Coordinates in Matlab. (ρ, φ, z) is given in cartesian coordinates by: Feb 17, 2016 · Representing 3D points inRepresenting 3D points in Cylindrical Coordinates. Review of Cylindrical Coordinates. 1) are not convenient in certain cases. Mar 14, 2015 · Coordinates for DNA are usually given in the ‘Cylindrical polar coordinate system’ because of its helical symmetry. Cylindrical polar coordinates The cylindrical polar coordinates ρϕ,,z are given, in terms of the rectangular cartesian coordinates x, y, z by z x y z x y | {
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ρϕ,,z are given, in terms of the rectangular cartesian coordinates x, y, z by z x y z x y zz = = = ρϕ ρϕ cos sin. This coordinate system works best when integrating cylinders or cylindrical-like objects. Unzip the folder. a. e) r = 2sinθ to Cartesian coordinates. to_cartesian These formulas are automatically used if we ask to plot the grid of spherical coordinates in terms of Cartesian coordinates: sage: spherical. After rectangular (aka Cartesian) coordinates, the two most common an useful coordinate systems in 3 dimensions are cylindrical coordinates (sometimes called cylindrical polar coordinates) and spherical coordinates (sometimes called spherical polar coordinates). 1213 0 -5. Dec 27, 2016 · How do you find the rectangular coordinates if you given the cylindrical coordinate #(5, pi/6, 5)#? See all questions in Converting Coordinates from Rectangular to Polar Impact of this question The $(x,y,z)$ ECEF cartesian coordinates can be expressed in the ellipsoidal coordinates $(\varphi, \lambda, h)$, where $\varphi$ and $\lambda$ are, respectively, the latitude and longitude from the ellipsoid, and $h$ the height above it. [THETA,RHO,Z] = cart2pol(X,Y,Z) transforms three-dimensional Cartesian coordinates stored in corresponding elements of arrays X, Y, and Z, into cylindrical coordinates. Jul 21, 2020 · Cylindrical coordinates are a generalization of two-dimensional polar coordinates to three dimensions by superposing a height (z) axis. Scale factors for each component's direction. Spherical coordinates in IR3. If a third axis, z (height), is added to polar coordinates, the coordinate system is referred to as cylindrical coordinates (r, θ, z). Sep 20, 2016 · While Cartesian 2D coordinates use x and y, polar coordinates use r and an angle, $\theta$. In this approach, you start with the divergence formula in Cartesian then convert each of its element into the cylindrical using proper conversion formulas. The 2d nodes do match exactly. Transform Cartesian coordinates | {
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using proper conversion formulas. The 2d nodes do match exactly. Transform Cartesian coordinates to polar or cylindrical coordinates. Its elements, however, are something of a cross between the polar and Cartesian coordinates systems. 30 Jan 2020 In the Cartesian coordinate system, the location of a point in space is described using an ordered triple in which each coordinate represents a 26 May 2020 The third equation is just an acknowledgement that the z z -coordinate of a point in Cartesian and polar coordinates is the same. 36: Cylindrical Coordinates 1. Jun 01, 2018 · Recall that cylindrical coordinates are really nothing more than an extension of polar coordinates into three dimensions. (1) The (orthogonal) base vectors in the two systems of coordinates are linked by er Figure B. A set of equations relating the Cartesian coordinates to cylindrical coordinates: 3. 7. Click to copy this Sketching the Surface in Cartesian Coordinates. [i] Fill in the blanks: • If I convert 1 = x2 - y2 from Cartesian to cylindrical coordinates, I get (Simplify. The coordinate systems allow the geometrical problems to be converted into a numerica When given Cartesian coordinates of the form to cylindrical coordinates of the form , the first and third terms are the most straightforward. \[x = r\cos \theta \hspace{0. Example 14. Let be a subset of . The chapter introduces functions to deal with elasticity coefficients, strain-displacement relations, constitutive relations, and equilibrium and Nov 08, 2011 · This is not a cork screw at all! The problem is that plot3 expects cartesian coordinates, but we plotted cylindrical coordinates. Sometimes, employing angles can make mathematical Cartesian coordinates. 1 c ylindrical coordinates a1. 5–2 Cylindrical transformation. Oct 12, 2018 · The Cartesian coordinate system plots a point as $(x,y,z)$, where $x,y$ and $z$ are perpendicular distances of the point measured from the planes $y-z, z-x$ and $x-y$ respectively as you may a Here we | {
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of the point measured from the planes $y-z, z-x$ and $x-y$ respectively as you may a Here we use the identity cos^2(theta)+sin^2(theta)=1. ΔA=(2rΔr+Δr2). Conversion between cylindrical and Cartesian coordinates Section 13. In this section, you will compare grid surfaces in Cartesian, cylindrical, and spherical coordinates. Before going through the Carpal-Tunnel causing calisthenics to . EX 4Make the required change in the given equation (continued). For example, to change the polar coordinate . Convert Cartesian coordinates (x, y, z) to cylindrical coordinates (radius, azimuth, z). (r +Δr)2−π(r)2). The global (X, Y, Z) coordinates of the two points defining the axis of the cylindrical system (points a and b as shown in Figure 2) must be given. Syntax [X,Y] = pol2cart(THETA,RHO) [X,Y,Z] = pol2cart(THETA,RHO,Z) Description [X,Y] = pol2cart(THETA,RHO) transforms the polar coordinate data stored in corresponding elements of THETA and RHO to two-dimensional Cartesian, or xy, coordinates. 6. Thus, is the perpendicular distance from the -axis, and the angle subtended between the projection of the radius vector (i. Now recall our first example, where we graphed the surface (in Cartesian coordinates) defined by the equation . However, there is a large discrepancy in performance: for a single Intel Xeon 4. scale_factors (self). ) • When I convert the point P(k, 0,0) for k > 0 from cylindrical coordinates to Cartesian coordinates, I get • Working with spherical coordinates, when I sketch the graph of p = k for k > 0, the shape of the graph is when I sketch the graph of o= k for 0 < k < TT, k # 1/2, the Using these infinitesimals, all integrals can be converted to cylindrical coordinates. Midpoint formula Cylindrical coordinates To get a third dimension, each point also has a height above the original coordinate system. Δθ2π. Then, polar coordinates (r; ) are de ned in IR2 f(0;0)g, and given by r= p x2 The Cartesian, or rectangular, coordinate system is the most widely used | {
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and given by r= p x2 The Cartesian, or rectangular, coordinate system is the most widely used coordinate system. Δθ2. 2. Convert quadric surfaces in cylindrical or spherical coordinates to Cartesian and identify. From polar coordinates. The true origin point (0, 0) may or may not be in the proximity of the map data you are using. Care should be taken, however, when calculating . Cartesian (double[] elements) Initializes a set of Cartesian coordinates from the first 3 consecutive elements in the provided array. Cartesian coordinates in the figure below: (2,3) A Polar coordinate system is determined by a fixed point, a origin or pole, and a zero direction or axis. First I’ll review spherical and cylindrical coordinate systems so you can have them in mind when we discuss more general cases. In this chapter we will describe a Cartesian coordinate system and a cylindrical coordinate system. 0000 2. Cylindrical coordinates is a method of describing location in a three-dimensional coordinate system. In this handout we will find the solution of this equation in spherical polar coordinates. Someone please help me do this? I don't need you to solve the question completely, I just want help in how to solve it and to be shown the right direction because I'm completely lost! Fluent environment supports cylindrical and Cartesian coordinates. E Figure 11. Elasticity equations in cylindrical polar coordinates 1. 1 Spherical coordinates Figure 1: Spherical coordinate system. Examples: planes parallel to coordinate planes, cylindrical parame- terization of cylinder, and spherical parameterization of sphere. Figure 2. 1416 The focus of this chapter is on the governing equations of the linearized theory of elasticity in three types of coordinate systems, namely, Cartesian, cylindrical, and spherical coordinates. x = [1 2. Conventions. Using cylindrical coordinates can greatly simplify a triple integral when the region this with cylindrical coordinates is much easier than it would be for | {
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integral when the region this with cylindrical coordinates is much easier than it would be for cartesian Abstract: Application peculiarities of the Green's functions method for Cartesian, cylindrical and spherical coordinate system are under consideration. Converts from Cartesian (x,y,z) to Cylindrical (ρ,θ,z) coordinates in 3-dimensions. Triple Integrals in Cylindrical Coordinates It is the same idea with triple integrals: rectangular (x;y;z) coordinates might not be the best choice. We want to rotate the above so that the h axis is aligned with the arbitrary axis (Ax, Ay, Az) in other words we want to lookat the point (Ax, Ay, Az) see lookat Next we have a diagram for cylindrical coordinates: And let's not forget good old classical Cartesian coordinates: These diagrams shall serve as references while we derive their Laplace operators. The cylindrical coordinates of a point in $$\R^3$$ are given by $$(r,\theta,z)$$ where $$r$$ and $$\theta$$ are the polar coordinates of the point $$(x, y)$$ and $$z$$ is the same $$z$$ coordinate as in Cartesian coordinates. We introduce cylindrical coordinates by extending polar coordinates with theaddition of a third axis, the z-axis,in a 3-dimensional right-hand coordinate system. Jul 23, 2020 · Spherical Coordinates. Also the axis vectors depend on the same variable (in this case φ) which makes for interesting derivatives as we will see in a moment. The next step is to develop a technique for transforming spherical coordinates into Cartesian coordinates, and vice-versa. 13 Compute $\ds \int_{0}^a\int_{-\sqrt{a^2-x^2}}^0 x^2y\,dy\,dx$ by converting to cylindrical coordinates. Figure 3. That supposed to be superposition of a vortex and source. Unfortunately, there are a number of different notations used for the other two coordinates. Since the graph of this equation is a surface formed by revolving a curve about the z-axis, it might be better to use cylindrical coordinates. The origin of the local coordinate system is at the | {
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might be better to use cylindrical coordinates. The origin of the local coordinate system is at the node of interest. ΔA=(π. Note. Get the free "Triple Integral - Cylindrical" widget for your website, blog, Wordpress, Blogger, or iGoogle. Another necessary information for Definition. If you pipe is aligned with the Z axis, you'll already have access to these. Cartesian coordinates. To use the plot3 function we must convert the cylindrical coordinates to cartesian coordinates. 1 Cylindrical Coordinates Free Cartesian to Polar calculator - convert cartesian coordinates to polar step by step This website uses cookies to ensure you get the best experience. Generally, x, y , and z are used in Cartesian coordinates and these are replaced by r, θ , and z . Using the Cartesian coordinate system, geometric shapes (such as curves) can be described by Cartesian equations — algebraic equations involving the coordinates of the points lying on the shape. What I take away from your answer is that with FEM I should stay in cartesian coordinates, b/c curvilinear coordinates introduce messy artifacts. Figure 1: A point expressed in cylindrical coordinates. Express A using cylindrical coordinates and cylindrical base vectors. For example, the mapping between spherical polar coordinates and Cartesian coordinates uses these equivalences: Cartesian coordinate system top: two-dimensional coordinate system bottom: three-dimensional coordinate system n. Vector. When transforming from Cartesian to cylindircal, x and y become their polar counterparts. Find the y value. Now we compute compute the Jacobian for the change of variables from Cartesian coordinates to spherical coordinates. In polar coordinates, if ais a constant, then r= arepresents a circle Hi, this is module four of two dimensional dynamics, our learning outcomes for today are to describe a rectangular Cartesian coordinate system, a cylindrical coordinate system and to describe the kinematic relationships of position and velocity | {
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a cylindrical coordinate system and to describe the kinematic relationships of position and velocity in a tangential and normal coordinate system, so the one you are probably most familiar with for studying curvilinear motion, or curvilinear motion of One example is the Z-coordinate, which exists in both Cartesian and cylindrical systems. Input array must have a length of 3 and be in the correct order. The radial, tangential, and axial directions must be defined based on the original coordinates of each node in the node set for which the transformation is invoked. µ is called the \polar angle", the \azimuthal angle". Transform polar or cylindrical coordinates to Cartesian. This cylindrical coordinates converter/calculator converts the rectangular (or cartesian) coordinates of a unit to its equivalent value in cylindrical coordinates, according to the formulas shown above. Triple integrals in spherical coordinates Our mission is to provide a free, world-class education to anyone, anywhere. cartesian laplacian. For a 2D vortex, uz=0. NumPy Random Object Exercises, Practice and Solution: Write a NumPy program to convert cartesian coordinates to polar coordinates of a random 10x2 matrix representing cartesian coordinates. The position vector in cylindrical coordinates becomes r = rur + zk. Is it implicitly set to y=0? 2) This is basically a test problem I wanted to understand before continuing with a more complicated two-phase flow problem in spherical coordinates. Vectors are defined in cylindrical coordinates by (ρ, φ, z), where ρ is the length of the vector projected onto the xy-plane, φ is the angle between the projection of the vector onto the xy-plane (i. Jan 09, 2016 · If (a,b) is a are the coordinates of a point in Cartesian Plane, u is its magnitude and alpha is its angle then (a,b) in Polar Form is written as (u,alpha). B. coordinates (x;y) to polar coordinates (r; ). Appreciate your help! I have actually already came across the links. Free Polar to Cartesian | {
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(r; ). Appreciate your help! I have actually already came across the links. Free Polar to Cartesian calculator - convert polar coordinates to cartesian step by step This website uses cookies to ensure you get the best experience. As we have seen earlier, in two-dimensional space a point with rectangular coordinates can be identified with in polar coordinates and vice versa, where and are the relationships between the variables. from publication: Effect of Cartesian and Polar coordinate converting. A coordinate system in which the coordinates of The term "Cartesian coordinates" is used to describe such systems, and the values of the three coordinates unambiguously locate a point in space. It presents equations for several concepts that have not been covered yet, but will be on later pages. In this activity we will show that a suitable change of coordinates can greatly imporve the look of a surface in three-space. The cylindrical coordinate system is convenient to use when there is a line of symmetry that is defined as the z axis. When we expanded the traditional Cartesian coordinate system from two dimensions to three, we simply added a new 26 Apr 2018 Cartesian and polar coordinates SVG has a coordinate system that has its origin in the top-left corner, with the X coordinate increasing as we 20 Mar 2017 Cylindrical and Spherical Coordinates θr θr (r,θ,z) 14 Example: Find the cylindrical coordinates of the point (1,2,3) in Cartesian Coordinates 20 Nov 2009 Its form is simple and symmetric in Cartesian coordinates. Likewise, if we To Cartesian coordinates. Convert the cylindrical coordinates to cartesian coordinates in Cylindrical vs. Cartesian Coordinates. Cylindrical Coordinates In the cylindrical coordinate system, , , and , where , , and , , are standard Cartesian coordinates. Cartesian Cylindrical Spherical Cylindrical Coordinates x = r cosθ r = √x2 + y2 y = r sinθ tan θ = y/x z = z z = z Jul 27, 2020 · concept of cartesian coordinates system, its axis, its | {
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r sinθ tan θ = y/x z = z z = z Jul 27, 2020 · concept of cartesian coordinates system, its axis, its variables and ranges of variables 5. Aug 10, 2016 · Integration in cylindrical coordinates (r, \\theta, z) is a simple extension of polar coordinates from two to three dimensions. In such a coordinate system you can calculate the distance between two points and perform operations like axis rotations without altering this value. We start from this step: From rectangular coordinates, the arc length of a parameterized function is. y = rsin Θ. A: The reason we only use Cartesian base vectors for constructing a position vector is that Cartesian base vectors are the only base vectors whose directions are fixed—independent Mar 02, 2013 · Cartesian Coordinates vs Polar Coordinates In Geometry, a coordinate system is a reference system, where numbers (or coordinates) are used to uniquely determine the position of a point or other geometric element in space. In the cylindrical coordinate system, a point P (x;y;z); whose Cartesian A general system of coordinates uses a set of parameters to define a vector. D. Think of coordinates as different ways of expressing the position of the vector. The variable ρ is the distance of a coordinate point from the z Cartesian axis, and φ is its azimuthal angle. Each point is determined by an angle and a distance relative to the zero axis and the origin. The coordinate systems allow the geometrical problems to be converted into a numerica Figure 1. zip. Cartesian coordinates: If we wanted to write rˆˆ, , θφˆ in terms of xyzˆˆˆ, , , we would need to use the angles of and . Either r or rho is used to refer to the radial coordinate and either phi or theta to the azimuthal coordinates. In cylindrical coordinates, they are i r, i, i z, and in spherical coordinates, i r, i, i. The following equations describe the relationship between a Cartesian coordinate and a cylindrical coordinate: x = · cos, y = · sin, z = z Elasticity equations in cylindrical | {
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and a cylindrical coordinate: x = · cos, y = · sin, z = z Elasticity equations in cylindrical polar coordinates 1. In the Cartesian system the coordinates are perpendicular to one another with the same unit length Convert from rectangular to spherical coordinates. Cylindrical coordinates are a generalization of two-dimensional polar coordinates to three dimensions by superposing a height (z) axis. For instance when integrating vector function in Cartesian coordinates we can take the unit vectors outside the integral, since they are constant. Cartesian coordinates consist of a set of mutually perpendicular axes, which intersect at a 1-1-2 Circular Cylindrical Coordinates . Purpose of use Too lazy to do homework myself. Therefore, a Cartesian coordinate system is used, where the origin (0, 0) is toward the lower left of the planar section. May 18, 2020 · Cartesian coordinates (Section 4. 7854 1. The level surface of points such that z ˆz z=zP define a plane. Thanks for your time. Find more Mathematics widgets in Wolfram|Alpha. Explanation: A polar coordinate is in the form (r,θ) , where r is the distance from the origin and θ is the corresponding angle. Working in cylindrical coordinates is essentialy the same as working in polar coordinates in two dimensions except we must account for the z-component of the system. The Newtonian Constitutive Equation in Cartesian, Cylindrical, and Spherical coordinates Prof. Cylindrical just adds a z-variable to polar. The cylindrical radial coordinate is the perpendicular distance from the point to the z axis. person_outline Anton schedule 2018-10-22 12:49:06 This calculator is intended for coordinates transformation from / to the following 3d coordinate systems: As in the case of Cartesian coordinates, analytical solutions are readily obtained for unidirectional problems in cylindrical and spherical coordinates. The vector k is introduced as the direction vector of the z-axis. The cylindrical system is closely 26 Feb 2018 The | {
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introduced as the direction vector of the z-axis. The cylindrical system is closely 26 Feb 2018 The calculation uses Cartesian coordinates. However the governing equations where i am using this velocity profile are written in spherical co ordinates. For example, there are times when a problem has Oct 22, 2019 · Although Cartesian coordinates can be used in three dimensions (x, y, and z), polar coordinates only specify two dimensions (r and θ). Thus, ! r V =ure ö r+u"e ö "+uze ö z=0e ö r+u"e ö "+0e ö z Processing Triple Integrals in Cartesian Coordinates Calculation of a triple integral in Cartesian coordinates can be reduced to the consequent calculation of three integrals of one variable. 2 Cylindrical Coordinates These are coordinates for a three-dimensional space. If we start with the Cartesian equation of the sphere and substitute, we get the spherical equation: \eqalign{ x^2+y^2+z^2&=2^2\cr \rho^2\sin^2\phi\cos^2\theta+ \rho^2\sin^2\phi\sin^2\theta+\rho^2\cos^2\phi&=2^2\cr \rho^2\sin^2\phi Convert the three-dimensional Cartesian coordinates defined by corresponding entries in the matrices x, y, and z to cylindrical coordinates theta, rho, and z. 4 Relations between Cartesian, Cylindrical, and Spherical Coordinates. c) ρ = 2cos φ to cylindrical coordinates. Fields in Cylindrical Coordinate Systems. 22 Oct 2019 Coordinate systems provide a way to specify a point in space. The coordinate system uses the standard polar coordinate system in the x-y plane, utilizing a distance from the origin (r) and an angle (θ) of Transforms 3d coordinate from / to Cartesian, Cylindrical and Spherical coordinate systems. In polar coordinates we specify a point using the distance rfrom the origin and the angle with the x-axis. The transformation from Cartesian To convert the Cartesian nabla to the nabla for another coordinate system, say… cylindrical coordinates. Later in the course, we will also see how cylindrical coordinates can be useful in calculus, when evaluating limits or | {
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we will also see how cylindrical coordinates can be useful in calculus, when evaluating limits or integrating in Cartesian coordinates is very difficult. I have vector in cartesian coordinate system: \\vec{a}=2y\\vec{i}-z\\vec{j}+3x\\vec{k} And I need to represent it in cylindrical and spherical coord. Slide 2 ’ & $% Polar coordinates in IR2 De nition 1 (Polar coordinates) Let (x;y) be Cartesian coordinates in IR2. When this is the case, Cartesian coordinates (x;y;z) are converted to cylindrical coordinates (r; ;z). ρ) and the positive x-axis (0 ≤ φ < 2π), z is the regular z-coordinate. An example is given below. May 26, 2020 · Section 1-12 : Cylindrical Coordinates As with two dimensional space the standard $$\left( {x,y,z} \right)$$ coordinate system is called the Cartesian coordinate system. Figure 1 illustrates the relation between Cartesian Oct 10, 2019 · Some of the Worksheets below are Cylindrical and Spherical Coordinates Worksheets, list of Formulas that you can use to switch between Cartesian and polar coordinates, identifying solids associated with spherical cubes, translating coordinate systems, approximating the volume of a spherical cube, … First off, the definition of your cylindrical co-ordinates is wrong. The conventional choice of coordinates is shown in Fig. Consider a cartesian, a cylindrical, and a spherical coordinate system, related as shown This converter/calculator converts a cartesian, or rectangular, coordinate to its equivalent cylindrical coordinate. 0000 Oct 13, 2010 · Homework Statement This seems like a trivial question (because it is), and I'm just not sure if I'm doing it right. 25in}y = r\sin \theta \hspace{0. The polar coordinates are defined in terms of r r r and θ \theta θ, where r r r is the distance of the point from the origin Using various functions, you can convert data between Spherical, Cartesian, and Cylindrical coordinate systems. For example, there are different languages in which the word "five" is said differently, | {
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systems. For example, there are different languages in which the word "five" is said differently, but it is five regardless of whether it is said in English or Spanish, say. This tutorial will denote vector quantities with an arrow atop a letter, except unit vectors that define coordinate systems which will have a hat. INSTRUCTIONS: Choose the preferred angle units and enter the following: (r) Polar radius (Θ) Polar angle (z) Vertical offset; Cartesian from Cylindrical: The calculator returns the Cartesian coordinates (x, y, z). The global coordinates of the two points defining the axis of the cylindrical system (points a and b as shown in Figure 2. Here's what they look like: The Cartesian Laplacian looks pretty straight forward. 9: Cylindrical and Spherical Coordinates In the cylindrical coordinate system, a point Pin space is represented by the ordered triple (r; ;z), where rand are polar coordinates of the projection of Ponto the xy-plane and zis the directed distance from the xy-plane to P. The chain rule relates the Cartesian operators Cartesian, the circular cylindrical, and the spherical. Solutions to steady unidimensional problems can be readily obtained by elementary methods as shown below. Translating Spherical Coordinates to Cartesian Coordinates. The most well-known coordinate system is the Cartesian coordinate to use, where every point has an x-coordinate and y-coordinate expressing its horizontal position, and The Wave Equation in Cylindrical Coordinates Overview and Motivation: While Cartesian coordinates are attractive because of their simplicity, there are many problems whose symmetry makes it easier to use a different system of coordinates. x = r cos θ = 4 cos 2 π 3 = −2 y = r sin θ = 4 sin 2 π 3 = 2 3 z = −2. Recall: A grid surface of a 3-d coordinate system is a surface generated by holding one of the coordinates constant while letting the other two vary. Dec 10, 2019 · CFX actually gives you cylindrical coordinates (r and theta) about the Z | {
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two vary. Dec 10, 2019 · CFX actually gives you cylindrical coordinates (r and theta) about the Z axis of any coordinate system. The cylindrical (left) and spherical (right) coordinates of a point. Cylindrical coordinates are an alternative to the more common Cartesian coordinate system. to a rectangular coordinate, follow these steps: Find the x value. A natural extension of the 2d polar coordinates are cylindrical coordinates, since they just add a height value out of the xy Conversion from cartesian to spherical coordinates: Cartesian [x, y, z] Spherical [r, θ, φ] Conversion from spherical to cylindrical coordinates: Spherical [r, θ, φ] Cylindrical [ρ, φ', z'] ρ = r sin θ φ' = φ z' = r cos θ Conversion from cylindrical to spherical coordinates: Cylindrical [ρ, φ, z] Spherical [r, θ, φ'] θ = arctan Relationships in Cylindrical Coordinates This section reviews vector calculus identities in cylindrical coordinates. Morrison Continuity Equation, Cartesian coordinates ∂ρ ∂t + vx ∂ρ ∂x +vy ∂ρ ∂y +vz ∂ρ ∂z +ρ ∂vx ∂x + ∂vy ∂y + ∂vz ∂z = 0 Continuity Equation, cylindrical coordinates ∂ρ ∂t + 1 r ∂(ρrvr) ∂r + 1 r ∂(ρvθ) ∂θ + ∂(ρvz) ∂z = 0 This article contains a download link for a script which converts cylindrical or spherical coordinates to xyz coordinates for use in Surfer. Overrides: fromReference in class CoordinateSystem Parameters: tuples - float array in Cartesian coordinates ordered as x, y, z Returns: float array containing the radius, azimuth and z values Next: An example Up: Cylindrical Coordinates Previous: Regions in cylindrical coordinates The volume element in cylindrical coordinates. b) 2 2x + y- z2 = 1 to spherical coordinates. 1;1/, which can be thought of as height Transformation between Cartesian and Cylindrical Coordinates; Velocity Vectors in Cartesian and Cylindrical Coordinates; Continuity Equation in Cartesian and Cylindrical Coordinates; Introduction to Conservation of Momentum; Sum of Forces on a Fluid Element; Expression of Inflow and Outflow | {
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to Conservation of Momentum; Sum of Forces on a Fluid Element; Expression of Inflow and Outflow of Momentum; Cauchy Momentum Equations and the Navier Jan 24, 2017 · The basic form of heat conduction equation is obtained by applying the first law of thermodynamics (principle of conservation of energy). He's associating, all of a sudden, these relationships with points on a coordinate plane. coordinates and back again anytime. com/ for more stuff. Cylindrical coordinates to Cartesian coordinates. (1) into Z Z R R R T T c ′′ + Φ Φ′′ + = ′′+ ′ ′′ 2 1 1 1 1 ρ ρ. P = ( r, ) x y r 0 0 Theorem (Cartesian-polar transformations) The Cartesian coordinates of a point P = (r,θ) in the first quadrant are given by x = r cos(θ), y = r sin(θ). The cylindrical coordinate system is similar to that of the spherical coordinate system, but is an alternate extension to the polar coordinate system. Regardless, one should be able, in principle, to write down the coordinate transformations in the following form: May 16, 2011 · I need to work out how to convert phi = pi/3 from spherical coordinates to cartesian and cylindrical coordinates. So, coordinates are written as (r,$\theta\$, z). Velocity And Acceleration In Cylindrical Coordinates Velocity of a physical object can be obtained by the change in an object's position in respect to time. The relations used for the conversion of the coordinates of the point from the Cartesian coordinate system to the cylindrical coordinate system are: In many problems involving cylindrical polar coordinates, it is useful to know the line and volume elements; these are used in integration to solve problems involving paths and volumes. which means that y = 1. When we get to triple integrals, some integrals are more easily evaluated in cylindrical coordinates and you will even have some integrals that can't be evaluated in rectangular coordinates but can be in cylindrical. The above result is another way of deriving the result dA=rdrd(theta). 1 4/6/13 a | {
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be in cylindrical. The above result is another way of deriving the result dA=rdrd(theta). 1 4/6/13 a ppendix 1 e quations of motion in cylindrical and spherical coordinates a1. ) This is intended to be a quick reference page. Rectangular coordinates are depicted by 3 values, (X, Y, Z). Circular Cylindrical Coordinates System concept of cylindrical coordinates system, its axis, its Cylindrical coordinates are a simple extension of the two-dimensional polar coordinates to three dimensions. Move the sliders to convert cylindrical coordinates to Cartesian coordinates for a comparison. 11 Jul 2018 about the Conversion between Cylindrical & Cartesian systems for Electromagnetism. Cylindrical coordinates definition, a member of a system of coordinates for locating a point in space by its polar coordinates and its perpendicular distance to the polar plane. 0000 the part of the solution depending on spatial coordinates, F(~r), satisfies Helmholtz’s equation ∇2F +k2F = 0, (2) where k2 is a separation constant. When we talk about the point with coordinates (x,y,z)'' or `the surface with equation f(x,y,z)'', we will always have in mind cartesian coordinates. Site: http://mathispower4u. Cylindrical Coordinates. Circular cylindrical coordinates use the plane polar coordinates ρ and φ (in place of x and y) and the z Cartesian coordinate. – Cartesian coordinates – Cylindrical coordinates – Spherical coordinates. 2GHz processor, the runtime of the cylindrical simulation is nearly 90 times shorter than the 3d simulation. Sponsored Links. [Hint: Think about the distance of any point ( x , y , z ) on the cylinder from the z -axis. What is more challenging is determining the velocity vector in Cylindrical coordinates if you have a position in Cylindrical coordinates as a function of time. The Cartesian Nabla: 2. What is dV in cylindrical coordinates? Well, a piece of the cylinder looks like so which tells us that We can basically think of cylindrical coordinates as polar coordinates plus | {
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so which tells us that We can basically think of cylindrical coordinates as polar coordinates plus z. There's three independent variables, x, y, and z. B-5 Feb 24, 2015 · Preliminaries. The z component does not change. For example, in the Cartesian coordinate system, the cross-section of a cylinder concentric with the $$z$$-axis requires two coordinates to describe: \(x Cylindrical coordinates are depicted by 3 values, (r, φ, Z). cartesian to cylindrical coordinates | {
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cbaxcjsr, 2kshboa zs0xeiw, 0ip9r1ovv, c37tgzajkey, u4t wr8nnwkgux, q i6 xh fgsk7v0vda5, | {
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# A conjectured closed form of $\int\limits_0^\infty\frac{x-1}{\sqrt{2^x-1}\ \ln\left(2^x-1\right)}dx$
Consider the following integral: $$\mathcal{I}=\int\limits_0^\infty\frac{x-1}{\sqrt{2^x-1}\ \ln\left(2^x-1\right)}dx.$$
I tried to evaluate $\mathcal{I}$ in a closed form (both manually and using Mathematica), but without success.
However, if WolframAlpha is provided with a numerical approximation $\,\mathcal{I}\approx 3.2694067500684...$, it returns a possible closed form: $$\mathcal{I}\stackrel?=\frac\pi{2\,\ln^2 2}.$$ Further numeric caclulations show that this value is correct up to at least $10^3$ decimal digits. So, I conjecture that this is the exact value of $\mathcal{I}$.
Question: Is this conjecture correct?
• After a substitution of y = 2^x-1, does this not reduce to knowledge about identities of polylogaritms ?
– mick
Oct 9 '13 at 21:41
Sub $u=\log{(2^x-1)}$. Then $x=\log{(1+e^u)}/\log{2}$, $dx = (1/\log{2}) (du/(1+e^{-u})$. The integral then becomes
\begin{align}\frac{1}{\log{2}} \int_{-\infty}^{\infty} \frac{du}{1+e^{-u}} e^{-u/2} \frac{\frac{\log{(1+e^u)}}{\log{2}}-1}{u} = \frac{1}{2\log^2{2}} \int_{-\infty}^{\infty} \frac{du}{\cosh{(u/2)}} \frac{\log{(1+e^u)}-\log{2}}{u}\\ = \underbrace{\frac{1}{2\log^2{2}} \int_{-\infty}^{0} \frac{du}{\cosh{(u/2)}} \frac{\log{(1+e^u)}-\log{2}}{u}}_{u\rightarrow -u} \\+ \frac{1}{2\log^2{2}} \int_{0}^{\infty} \frac{du}{\cosh{(u/2)}} \frac{\log{(1+e^u)}-\log{2}}{u}\\ = \underbrace{-\frac{1}{2\log^2{2}} \int_{0}^{\infty} \frac{du}{\cosh{(u/2)}} \frac{\log{(1+e^{-u})}-\log{2}}{u}}_{\log{(1+e^{-u})} = \log{(1+e^u)}-u}\\+ \frac{1}{2\log^2{2}} \int_{0}^{\infty} \frac{du}{\cosh{(u/2)}} \frac{\log{(1+e^u)}-\log{2}}{u}\\ \end{align}
The nasty pieces of the integral cancel, and we are left with
$$\frac{1}{2\log^2{2}}\int_{0}^{\infty} \frac{du}{\cosh{(u/2)}} = \frac{\pi}{2 \log^2{2}}$$
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as correctly conjectured.
With the change of variables $z \equiv 2^{x} - 1\yy x = \ln\pars{1 + z}/\ln\pars{2},\ {\cal I}$ is reduced to $${\cal I} = {1 \over \ln^{2}\pars{2}} \int\limits_{0}^{\infty}{\ln\pars{1 + z} - \ln\pars{2} \over z^{1/2}\,\pars{1 + z}\,\ln\pars{z}} \,\dd z$$ Now, we split the integral from $\pars{0, 1}$ and from $\pars{1, \infty}$. In the second one, we makes the change $z \to 1/z$ such that we are left with an integration over $\pars{0, 1}$: \begin{align} {\cal I} &= {1 \over \ln^{2}\pars{2}} \int\limits_{0}^{1}{\ln\pars{1 + z} - \ln\pars{2} \over z^{1/2}\,\pars{1 + z}\,\ln\pars{z}} \,\dd z + {1 \over \ln^{2}\pars{2}}\int\limits_{0}^{1} {\ln\pars{1 + 1/z} - \ln\pars{2} \over z^{-1/2}\,\pars{1 + 1/z}\,\bracks{-\ln\pars{z}}} \,{\dd z \over z^{2}} \\[3mm]&= {1 \over \ln^{2}\pars{2}} \int\limits_{0}^{1}{\ln\pars{1 + z} - \ln\pars{2} \over z^{1/2}\,\pars{1 + z}\,\ln\pars{z}} \,\dd z - {1 \over \ln^{2}\pars{2}}\int\limits_{0}^{1} {\ln\pars{1 + z} - \ln\pars{z} - \ln\pars{2} \over z^{1/2}\,\pars{1 + z}\,\ln\pars{z}}\,\dd z \\[3mm]&= {1 \over \ln^{2}\pars{2}} \int\limits_{0}^{1}{1 \over z^{1/2}\,\pars{1 + z}} \,\dd z\,, \quad \pars{~\mbox{Let's}\quad r \equiv z^{1/2}\yy\ z = r^{2}~} \\[3mm]&= {2 \over \ln^{2}\pars{2}} \underbrace{\quad\int\limits_{0}^{1}{\dd r \over r^{2} + 1}\quad} _{\ds{\arctan\pars{1}\ =\ {\pi \over 4}}} = \color{#ff0000}{\Large{\pi \over 2\ln^{2}\pars{2}}} \end{align} | {
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Let $\tan^2t=2^x-1\;\Rightarrow\;x=\dfrac{\ln(1+\tan^2t)}{\ln 2}\;\Rightarrow\;dx=\dfrac{2\tan t\sec^2t\ dt}{(1+\tan^2t)\ln 2}$ and the corresponding region is $0<t<\dfrac\pi2$. Using identity $\sec^2t=1+\tan^2t$, then the integral turns out to be $$\mathcal{I}=\frac{1}{\ln^22}\int_0^{\Large\frac\pi2}\frac{2\ln(\sec t)-\ln2}{\ln (\tan t)}\ dt.\tag1$$ Now, using property $$\int_b^af(x)\ dx=\int_b^af(a+b-x)\ dx$$ equation $(1)$ becomes $$\mathcal{I}=\frac{1}{\ln^22}\int_0^{\Large\frac\pi2}\frac{2\ln(\csc t)-\ln2}{\ln (\cot t)}\ dt.\tag2$$ Adding $1$ and $2$ yields \begin{align} 2\mathcal{I}&=\frac{1}{\ln^22}\int_0^{\Large\frac\pi2}\left(\frac{2\ln(\sec t)-\ln2}{\ln (\tan t)}+\frac{2\ln(\csc t)-\ln2}{\ln (\cot t)}\right)\ dt\\ &=\frac{1}{\ln^22}\int_0^{\Large\frac\pi2}\left(\frac{2\ln\left(\dfrac{1}{\cos t}\right)-\ln2}{\ln (\tan t)}+\frac{2\ln\left(\dfrac{1}{\sin t}\right)-\ln2}{\ln \left(\dfrac{1}{\tan t}\right)}\right)\ dt\\ &=\frac{1}{\ln^22}\int_0^{\Large\frac\pi2}\left(\frac{-2\ln(\cos t)-\ln2}{\ln (\tan t)}+\frac{2\ln(\sin t)+\ln2}{\ln (\tan t)}\right)\ dt\\ &=\frac{2}{\ln^22}\int_0^{\Large\frac\pi2}\frac{\ln(\sin t)-\ln(\cos t)}{\ln \left(\dfrac{\sin t}{\cos t}\right)}\ dt\\ \mathcal{I}&=\frac{1}{\ln^22}\int_0^{\Large\frac\pi2}\ dt\\\\ \int_0^\infty\frac{x-1}{\sqrt{2^x-1}\ \ln\left(2^x-1\right)}dx&=\large\color{blue}{\frac{\pi}{2\ln^22}}.\qquad\qquad\qquad\blacksquare \end{align}
• Wow, this is remarkably brilliant! I'm impressed. Jun 10 '16 at 12:06
• This is really nice thanks
– Alex
Sep 4 '20 at 10:22
Substitute $(2^x-1) = t^2$ to get,
$\text{I} = \displaystyle \dfrac{1}{\ln^2 2} \int_{0}^{\infty} \left( \dfrac{\ln (t^2+1) - \ln 2}{(t^2 + 1) \ln t} \right) \mathrm{d}t$
Substitute $t \mapsto \dfrac{1}{t}$
$\implies \text{I} = -\displaystyle \dfrac{1}{\ln^2 2} \int_{0}^{\infty} \left( \dfrac{\ln (t^2+1) - \ln 2}{(t^2 + 1) \ln t} \right) \mathrm{d}t + \dfrac{2}{\ln^2 2} \int_{0}^{\infty} \dfrac{\mathrm{d}t}{t^2+1}$ | {
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$\implies \text{I} = -\text{I} + \dfrac{\pi}{\ln^2 2}$
$\implies \text{I} = \dfrac{\pi}{2 \ln^2 2}$
• good one $(+1)$ Jul 9 '17 at 20:48
• And with the right substitution the integral just dissolves away. Nice work. Jan 25 '19 at 1:04
• It looks easy this way
– Alex
Sep 4 '20 at 10:22 | {
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# Drawing n intervals uniformly randomly, probability that at least one interval overlaps with all others
Randomly draw $n$ intervals from $[0,1]$, where each end point A,B are selected from the uniform distribution between $[0,1]$.
What's the probability that at least one interval overlaps with all others?
• You can look at the probability that the last drawn $A_n$ is smaller than the minimum of all previously drawn $A$, and the probability that the last $B_n$ is greater than the maximum of all previously drawn $B$. This should be helpful. Then inflate the probability to account for the fact that we don't need the last one, but any one. (I don't have the time to work through it, but it looks like a fun little problem. Good luck!) Apr 20, 2015 at 7:02
• It may be somewhat surprising that (1) the answer does not depend on the distribution (only that it be continuous) and (2) for $n\gt 1$ it is constant!
– whuber
Apr 22, 2015 at 15:41
• Is this how the nth interval is construted: i) draw two numbers uniformly at random from [0,1], ii) let the smaller one be $A_n$ and the larger one $B_n$?
– KOE
Apr 23, 2015 at 21:36
This post answers the question and outlines partial progress toward proving it correct.
For $n=1$, the answer trivially is $1$. For all larger $n$, it is (surprisingly) always $2/3$.
To see why, first observe that the question can be generalized to any continuous distribution $F$ (in place of the uniform distribution). The process by which the $n$ intervals are generated amounts to drawing $2n$ iid variates $X_1, X_2, \ldots, X_{2n}$ from $F$ and forming the intervals
$$[\min(X_1,X_2), \max(X_1,X_2)], \ldots, [\min(X_{2n-1}, X_{2n}), \max(X_{2n-1}, X_{2n})].$$
Because all $2n$ of the $X_i$ are independent, they are exchangeable. This means the solution would be the same if we were randomly to permute all of them. Let us therefore condition on the order statistics obtained by sorting the $X_i$:
$$X_{(1)} \lt X_{(2)} \lt \cdots \lt X_{(2n)}$$ | {
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$$X_{(1)} \lt X_{(2)} \lt \cdots \lt X_{(2n)}$$
(where, because $F$ is continuous, there is zero chance that any two will be equal). The $n$ intervals are formed by selecting a random permutation $\sigma\in\mathfrak{S}_{2n}$ and connecting them in pairs
$$[\min(X_{\sigma(1)},X_{\sigma(2)}), \max(X_{\sigma(1)},X_{\sigma(2)})], \ldots, [\min(X_{\sigma(2n-1)}, X_{\sigma(2n)}), \max(X_{\sigma(2n-1)}, X_{\sigma(2n)})].$$
Whether any two of these overlap or not does not depend on the values of the $X_{(i)}$, because overlapping is preserved by any any monotonic transformation $f:\mathbb{R}\to\mathbb{R}$ and there are such transformations that send $X_{(i)}$ to $i$. Thus, without any loss of generality, we may take $X_{(i)}=i$ and the question becomes:
Let the set $\{1,2,\ldots, 2n-1, 2n\}$ be partitioned into $n$ disjoint doubletons. Any two of them, $\{l_1,r_1\}$ and $\{l_2,r_2\}$ (with $l_i \lt r_i$), overlap when $r_1 \gt l_2$ and $r_2 \gt l_1$. Say that a partition is "good" when at least one of its elements overlaps all the others (and otherwise is "bad"). As a function of $n$, what is the proportion of good partitions?
To illustrate, consider the case $n=2$. There are three partitions,
$$\color{gray}{\{\{1,2\},\{3,4\}\}},\ \color{red}{\{\{1,4\},\{2,3\}\}},\ \color{red}{\{\{1,3\},\{2,4\}\}},$$
of which the two good ones (the second and third) have been colored red. Thus the answer in the case $n=2$ is $2/3$.
We may graph such partitions $\{\{l_i,r_i\},\,i=1,2,\ldots,n\}$ by plotting the points $\{1,2,\ldots,2n\}$ on a number line and drawing line segments between each $l_i$ and $r_i$, offsetting them slightly to resolve visual overlaps. Here are plots of the preceding three partitions, in the same order with the same coloring:
From now on, in order to fit such plots easily in this format, I will turn them sideways. For instance, here are the $15$ partitions for $n=3$, once again with the good ones colored red: | {
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Ten are good, so the answer for $n=3$ is $10/15=2/3$.
The first interesting situation occurs when $n=4$. Now, for the first time, it is possible for the union of the intervals to span $1$ through $2n$ without any single one of them intersecting the others. An example is $\{\{1,3\},\{2,5\},\{4,7\},\{6,8\}\}$. The union of the line segments runs unbroken from $1$ to $8$ but this is not a good partition. Nevertheless, $70$ of the $105$ partitions are good and the proportion remains $2/3$.
The number of partitions increases rapidly with $n$: it equals $1\cdot 3\cdot 5 \cdots \cdot 2n-1 = (2n)!/(2^nn!)$. Exhaustive enumeration of all possibilities through $n=7$ continues to yield $2/3$ as the answer. Monte-Carlo simulations through $n=100$ (using $10000$ iterations in each) show no significant deviations from $2/3$.
I am convinced there is a clever, simple way to demonstrate there is always a $2:1$ ratio of good to bad partitions, but I have not found one. A proof is available through careful integration (using the original uniform distribution of the $X_i$), but it is rather involved and unenlightening. | {
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• Very cool. I have a hard time following what it means to "condition on the order statistics", would it be possible to add a line of intuition? Seems like a useful technique. I understand up to that the $X_i$ are exchangeable, indeed even $iid$, that that this allows us to consider any permutation.
– KOE
Apr 24, 2015 at 16:10
• @Student To "condition on" means to say, let's temporarily hold these values fixed and consider what we can learn from that. Later, we will let those values vary (according to their probability distribution). In this case, once we find that the answer is $2/3$ regardless of the fixed values of the order statistics, then we no longer have to carry out the second step of varying the order statistics. Mathematically, the order stats are a vector-valued variable $\mathbf{X}$ and the indicator of being good is $Y$, so $$\mathbb{E}(Y)=\mathbb{E}(\mathbb{E}(Y|\mathbf{X}))=\mathbb{E}(2/3)=2/3.$$
– whuber
Apr 24, 2015 at 16:36
• This very question came back on FiveThirtyEight The Riddler. Jun 2, 2020 at 9:22 | {
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+0
# last question
0
38
3
+81
How many different rational numbers between 1/1000 and 1000 can be written either as a power of 2 or as a power of 3, where the exponent is a (possibly negative) integer?
I got 31; is this correct?
mathbum Oct 6, 2018
#1
+1
$$2^9=512$$
$$3^6=729$$
Those are the largest positive powers of 2 and 3 under 1000.
$$2^{-9}=0.001953.....$$
$$3^{-6}=0.0013717.....$$
These are the smallest negative powers greater than $$\frac{1}{1000}$$
Therefore, we have $$9-(-9)+1=19$$ different powers of 2 between $$\frac{1}{1000}$$ and 1000.
We also have $$6-(-6)+1=13$$ different powers of 3 between $$\frac{1}{1000}$$ and 1000.
This gives us $$19+13=32$$ total integers...
32 is the answer I got but someone is going to have to check over my work :b
Guest Oct 6, 2018
#2
+2362
+1
The problem is symmetric in x vs. 1/x so we can find the number of integers between 0 and 1000 that are either a power of two or 3 and just double the answer.
Note that any non-zero power of 2 is even and any non-zero power of 3 is odd so these two sets are disjoint.
There are 10 powers of 2, 0-9, and 6 non-zero powers of 3 that are less than 100.
We use non-zero power for 3 since we only want to count 1 = 20 = 30 once
That gets us 16 from 0 to 1000. Doubling this we get 32 but we don't want to count $$1 = \dfrac{1}{1}$$
twice so we subtract 1 from this getting 31 as you found.
Rom Oct 6, 2018
edited by Rom Oct 6, 2018
edited by Rom Oct 6, 2018
edited by Rom Oct 6, 2018
#3
+93644
0
That is an interesting question and good answers from both guest and Rom.
Thanks :)
Melody Oct 6, 2018 | {
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Probability of drawing cards that sum to 10, given a starting card
Two people are playing a card game. They are using a reduced deck of cards, consisting of A, 2, 3, ..., 9 for each of the four suits (i.e. 36 cards). In this game an ace has a value of 1.
Player A deals a single card to themselves and Player B.
Player A has a 5 of diamonds.
Player B has a 7 of spades.
Player A then draws one card at a time trying to reach a sum of 10, with the starting card. If they go over 10, they lose, and the cards drawn are shuffled back into the deck.
What is the probability that A wins, and what is the probability that B wins. (Winning means they can form a sum of 10).
The answers given are 0.1536 for A, and 0.1468 for B.
I can get the answer for B as follows: $$\frac{nCr(4,1)}{nCr(34,1)}+\frac{nCr(4,1)}{nCr(34,1)}\times\frac{nCr(4,1)}{nCr(33,1)}\times2+\frac{nCr(4,3)}{nCr(34,3)}$$ which is the probability of a 3 + probability of a 2 and an A + probability of three Aces.
However, I can't get the answer for A, even trying very similar techniques.
• Do the players draw the cards alternatively? Does the game end if A or B first reach 10? Does the game end (without a winner) if they both go over 10?
– user
Apr 25, 2021 at 13:23
The answer for $$A$$ seems to have approximation error. Here is how I look at $$A$$ getting to sum of $$10$$.
In one draw - gets one of the remaining $$3$$ cards with face value $$5$$.
In two draws - $$(4,1)$$ or $$(3,2)$$
In three draws - $$(1, 1, 3)$$ or $$(2, 2, 1)$$
In four draws - $$(1, 1, 1, 2)$$
In five draws - $$(1, 1, 1, 1, 1)$$.
So desired probability $$= \displaystyle \small \frac{3}{34} + 2 \cdot 2! \big(\frac{4}{34}\big)^2 + 2 \cdot \frac{3!}{2!} \big(\frac{4}{34}\big)^3 + \frac{4!}{3!} \big(\frac{4}{34}\big)^4 + \big(\frac{4}{34}\big)^5$$
$$= \displaystyle \small \frac{437763}{2839714} \approx 0.154$$ | {
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$$= \displaystyle \small \frac{437763}{2839714} \approx 0.154$$
• Thanks for your help. Though as the game is involving a deck of cards it's not possible to have five aces. Apr 26, 2021 at 4:42
• you are right but I am going by the question that "cards drawn are shuffled back into the deck" so I can draw $5$ Aces in $5$ draws with probability $(\frac{4}{34})^5$. Apr 26, 2021 at 7:00 | {
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# Help explaining Structural Induction
I am trying to wrap my head around structural induction. Can someone break it down and explain it around this problem?
Let S, a subset of $$\mathbb{N}*\mathbb{N}$$, be defined recursively by:
Base case: $$(0,0)$$ $$\in S$$
Constructor case: If $$(m,n) \in S$$, then $$(m+5,n+1) \in S$$
Prove that if $$(m,n) \in S$$, then m+n is a multiple of 3.
How is it different than normal induction (using this example please) and what is the point of a Constructor case? Can someone wright the proof out so i can see what this structural induction proof looks like?
• Her we are "performing induction" not on $\mathbb N$ but of $\mathbb N \times \mathbb N$, and not all pairs $(n,m)$ will satisfy bthe property : $(1,1) \notin S$ because $1+1$ is not a multiple of $3$. Oct 18, 2016 at 10:08
• Now for the inductive step (here the "constructor case") ; assume that the property holds for $(m,n)$ and show that it holds for $(m+5,n+1)$. Oct 18, 2016 at 10:12
• Here is a video attempting to explain structural induction: youtu.be/u21QV-MlVDY Aug 8 at 15:44
The set $$S$$ is defined recursively: certain base elements of $$S$$ are specified, in this case just the ordered pair $$\langle 0,0\rangle$$, and a rule is given that allows ‘new’ elements of $$S$$ to be constructed from ‘old’ ones. Here each ‘old’ element $$\langle m,n\rangle$$ gives rise to just one ‘new’ one, $$\langle m+5,n+1\rangle$$. Thus, in this case
$$S=\{\langle 0,0\rangle,\langle 5,1\rangle,\langle 10,2\rangle,\langle 15,3\rangle,\ldots\}\;.$$
There is also a rule, often (as in this case) left unstated, to the effect that the only members of $$S$$ are the objects that can be obtained by repeatedly applying the constructor rule(s) to the base elements. | {
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We can show that every member of $$S$$ has some property $$P$$ by first verifying that each of the base elements has $$P$$ and then showing that the construction process preserves the property $$P$$: that is, if we apply the construction process to objects that have $$P$$, the new objects also have $$P$$. If we can do this, we can conclude by structural induction that every member of $$S$$ has $$P$$.
In your problem an ordered pair $$\langle m,n\rangle$$ has the property $$P$$ if and only if $$m+n$$ is a multiple of $$3$$. This is clearly the case for the one base element $$\langle 0,0\rangle$$: $$0+0=0=3\cdot 0$$ is a multiple of $$3$$. That’s the base case of your structural induction. For the induction step assume that $$\langle m,n\rangle\in S$$ has $$P$$, i.e., that $$m+n$$ is a multiple of $$3$$. When we apply the construction process to $$\langle m,n\rangle$$, we get the pair $$\langle m+5,n+1\rangle\in S$$, and we want to show that it also has $$P$$, i.e., that $$(m+5)+(n+1)$$ is a multiple of $$3$$. By hypothesis $$m+n=3k$$ for some integer $$k$$, so
$$(m+5)+(n+1)=m+n+6=3k+6=3(k+2)\;;$$
and $$k+2$$ is an integer, so $$(m+5)+(n+1)$$ is indeed a multiple of $$3$$. We’ve now shown
• that the base element $$\langle 0,0\rangle$$ has the desired property, and
• that the construction process preserves this property: when applied to a pair $$\langle m,n\rangle$$ such that $$m+n$$ is a multiple of $$3$$, it produces another pair whose components sum to a multiple of $$3$$.
These are the base case and induction step of a proof by structural induction; between them they constitute a proof that $$m+n$$ is a multiple of $$3$$ for each $$\langle m,n\rangle\in S$$.
• Thank you, that was a great explanation. It made it immediately obvious that induction on the natural numbers is a special case of structural induction where the constructor case is $k+1$ for every $k$ in $S$. Jan 9, 2017 at 22:05
• @jeremy: You’re welcome; I’m glad that it helped. Jan 9, 2017 at 22:07 | {
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Mathematical induction is defined over natural number and it is based on two fundamental facts :
• there is an "initial" number : $0$
• every number $n$ has a unique successor : $n+1$.
Structural induction generalize this type of proof to "structures" on which a well-founded partial order is defined, i.e.
• that have an "initial" or minimal element
and
• they have a partial order.
It applies to structures recursively defined (such as lists or trees). | {
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# determining if sequence has upper bound
I am somewhat stuck in my calculations when determining if sequence has an upper bound.
The sequence $$x_n = \frac{1}{n+1}+\frac{1}{n+2}+..+\frac{1}{2n-1}+\frac{1}{2n}$$ Is equal to $$\frac{1}{n}(\frac{1}{1+\frac{1}{n}}+\frac{1}{1+\frac{2}{n}}+..+\frac{1}{1+\frac{n}{n}})$$
And so I notice that all the denominators are greater than 1, which means that all terms in the parentheses are less than 1.
But how can I determine further if there is an upper bound?
• ... and there are $n$ of them. – Zeekless Feb 4 at 17:54
• @Zeekless so the sequence should be less than (1/n) * n = 1 is that correct? – F Wi Feb 4 at 18:02
• – Martin Sleziak Feb 5 at 8:22
The largest term is the first, so an obvious upper bound is to set all terms equal to the first one and get $$x_n < \frac{n}{n+1} <1.$$
You could also say that, since the last term is the smallest, one has $$x_n > \frac{n}{2n} = \frac 12,$$
which means that $$\frac 12 < x_n < 1, n \in \mathbb{N}$$.
By C-S $$\sum_{i=1}^n\frac{1}{n+i}=1+\sum_{i=1}^n\left(\frac{1}{n+i}-\frac{1}{n}\right)=1-\frac{1}{n}\sum_{i=1}^n\frac{i}{n+i}=$$ $$=1-\frac{1}{n}\sum_{i=1}^n\frac{i^2}{ni+i^2}\leq1-\frac{1}{n}\frac{\left(\sum\limits_{i=1}^ni\right)^2}{\sum\limits_{i=1}^n(ni+i^2)}=1-\frac{1}{n}\frac{\frac{n^2(n+1)^2}{4}}{\frac{n^2(n+1)}{2}+\frac{n(n+1)(2n+1)}{6}}=$$ $$=1-\frac{3(n+1)}{2(5n+1)}=\frac{7n-1}{10n+2}<\frac{7}{10}.$$ Actually, $$\ln2=0.6931...$$ Cauchy-Schwarz forever!
Actually, by calculus we can show that $$\lim_{n\rightarrow+\infty}\sum_{i=1}^n\frac{1}{n+i}=\ln2.$$
Notice the Riemann sum $$\frac1n\sum_{k=1}^n \frac1{1+k/n} < \int_0^1\frac{dt}{1+t} = \log 2$$
hint
For each $$n\ne 0$$,
$$\frac{1}{n+1}\le \frac{1}{n}$$ $$\frac{1}{n+2}\le \frac{1}{n}$$ ...
$$\frac{1}{2n}\le \frac 1n$$
You can finish. | {
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# How many solutions does the equation $x+y+z=11$ have? [duplicate]
How many solution does $x+y+z=11$ have where $x, y, z$ are non-negative integers. In light of the restrictions, its clear that $x,y,z \in \{0,1,2,..11\}$. So, at face value I would assign a value for $x$ and determine the different combinations that $y$ and $z$ can hold. For example,
For $x=0$, we have $y+z=11$. With writing them out I found that there are $12$ different assigned combinations for $y$ and $z$ that satisfy the equation. For $x=1$, I got $11$. Consequently, the pattern becomes clear whereby each one takes a value less by one. Hence, the number of solutions is $1+2+3+4+5+6+7..+12=78$. I was wondering if there is an easier method perhaps with combinations equation $C(a,b)$..? | {
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• We have had too many questions of this type. – evil999man May 3 '14 at 12:13
• See my answer here:math.stackexchange.com/questions/689975/… – evil999man May 3 '14 at 12:15
• Guys, stop upvoting duplicate homework questions, seriously! – Alec Teal May 3 '14 at 12:18
• @Alec Lighten up. The OP shows plenty of effort, and arrives at the correct answer in doing so. It deserves an upvote. John likely didn't know that this is a classic sort of problem, and probably hasn't encountered it before. Nor that many similar questions can be answered by the same method. And if you believe it is a duplicate and should be slammed shut because of it, then why'd you answer it? Besides, stop playing the homework police on questions showing commendable levels of effort! – amWhy May 3 '14 at 12:24
• @amWhy Not thinking something deserves an upvote isn't the same as saying it deserves to be closed. When I upvote something, what I personally mean is "this is the kind of content I come to the site for". My preferred policy would for routine problems to be dealt with quickly and without fanfare (positive or negative), and to save upvotes (and therefore time on the front page) for the more original, thought-provoking questions. – Jack M May 3 '14 at 12:28
This is a version of the classic stars-and-bars problem in combinatorics.
For any pair of natural numbers $n$ and $k$, the number of distinct $n$-tuples of non-negative integers whose sum is $k$ is given by the binomial coefficient $$\binom{n + k - 1}{k}$$
Here, $n = 3$, and $k = 11$, giving you $$\binom{3 + 11 - 1}{11} = \binom{13}{11} = \dfrac{13\cdot 12}{2} = 6\cdot 13 = 78$$ | {
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• The link takes you to Wikipedia's Stars and Bars entry, where you'll find a nice explanation of why this works, using an example that's fleshed out nicely. – amWhy May 3 '14 at 12:17
• Using the formula makes things easy, but I don't see what the connection is. Even with using the stars and bars method, it seems rather inappropriate, due to the dependency of the variables. In other terms, if x takes a value it creates a dependency on the other variables. I would appreciate it if you could make the connection clearer. – John May 3 '14 at 13:41
• Did you read the Wikipedia entry? It elaborates on the "why's" of this formula. – amWhy May 3 '14 at 13:45
• I read it and I understood everything but when I went to this question, math.stackexchange.com/questions/322369/…, with the additional restrictions I started to doubt everything. None of the answers that used combinations were clear. – John May 3 '14 at 13:49
• Yes. I see you used $x, y, z$, so if only $x$ needs to be greater than or equal to 2, then the solution (with k=11 - 2 = 9) is $\binom{3 + 9 - 1}{9}$. – amWhy May 3 '14 at 15:17
imagine 11 balls in a row and two blocks which you will place somewhere. you insert the blocks before, after or between the balls and then you assign values to $x,y,z$ in the following way: $x$ is the number of balls from the beginning of the row up to the first block, $y$ the number of balls between the two blocks and $z$ number of balls from the second block up until the end of the row. you will easily see that the number of ways in which you can place the blocks is equal to the number of different triplets $x,y,z$. Do you know how to compute the number of possible distributions of blocks? | {
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• call it stars and bars if you don't like blocks and balls, but this is the usual way to approach such problems – Alessandro Codenotti May 3 '14 at 12:15
• it's an answer explained in a tangible way. actually it's pretty much the same thing you gave as an answer although it's explained with words rather than a picture. – mm-aops May 3 '14 at 12:15
• Oh right! You could have laid it out nicer, I thought you did some weird thing that'd result in a 3! somewhere – Alec Teal May 3 '14 at 12:16
• Could you edit the answer so I may at least remove my DV? – Alec Teal May 3 '14 at 12:17
Okay let us write a solution to $a+b+c+d+e=10$ a different question, just incase it is homework.
Each solution will have the form:
||||-|---||||| <-> 4As 1B 0Cs, 0Ds, 5Es
How many different ways can we arrange 10 |s and 4 (4=5-1) -s? Each arrangement of these |s and -s is a valid solution.
$$\frac{(10+4)!}{4!10!}=\frac{14!}{10!4!}$$ | {
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# Rational Canonical form from given minimal and characteristic polynomial
$A$ is a $4\times 4$ matrix over $F$ with characteristic polynomial $(x-1)^4$ and minimal polynomial $(x-1)^2$. What is the rational canonical form of $A$?
My answer was the following: it is one of the following: $$\begin{bmatrix} 0 & -1 & & \\ 1 & 2 & & \\ & & 1 & \\ & & & 1 \end{bmatrix} or \begin{bmatrix} 0 & -1 & & \\ 1 & 2 & & \\ & & 0 & -1\\ & & 1 & 2 \end{bmatrix}.$$ While our teacher finally reached at only second form. I am not satisfied with that answer.
My question is that whether the first matrix here can also be a rational form?
In general, to write rational canonical form of a matrix, I will proceed as follows: let $$m_A(x)=(x-a_1)^{k_1}(x-a_2)^{k_2}\cdots.$$ For each factor $(x-a_i)^{k_i}$ write one block diagonal companion matrix.
If this fills up the matrix size (i.e. if $m_A(x)$ equals characteristic polynomial, then this is required form.
Otherwise, fill up remaining parts (diagonal blocks) by writing companion matrix of factors $(x-a_i)^{l_i}$ where $l_i \leq k_i$.
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Is this correct way?
Yes, you have two possible rational canonical forms given the information you have. Both the matrices you wrote have minimal polynomial $(x-1)^2$ and characteristic polynomial $(x-1)^4$.
To justify that $A$ has one of the two possible canonical forms above, let $a_1 \, | \, a_2 \, | \, \ldots \, | \, a_k$ denote the invariant factors of $A$. The highest invariant factor is always the minimal polynomial so $a_k = (x-1)^2$. The characteristic product of the matrix is the product of the invariant factors so we have a priori two options:
$$a_1(x) = (x-1), a_2(x) = (x-1), a_3(x) = (x-1)^2, \\ a_1(x) = (x-1)^2, a_2(x) = (x-1)^2.$$
• Determining (all possible) rational canonical forms I mean the following: suppose characteristic pol. is $(x-1)^6$ and minimal polynomial is $(x-1)^3$. Then possible forms are obtained by putting Companion matrices of size $\leq 3$; the possibilities will be $3+3$, $3+2+1$, $3+1+1+1$. So there will be three possible rational canonical forms when min. pol. is $(x-1)^3$ and char. poly. is $(x-1)^6$. (This situation is almost similar to that in Jordan theory, in which we consider Jordan blocks; in Rational form, we consider Companion blocks. I would like to ensure whether this is correct.) – Beginner Dec 5 '15 at 6:05 | {
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These properties concern its sides, angles, and diagonals. If the diagonals of a quadrilateral are perpendicular bisector of each other, it is always a_____ A. Rectangle . Every square is a rhombus. B. Rhombus. The diagonals are perpendicular bisectors of each other. ToProve: if the diagonals of parallelograms are perpendicular, then the parallelogram is a rhombus.. The longer diagonal of a parallelogram measures 62 cm and makes an angle of 30 degrees with the base. We have (2)The diagonals of a square are perpendicular to each other. 5. If the diagonals of a parallelogram are perpendicular then the parallelogram will be a rectangle. If ABCD is a parallelogram, what is the length of BD? Answer. Yes, because a rhombus has two sets of parallel sides and all sides are congruent. Subscribe to bartleby learn! Bob R. Lv 6. "D" is the best answer. Plus, you’ll have access to millions of step-by-step textbook answers! Answer. Then we have the two diagonals are A + B and A − B. Just so, do the diagonals of a trapezium bisect each other at 90 degrees? The diagonals of a parallelogram_____bisect the angles of the parallelogram Sometimes A quadrilateral with one pair of sides congruent and on pair parallel is_______a parallelogram 10. C. Every trapezoid is a parallelogram. In a parallelogram, the diagonals bisect each other, so you can set the labeled segments equal to one another and then solve for . Median response time is 34 minutes and may be longer for new subjects. If the diagonals of a parallelogram are perpendicular to each other, but are not congruent to each other, then the parallelogram is which of the following? Get a free answer to a quick problem. We've seen that one of the properties of a rhombus is that its diagonals are perpendicular to each other. The diagonals meet each other at 90°, this means that they form a perpendicular bisection. If a quadrilateral has 2 pairs of opposite sides that are congruent, then it is a parallelogram. If the diagonals of a | {
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has 2 pairs of opposite sides that are congruent, then it is a parallelogram. If the diagonals of a quadrilateral both bisect each other, then the quadrilateral is a parallelogram. Sue. For example, the diagonals of a rhombus, kite or square are perpendicular, but those of a rectangle or general parallelogram are not. This is the currently selected item. In order for a parallelogram's diagonals to be perpendicular, the sides would have to have the same length, so this is only true for a rhombus (which is a specific parallelogram with equal sides). by Jennifer Kahle. The diagonals of a parallelogram bisect each other. If the diagonals of a parallelogram are perpendicular to each other, then it is a rhombus ; If the diagonals of a parallelogram are equal and perpendicular, then it is a square ∵ In a parallelogram, its diagonals bisect each other at right angles ∴ Its diagonals are perpendicular ∵ Its diagonals are equal → By using rule 3 above ∴ The parallelogram is a square. d.trapezoid. Opposite sides are congruent in parallelogram. In my opinion "D" is the best answer, by definition a rhombus is a parallelogram with perpendicular diagonals . 11. Stephen K. If you just look […] faiqaferoz646 faiqaferoz646 22.06.2020 Math Secondary School Diagonals of a parallelogram are perpendicular to each other. Opposite angles are congruent. If the diagonals of a parallelogram are perpendicular and not congruent, then the parallelogram is. The parallelogram has the following properties: Opposite sides are parallel by definition. The parallelogram has the following properties: Opposite sides are parallel by definition. 10. If all the angles of the rhombus are right angles then you have a special rhombus which is a square A rhombus is a special kind of parallelogram, in which all the sides are equal. Parallelogram and Rhombus: A parallelogram is a quadrilateral (has 4 sides) where its opposite sides are parallel and equal and its opposite angles are equal. When the diagonals of a | {
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its opposite sides are parallel and equal and its opposite angles are equal. When the diagonals of a parallelogram are perpendicular to each other then it is called. So I'm thinking of a parallelogram that is both a rectangle and a rhombus. Start here or give us a call: (312) 646-6365, © 2005 - 2021 Wyzant, Inc. - All Rights Reserved, a Question Trapezoid Midsegment Theorem. Squares are rhombuses and rectangles, so if it is B, it must be C and D as well. The diagonals bisect each other. In a parallelogram a diagonal of the length 20 cm is perpendicular to one of the sides. . answered 10/08/20, If the diagonals of a parallelogram are perpendicular they divide the figure into 4 congruent triangles so all four sides are of equal length. Here we will show the converse- that if a parallelogram has perpendicular diagonals, it is a rhombus - all its sides are equal. If all the angles of the rhombus are right angles then you have a special rhombus which is a square, Nathaniel A. If the angle at which they meet is a right angle, then a right triangle is formed whose legs are half the length of each of the diagonals, and whose hypotenuse is the length of one side of the parallelogram (rhombus). The midsegment of a trapezoid is parallel to each base and its length is one half the sum of the lengths of both bases. Feb 18, 2016 . In this case, the diagonals divide the rhombus in four congruent right-angled triangles. Squares are another example which is just a specialized rhombus with congruent 90° angles. Answer: 3. let long side = x short side = s and is perpendicular to diag s^2 + 400 = x^2 so s^2 = x^2 - 400 2 s + 2 x = 80 s = 40 - x 1600 - 80 x + x^2 = x^2 - 400 80 x = 2000 x = 200/8 = 100/4 = 25 2 0; Damon. Proof: Figure is made having diagonals AC and BD. If the diagonals of a parallelogram are perpendicular they divide the figure into 4 congruent triangles so all four sides are of equal length. The diagonals bisect each other. Ask subject matter experts 30 homework | {
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sides are of equal length. The diagonals bisect each other. Ask subject matter experts 30 homework questions each month. 1 0 736; Sam. A.) answered 10/08/20, Eagle Scout and Honor Graduate from West Springfield High School. No packages or subscriptions, pay only for the time you need. Maths. Thus you have a rhombus. EASY. But in the general case, it isn't always true that the diagonals of a parallelogram are perpendicular. (1)The diagonals of a parallelogram are equal. The properties of the parallelogram are simply those things that are true about it. Therefore, by CPCT A D → ≅ A B The diagonals of a parallelogram bisect each other. If either diagonal of a parallelogram bisects two angles, then it’s a rhombus (neither the reverse of the definition nor the converse of a property). Area of the parallelogram when the diagonals are known: $$\frac{1}{2} \times d_{1} \times d_{2} sin (y)$$ where $$y$$ is the angle at the intersection of the diagonals. C. Square. Find the area of the parallelogram if the diagonals intersect at angle of 70 degrees. Parallelogram. In a parallelogram diagonals bisect each other. Consecutive angles are supplementary. No, diagonals of a parallelogram are not perpendicular to each other, because they only bisect each other. Find the longer side of parallelogram if its perimeter is 80 cm. AB→≅DC→ & AD→≅BC→ ∴(AB→≅AD→)⇒AD→≅DC→≅AB→≅BC→. So let me see. Proof: The diagonals of a kite are perpendicular. Given that, we want to prove that this is a parallelogram. So A is out. If diagonals of a parallelogram are perpendicular, then it is a . bisects. a.rectangle. Remember a square is a special rectangle with all side lengths equivalent however we have no information regarding the side lengths of this problem. Opposite sides are congruent. Mar 3, 2019 . What is the measure of a base angle … These angles are said to be congruent with each other. If you just look […] Show that if the diagonals of a quadrilateral are equal and bisect each other at | {
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you just look […] Show that if the diagonals of a quadrilateral are equal and bisect each other at right angles, then it is a square. Proof: Opposite angles of a parallelogram. Let's think. And you see the diagonals intersect at a 90-degree angle. The diagonals of parallelograms are perpendicular. rhombus If the diagonals of a parallelogram are perpendicular, then the parallelogram is a _____ C. Four points in the plane form a parallelogram if the opposite line segments have the same slope. Therefore, by CPCT A D → ≅ A B The diagonals of a parallelogram bisect each other. trapezoid. (Points : 5) rectangle rhombus square trapezoid perpendicular. The only parallelogram that satisfies that description is a square. If the diagonals of a quadr... maths. Opposite angles are congruent. The adjacent sides of parallelogram are 26cm and 28cm and one of its diagonal is … If the diagonals of a parallelogram are perpendicular then the parallelogram will be a rectangle. 0 0. Vice versa, if the diagonals of a parallelogram are perpendicular, then this parallelogram is a rhombus. If a parallelogram has (at least) one right angle, then it is a rectangle. A parallelogram, the diagonals bisect each other. Proof: Figure is made having diagonals AC and BD. In a parallelogram, the diagonals bisect each other, so you can set the labeled segments equal to one another and then solve for . (4)Every quadrilateral is either a trapezium or a parallelogram or a kite. Prove that, if the diagonals of a parallelogram are perpendicular to each other, the parallelogram is a rhombus. b.rhombus.....Ans. The diagonals are perpendicular bisectors of each other. Theorem 16.8: If the diagonals of a parallelogram are congruent and perpendicular, the parallelogram is a square. Answer: Let two adjacent sides of the parallelogram be the vectors A and B (as shown in the figure). A. quadrilateral. If the diagonals of a parallelogram are perpendicular to each other, but are not congruent to each other, then the | {
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