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3.9. EXERCISES FOR CHAPTER 3 73 Example 3.27. In Example 3.20, we showed that the function given by f(x, y) = x3+y3 x2+y2 , which is defined for (x, y) ̸= (0, 0), could be made continuous at (0, 0) by setting f(0, 0) = 0. (Strictly speaking, this extended function is the one called ef in the definition.) Hence, by defi... | Multivariable_Calculus_Shimamoto_Page_85_Chunk3301 |
74 CHAPTER 3. REAL-VALUED FUNCTIONS: PRELIMINARIES (a) Sketch the level set corresponding to c = 0. (Hint: x3 −3xy2 = x(x + √ 3y)(x − √ 3y).) (b) Draw the region of the xy-plane in which f(x, y) > 0 and the region in which f(x, y) < 0. (Hint: Part (a) might help.) (c) Use the information from parts (a) and (b) to make ... | Multivariable_Calculus_Shimamoto_Page_86_Chunk3302 |
3.9. EXERCISES FOR CHAPTER 3 75 Section 3 The equation of a plane 3.1. Find an equation of the plane through the point p = (1, 2, 3) with normal vector n = (4, 5, 6). 3.2. Find an equation of the plane through the point p = (2, −1, 3) with normal vector n = (1, −4, 5). 3.3. Find a point that lies on the plane x + 3y + ... | Multivariable_Calculus_Shimamoto_Page_87_Chunk3303 |
76 CHAPTER 3. REAL-VALUED FUNCTIONS: PRELIMINARIES (b) Show that the perpendicular distance d between π1 and π2 is given by: d = |D2 −D1| √ A2 + B2 + C2 . (c) Find the perpendicular distance between the planes x + y + z = 1 and x + y + z = 5. 3.16. Let α be a path in R3 that has constant torsion τ = 0. You may assume t... | Multivariable_Calculus_Shimamoto_Page_88_Chunk3304 |
3.9. EXERCISES FOR CHAPTER 3 77 (b) Use the ϵδ-definition of continuity to determine whether f is continuous at (0, 0). 5.2. Consider the function f(x, y) defined by: f(x, y) = x2y4 (x2 + y4)2 if (x, y) ̸= (0, 0). Determine what happens to the value of f(x, y) as (x, y) approaches the origin along: (a) the x-axis, (b) ... | Multivariable_Calculus_Shimamoto_Page_89_Chunk3305 |
78 CHAPTER 3. REAL-VALUED FUNCTIONS: PRELIMINARIES Section 7 The Cauchy-Schwarz and triangle inequalities 7.1. Let v and w be vectors in Rn. (a) Show that ∥v∥−∥w∥≤∥v −w∥. (Hint: v = (v −w) + w.) (b) Show that ∥v∥−∥w∥ ≤∥v −w∥. 7.2. (a) Let a be a point of Rn. If r > 0, show that B(a, r) ⊂B(0, ∥a∥+r). Draw a picture that... | Multivariable_Calculus_Shimamoto_Page_90_Chunk3306 |
3.9. EXERCISES FOR CHAPTER 3 79 Section 8 Limits 8.1. Prove parts 1 and 3 of Proposition 3.28 about limits of sums and products. (Hint: By using the corresponding properties of continuous functions in Proposition 3.21, you should be able to avoid ϵ’s and δ’s entirely.) 8.2. Let U be an open set in Rn, x0 a point of U, ... | Multivariable_Calculus_Shimamoto_Page_91_Chunk3307 |
80 CHAPTER 3. REAL-VALUED FUNCTIONS: PRELIMINARIES Addendum: Folding a hypar (Exercise 1.9) 1. Take your square piece of paper, and fold and unfold along each of the diagonals. Step 1 2. Turn the paper over. 3. Fold the bottom edge of the paper to the center point, but only crease the part between the diagonals. Unfold... | Multivariable_Calculus_Shimamoto_Page_92_Chunk3308 |
3.9. EXERCISES FOR CHAPTER 3 81 Step 7 Step 9 10. Turn the paper over. The creases should alternate as you move from one concentric square to the next: mountain, valley, mountain, valley, . . . . 11. Now, try to fold all the creases. It’s easiest to start from the outer ring and work your way in, pinching in at the cor... | Multivariable_Calculus_Shimamoto_Page_93_Chunk3309 |
82 CHAPTER 3. REAL-VALUED FUNCTIONS: PRELIMINARIES | Multivariable_Calculus_Shimamoto_Page_94_Chunk3310 |
Chapter 4 Real-valued functions: differentiation We now extend the notion of derivative from real-valued functions of one variable, as in first-year calculus, to real-valued functions of n variables f : U →R, where U is an open set in Rn. The most natural choice would be to copy the one-variable definition verbatim and... | Multivariable_Calculus_Shimamoto_Page_95_Chunk3311 |
84 CHAPTER 4. REAL-VALUED FUNCTIONS: DIFFERENTIATION By definition, f′(a) = limx→a f(x)−f(a) x−a , assuming the limit exists. This can be rewritten as 0 = limx→a | Multivariable_Calculus_Shimamoto_Page_96_Chunk3312 |
4.1. THE FIRST-ORDER APPROXIMATION 85 If h < 0, then the denominator in equation (4.4) is |h| = −h. The minus sign can be factored out and canceled, so equation (4.5) still holds. This can be solved for T(e1): m1 = T(e1) = lim h→0 f(a + he1) −f(a) h = lim h→0 f(a1 + h, a2, . . . , an) −f(a1, a2, . . . , an) h . This li... | Multivariable_Calculus_Shimamoto_Page_97_Chunk3313 |
86 CHAPTER 4. REAL-VALUED FUNCTIONS: DIFFERENTIATION Definition. Let U be an open set in Rn, f : U →R a real-valued function, and a a point of U. Then f is said to be differentiable at a if: lim x→a f(x) −f(a) −Df(a) · (x −a) ∥x −a∥ = 0. When this happens, the matrix Df(a) is called the derivative of f at a. It is also... | Multivariable_Calculus_Shimamoto_Page_98_Chunk3314 |
4.1. THE FIRST-ORDER APPROXIMATION 87 The function in this last expression is continuous, as it is a composition of sums and products of continuous pieces. Thus the value of the limit is simply the value of the function at the point (1, 2). In other words: lim (x,y)→(1,2) x2 + y2 −5 − 2 4 x −1 y −2 p (x −1)2 + (... | Multivariable_Calculus_Shimamoto_Page_99_Chunk3315 |
88 CHAPTER 4. REAL-VALUED FUNCTIONS: DIFFERENTIATION Moreover, x −a = x −0 y −0 = x y , and ∥x −a∥= p x2 + y2. Thus, according to the definition of differentiability, we need to check if: lim (x,y)→(0,0) xy x2+y2 −0 − 0 0 x y p x2 + y2 = 0, i.e., if lim(x,y)→(0,0) xy (x2+y2)3/2 = 0. For this, suppose we ap... | Multivariable_Calculus_Shimamoto_Page_100_Chunk3316 |
4.3. THE MEAN VALUE THEOREM 89 Example 4.5. Returning to function (4.7) above, we could have simply said from the start that f is not continuous at (0, 0) and therefore is not differentiable there either, avoiding the definition of differentiability entirely. Conversely, we know from first-year calculus that a function... | Multivariable_Calculus_Shimamoto_Page_101_Chunk3317 |
90 CHAPTER 4. REAL-VALUED FUNCTIONS: DIFFERENTIATION Figure 4.3: Going from a to b by way of c See Figure 4.3. To prove this, we introduce the intermediate point c = (b1, a2) and think of going from a to b by first going from a to c and then from c to b, as in Figure 4.3. Only one variable varies along each of the segm... | Multivariable_Calculus_Shimamoto_Page_102_Chunk3318 |
4.4. THE C1 TEST 91 We use the definition of differentiability. Let a be a point of U. We first analyze the error f(x) −f(a) −Df(a) · (x −a) in the first-order approximation of f at a. Since U is an open set and we are interested only in what happens when x is near a, we may assume that x lies in an open ball about a t... | Multivariable_Calculus_Shimamoto_Page_103_Chunk3319 |
92 CHAPTER 4. REAL-VALUED FUNCTIONS: DIFFERENTIATION of (4.7). We saw that f is not differentiable at the origin. On the other hand, at points other than the origin, the partial derivatives can be found using the quotient rule. This gives: ∂f ∂x = (x2 + y2) · y −xy · 2x (x2 + y2)2 = y3 −x2y (x2 + y2)2 = y(y2 −x2) (x2 +... | Multivariable_Calculus_Shimamoto_Page_104_Chunk3320 |
4.6. DIRECTIONAL DERIVATIVES 93 direction. Let a be a point of an open set U in Rn. To represent a direction going away from a, we choose a unit vector u pointing in that direction. We travel in that direction on the line through a and parallel to u, which is parametrized by α(t) = a + tu. The parametrization has veloc... | Multivariable_Calculus_Shimamoto_Page_105_Chunk3321 |
94 CHAPTER 4. REAL-VALUED FUNCTIONS: DIFFERENTIATION 4.7 ∇f as normal vector Let f : U →R be a differentiable real-valued function of n variables defined on an open set U in Rn, and let S be the level set of f corresponding to f = c. Let a be a point of S, and let v be a vector tangent to S at a. We take this to mean t... | Multivariable_Calculus_Shimamoto_Page_106_Chunk3322 |
4.7. ∇f AS NORMAL VECTOR 95 Figure 4.5: The tangent plane to z = x2 + y2 at p = (1, 2, 5) In other words, geometrically, the approximation we get by using the tangent plane at (a, f(a)) to approximate the graph coincides with the first-order approximation at a. This extends a familiar idea from first-year calculus, whe... | Multivariable_Calculus_Shimamoto_Page_107_Chunk3323 |
96 CHAPTER 4. REAL-VALUED FUNCTIONS: DIFFERENTIATION 4.8 Higher-order partial derivatives Once they start partial differentiating, most people can’t stop. Example 4.20. Let f(x, y) = x3 −3x2y4 + exy. Then: ∂f ∂x = 3x2 −6xy4 + yexy and ∂f ∂y = −12x2y3 + xexy. Both of these are again real-valued functions of x and y, so ... | Multivariable_Calculus_Shimamoto_Page_108_Chunk3324 |
4.8. HIGHER-ORDER PARTIAL DERIVATIVES 97 This expression is symmetric enough in h and k that perhaps you are willing to believe that you would get the same thing when the order of differentiation is reversed. But if you are nervous about it, here is the parallel calculation for ∂2f ∂y ∂x: ∂2f ∂y ∂x(c, d) = ∂ ∂y | Multivariable_Calculus_Shimamoto_Page_109_Chunk3325 |
98 CHAPTER 4. REAL-VALUED FUNCTIONS: DIFFERENTIATION Example 4.22. Find ∂4f ∂x ∂y2 ∂z if f(x, y, z) = | Multivariable_Calculus_Shimamoto_Page_110_Chunk3326 |
4.10. MAX/MIN: CRITICAL POINTS 99 this for functions of more than one variable. This is a big subject that overflows with applications. Our treatment is not comprehensive. Instead, we have selected a couple of topics that illustrate some of the similarities and differences that pertain to moving to the multivariable ca... | Multivariable_Calculus_Shimamoto_Page_111_Chunk3327 |
100 CHAPTER 4. REAL-VALUED FUNCTIONS: DIFFERENTIATION It is called the Hessian matrix of f at a. By the equality of mixed partials, the two off-diagonal terms, ∂2f ∂x ∂y(a) and ∂2f ∂y ∂x(a), are actually equal. Theorem 4.24 (Second derivative test for functions of two variables). Let a be a critical point of f such tha... | Multivariable_Calculus_Shimamoto_Page_112_Chunk3328 |
4.10. MAX/MIN: CRITICAL POINTS 101 (b) By the same calculation, (0, 0) is the only critical point, only this time: H(0, 0) = −2 0 0 −2 . Hence det H(0, 0) = 4 > 0 and ∂2f ∂x2 (0, 0) = −2 < 0, so f has a local maximum at (0, 0). (c) Again, (0, 0) is the only critical point, but: H(0, 0) = 2 0 0 −2 . Since det H(0,... | Multivariable_Calculus_Shimamoto_Page_113_Chunk3329 |
102 CHAPTER 4. REAL-VALUED FUNCTIONS: DIFFERENTIATION Figure 4.10: The graph of f(x, y) = x3 + 8y3 −3xy near (0, 0) (left) and near (1 2, 1 4) (right) 4.11 Classifying nondegenerate critical points We now say a few words about why the second derivative test for functions of two variables works (Theorem 4.24 of the prec... | Multivariable_Calculus_Shimamoto_Page_114_Chunk3330 |
4.11. CLASSIFYING NONDEGENERATE CRITICAL POINTS 103 finding an approximation for f(x) is the same as approximating g(1). As a real-valued function of one variable, g(t) has a second-order Taylor approximation (4.16) at 0, which for g(1) gives: f(x) = g(1) ≈g(0) + g′(0) · (1 −0) + g′′(0) 2 · (1 −0)2 ≈f(a) + g′(0) + g′′(... | Multivariable_Calculus_Shimamoto_Page_115_Chunk3331 |
104 CHAPTER 4. REAL-VALUED FUNCTIONS: DIFFERENTIATION We remark that one can continue in this way to obtain higher-order approximations of f by using higher-order Taylor approximations of g. The coefficients involve the higher-order derivatives g(n)(0) of g. The idea is that, by the Little Chain Rule, each additional d... | Multivariable_Calculus_Shimamoto_Page_116_Chunk3332 |
4.12. MAX/MIN: LAGRANGE MULTIPLIERS 105 about the signs of the coefficients A and AC−B2 A then gives the second-derivative test as stated in Theorem 4.24. The key idea of the argument was to complete the square to turn f into one of the prototypes of being a sum and/or difference of squares, at least up to second order... | Multivariable_Calculus_Shimamoto_Page_117_Chunk3333 |
106 CHAPTER 4. REAL-VALUED FUNCTIONS: DIFFERENTIATION Figure 4.12: Intersecting S with a level set f = c (left) and a family of intersections (right) • The level set f = cmax must be tangent to S. Otherwise we would be able to cross the level curve at a and make the value of f larger while remaining on S. • ∇f(a) is a ... | Multivariable_Calculus_Shimamoto_Page_118_Chunk3334 |
4.12. MAX/MIN: LAGRANGE MULTIPLIERS 107 occur at points where ∇f = λ∇g, that is, where (1, 2, 3) = λ(1 2x, 2 9y, 2z). This gives a system of equations: 1 = 1 2λx 2 = 2 9λy 3 = 2λz. There are four unknowns and only three equations, which may seem like not enough information, but we must remember th... | Multivariable_Calculus_Shimamoto_Page_119_Chunk3335 |
108 CHAPTER 4. REAL-VALUED FUNCTIONS: DIFFERENTIATION It’s worth stepping back to review this calculation, as there is more here than meets the eye. A partial derivative can be viewed as a one-variable rate of change. For instance, in the previous example, ∂f ∂x is the rate at which the output changes per dollar spent ... | Multivariable_Calculus_Shimamoto_Page_120_Chunk3336 |
4.13. EXERCISES FOR CHAPTER 4 109 (a) Find ∂f ∂x(a), ∂f ∂y (a), and ∂f ∂z (a). (b) Find Df(a) and ∇f(a). (c) Find the first-order approximation ℓ(x, y, z) of f(x, y, z) at a. (You may assume that f is differentiable at a.) (d) Compare the values of f(1.05, −1.1, 0.95) and ℓ(1.05, −1.1, 0.95). In Exercises 1.3–1.8, find... | Multivariable_Calculus_Shimamoto_Page_121_Chunk3337 |
110 CHAPTER 4. REAL-VALUED FUNCTIONS: DIFFERENTIATION (c) Use the definition of differentiability to determine whether f is differentiable at (0, 0). (Hint: Polar coordinates might be useful.) 1.18. Consider the function f : R2 →R defined by: f(x, y) = ( x3+2y3 x2+y2 if (x, y) ̸= (0, 0), 0 if (x, y) = (0, 0). (a) Find ... | Multivariable_Calculus_Shimamoto_Page_122_Chunk3338 |
4.13. EXERCISES FOR CHAPTER 4 111 (c) If f is differentiable at a, prove that f is continuous at a. (Hint: Use the triangle inequality and the Cauchy-Schwarz inequality to show that: |f(x) −f(a)| ≤|Q(x)|∥x −a∥+ ∥∇f(a)∥∥x −a∥.) Section 3 The mean value theorem 3.1. Let B = B(a, r) be an open ball in R2 centered at the p... | Multivariable_Calculus_Shimamoto_Page_123_Chunk3339 |
112 CHAPTER 4. REAL-VALUED FUNCTIONS: DIFFERENTIATION (a) by using the Little Chain Rule, (b) by substituting for x and y in terms of t in the formula for f to obtain (f ◦α)(t) directly and differentiating the result. 5.2. Let f(x, y, z) = xyz, and let α(t) = (cos t, sin t, t). Calculate (f ◦α)′(π 6 ) in two different ... | Multivariable_Calculus_Shimamoto_Page_124_Chunk3340 |
4.13. EXERCISES FOR CHAPTER 4 113 5.7. Let f : R3 →R be a differentiable function with the property that ∇f(x) points in the same direction as x for all nonzero x in R3. If a > 0, prove that f is constant on the sphere x2 +y2 +z2 = a2. (Hint: If p and q are any two points on the sphere, there is a differentiable path α... | Multivariable_Calculus_Shimamoto_Page_125_Chunk3341 |
114 CHAPTER 4. REAL-VALUED FUNCTIONS: DIFFERENTIATION (b) If u is a unit vector in Rn, find a formula for the directional derivative (Duf)(x). 6.11. You discover that your happiness is a function of your location in the plane. At the point (x, y), your happiness is given by the formula: H(x, y) = x3e2y happs, where “ha... | Multivariable_Calculus_Shimamoto_Page_126_Chunk3342 |
4.13. EXERCISES FOR CHAPTER 4 115 Figure 4.14: The surface (9x2 + y2 + z2 −1)3 −y2z3 −2 5x2z3 = 0 7.6. Find all values of c such that, at every point of intersection of the spheres (x −c)2 + y2 + z2 = 3 and x2 + (y −1)2 + z2 = 1, the respective tangent planes are perpendicular to one another. Section 8 Higher-order par... | Multivariable_Calculus_Shimamoto_Page_127_Chunk3343 |
116 CHAPTER 4. REAL-VALUED FUNCTIONS: DIFFERENTIATION Figure 4.15: The rectangle with vertices a, u, v, w 8.7. This exercise gives the details of a full proof of the equality of mixed partials (Theorem 4.21). Let a = (c, d) be a point of R2, and let f : B →R be a real-valued function defined on an open ball B = B(a, r)... | Multivariable_Calculus_Shimamoto_Page_128_Chunk3344 |
4.13. EXERCISES FOR CHAPTER 4 117 10.8. Let f(x, y) = 3x4 −4x2y + y2. (a) Make a sketch in R2 indicating the set of points (x, y) where f(x, y) > 0 and the set where f(x, y) < 0. (Hint: Factor f.) (b) Explain why f does not have a local maximum or minimum at (0, 0). (c) On the other hand, show that f does have a local ... | Multivariable_Calculus_Shimamoto_Page_129_Chunk3345 |
118 CHAPTER 4. REAL-VALUED FUNCTIONS: DIFFERENTIATION (b) Let (x1, y1), (x2, y2), . . . , (xn, yn) be given data points. Show that the Hessian of the total squared error E is given by: H(m, b) = 2 | Multivariable_Calculus_Shimamoto_Page_130_Chunk3346 |
4.13. EXERCISES FOR CHAPTER 4 119 (b) Show that the quadratic term Ah2 + 2Bhk + Ck2 = 2Bhk + Ck2 in the second-order approximation can be written as a difference of two perfect squares. (Hence, based on our prototypical models, we predict a to be a saddle point. Hint: In the case that C = 0, too, consider the expansion... | Multivariable_Calculus_Shimamoto_Page_131_Chunk3347 |
120 CHAPTER 4. REAL-VALUED FUNCTIONS: DIFFERENTIATION Ax = λx for some scalar λ and that the maximum or minimum value is the corresponding value of λ. (In the language of linear algebra, x is called an eigenvector of A and λ is called the corresponding eigenvalue.) In fact, maximum and minimum values do exist, so the c... | Multivariable_Calculus_Shimamoto_Page_132_Chunk3348 |
Chapter 5 Real-valued functions: integration 5.1 Volume and iterated integrals We now study the integral of real-valued functions of more than one variable. Most of our time is devoted to functions of two variables. In this section, we introduce the integral informally by thinking about how to visualize it. This uses a... | Multivariable_Calculus_Shimamoto_Page_133_Chunk3349 |
122 CHAPTER 5. REAL-VALUED FUNCTIONS: INTEGRATION Figure 5.2: The region under z = 4 −x + 2y and above the square 1 ≤x ≤3, 2 ≤y ≤4 Here, f(x, y) = 4 −x + 2y, and, if D denotes the specified square, we are looking for RR D(4 − x + 2y) dA. As above, let W denote the region in question, below the plane and above the squar... | Multivariable_Calculus_Shimamoto_Page_134_Chunk3350 |
5.1. VOLUME AND ITERATED INTEGRALS 123 Alternatively, we could have used cross-sections perpendicular to the y-axis, in which case the cross-sections occur from y = 2 to y = 4 (Figure 5.4): Volume = Z 4 2 A(y) dy. (5.2) Figure 5.4: A cross-section perpendicular to the y-axis Each cross-section lies below the graph and ... | Multivariable_Calculus_Shimamoto_Page_135_Chunk3351 |
124 CHAPTER 5. REAL-VALUED FUNCTIONS: INTEGRATION Figure 5.5: The cylinders x2 + z2 = 1 (left), y2 + z2 = 1 (middle), and the region contained in both (right) Figure 5.6: The portion in the first octant (left) and the portion under only x2 + z2 = 1 (right) The volume of the original solid is 8 times the volume of the p... | Multivariable_Calculus_Shimamoto_Page_136_Chunk3352 |
5.1. VOLUME AND ITERATED INTEGRALS 125 Hence A(x) = R x 0 √ 1 −x2 dy, so: Volume = 8 Z 1 0 Z x 0 p 1 −x2 dy dx = 8 Z 1 0 p 1 −x2 · y y=x y=0 dx = 8 Z 1 0 x p 1 −x2 dx (let u = 1 −x2, du = −2x dx) = 8 −1 2 · 2 3(1 −x2)3/2 1 0 = −8 3(03/2 −13/2) = 8 3. Alternatively, suppose we had used cross-sections perpendic... | Multivariable_Calculus_Shimamoto_Page_137_Chunk3353 |
126 CHAPTER 5. REAL-VALUED FUNCTIONS: INTEGRATION Figure 5.9: The domain of integration: cross-sections perpendicular to the x-axis (b) To evaluate the integral as presented, we would antidifferentiate first with respect to y, treating x as constant: Z 2 0 Z 4 x2 x3ey3 dy dx = Z 2 0 x3 Z 4 x2 ey3 dy dx. The inne... | Multivariable_Calculus_Shimamoto_Page_138_Chunk3354 |
5.2. THE DOUBLE INTEGRAL 127 • D is a bounded subset of R2: this means that there is a rectangle R in R2 such that D ⊂R. • f is a bounded function: this means that there is a scalar M such that |f(x)| ≤M for all x in D. To define the integral, we take as a guiding principle that it is a limit of weighted sums. This is ... | Multivariable_Calculus_Shimamoto_Page_139_Chunk3355 |
128 CHAPTER 5. REAL-VALUED FUNCTIONS: INTEGRATION Figure 5.12: The rectangle R = [a, b] × [c, d] in R2 Figure 5.13: A subdivision of a rectangle into 4 subrectangles Definition. Let R = [a, b] × [c, d] be a rectangle in R2. The integral of a bounded function f : R →R is defined by: ZZ R f(x, y) dA = lim △xi→0 △yj→0 X i... | Multivariable_Calculus_Shimamoto_Page_140_Chunk3356 |
5.2. THE DOUBLE INTEGRAL 129 Theorem 5.5. Let f : R →R be a bounded function defined on a rectangle R = [a, b] × [c, d]. If f is continuous on R, except possibly at a finite number of points or on a finite number of smooth curves, then f is integrable on R. This is actually a special case of a more general theorem that... | Multivariable_Calculus_Shimamoto_Page_141_Chunk3357 |
130 CHAPTER 5. REAL-VALUED FUNCTIONS: INTEGRATION An example of a graph of ef is shown in Figure 5.15. Figure 5.15: The general case: the graph z = ef(x, y) The integral of ef, as a function defined on a rectangle, has been covered in Step 1. Definition. We say that f : D →R is integrable on D if ef is integrable on R.... | Multivariable_Calculus_Shimamoto_Page_142_Chunk3358 |
5.3. INTERPRETATIONS OF THE DOUBLE INTEGRAL 131 2. RR D cf dA = c RR D f dA for any scalar c. 3. If f(x, y) ≤g(x, y) for all (x, y) in D, then RR D f dA ≤ RR D g dA. 5.3 Interpretations of the double integral We have defined the integral of a bounded real-valued function f(x, y) of two variables over a bounded subset D... | Multivariable_Calculus_Shimamoto_Page_143_Chunk3359 |
132 CHAPTER 5. REAL-VALUED FUNCTIONS: INTEGRATION Figure 5.17: Visualizing a Riemann sum as approximating the volume under a graph This could also be written as: 2 · 1 5 + 4 · 1 5 + 7 · 1 5 + 4 · 1 5 + 4 · 1 5 or 2 · 1 5 + 4 · 3 5 + 7 · 1 5. In other words, it’s a sum in which each value is weighted by its relative fre... | Multivariable_Calculus_Shimamoto_Page_144_Chunk3360 |
5.3. INTERPRETATIONS OF THE DOUBLE INTEGRAL 133 Figure 5.18: The graph of f(x, y) = y2 −x2 over the triangle D Figure 5.19: The triangle D To set up the double integral RR D(y2 −x2) dA as an iterated integral, note that D is bounded on the left by y = −x, on the right by y = x, and on top by y = 2. We use cross-section... | Multivariable_Calculus_Shimamoto_Page_145_Chunk3361 |
134 CHAPTER 5. REAL-VALUED FUNCTIONS: INTEGRATION Figure 5.20: Reversing the order of integration to integrate over D Actually, we could try to take advantage of symmetry properties of D and f to avoid having to evaluate both halves. We take up this idea more fully when we discuss the change of variables theorem in Cha... | Multivariable_Calculus_Shimamoto_Page_146_Chunk3362 |
5.4. PARAMETRIZATIONS OF SURFACES 135 but, in our examples, D will often be a rectangle or a closed disk in R2, in which case all boundary points are included. The coordinates in D are the parameters. There are often natural choices for what to call them depending on the surface, though generically we call them s and t... | Multivariable_Calculus_Shimamoto_Page_147_Chunk3363 |
136 CHAPTER 5. REAL-VALUED FUNCTIONS: INTEGRATION In this example, we used the x and y coordinates of the xyz-coordinate system as parameters for the surface. This is always an option for surfaces that are graphs z = f(x, y) of functions of two variables. We next introduce some other coordinate systems that can be used... | Multivariable_Calculus_Shimamoto_Page_148_Chunk3364 |
5.4. PARAMETRIZATIONS OF SURFACES 137 Figure 5.24: The polar coordinate transformation from the rθ-plane to the xy-plane Figure 5.25: A parametrization of a cone using polar coordinates r and θ as parameters 5.4.2 Cylindrical coordinates (r, θ, z) in R3 Cylindrical coordinates are polar coordinates in the xy-plane toge... | Multivariable_Calculus_Shimamoto_Page_149_Chunk3365 |
138 CHAPTER 5. REAL-VALUED FUNCTIONS: INTEGRATION Again r = p x2 + y2. All of R3 is covered by choosing r ≥0, 0 ≤θ ≤2π, −∞< z < ∞. Example 5.15 (Circular cylinders). We parametrize the circular cylinder of radius a and height h given by the conditions: x2 + y2 = a2, 0 ≤z ≤h. The axis of the cylinder is the z-axis. In c... | Multivariable_Calculus_Shimamoto_Page_150_Chunk3366 |
5.4. PARAMETRIZATIONS OF SURFACES 139 Figure 5.28: Spherical coordinates (ρ, ϕ, θ) about the positive z-axis by the angle θ. This lands at the point (ρ sin ϕ cos θ, ρ sin ϕ sin θ, ρ cos ϕ). The coordinates are labeled in Figure 5.28. Thus the conversions from spherical to rectangular coordinates are: x = ρ si... | Multivariable_Calculus_Shimamoto_Page_151_Chunk3367 |
140 CHAPTER 5. REAL-VALUED FUNCTIONS: INTEGRATION Figure 5.29: A parametrization of a sphere using spherical coordinates ϕ and θ as parameters Here, along the cone, the angle ϕ down from the positive z-axis is fixed at π 4 . To see this, note, for instance, that the cross-section with the yz-plane is z = |y|. On the ot... | Multivariable_Calculus_Shimamoto_Page_152_Chunk3368 |
5.5. INTEGRALS WITH RESPECT TO SURFACE AREA 141 two-dimensional partition of small pieces of surface area △Sij, choose a sample point pij in each piece, form a sum P i,j f(pij) △Sij, and take the limit as the size of the pieces goes to zero. We use a parametrization of S to convert the calculation into a double integra... | Multivariable_Calculus_Shimamoto_Page_153_Chunk3369 |
142 CHAPTER 5. REAL-VALUED FUNCTIONS: INTEGRATION Figure 5.32: A blowup of the previous figure: σ sends a small △si by △tj subrectangle to a curvy quadrilateral whose area △Sij is approximately ∂σ ∂s × ∂σ ∂t △si △tj. We should acknowledge a point that was also glossed over in connection with integrals with re- spect to... | Multivariable_Calculus_Shimamoto_Page_154_Chunk3370 |
5.5. INTEGRALS WITH RESPECT TO SURFACE AREA 143 so: ∂σ ∂ϕ × ∂σ ∂θ = a2 sin ϕ q sin2 ϕ cos2 θ + sin2 ϕ sin2 θ + cos2 ϕ = a2 sin ϕ q sin2 ϕ + cos2 ϕ = a2 sin ϕ. (5.4) Hence: Area (S) = ZZ D a2 sin ϕ dϕ dθ = Z 2π 0 Z π 0 a2 sin ϕ dϕ dθ = Z 2π 0 −a2 cos ϕ ϕ=π ϕ=0 dθ = Z 2π 0 | Multivariable_Calculus_Shimamoto_Page_155_Chunk3371 |
144 CHAPTER 5. REAL-VALUED FUNCTIONS: INTEGRATION (c) This time, f(x, y, z) = x2 + y2 + z2. We use a trick! On the sphere S, x2 + y2 + z2 = a2, so RR S (x2 + y2 + z2) dS = RR S a2 dS = a2 RR S 1 dS = a2Area (S). Using the formula for the area from part (a) then gives: ZZ S (x2 + y2 + z2) dS = a2 · 4πa2 = 4πa4. The tric... | Multivariable_Calculus_Shimamoto_Page_156_Chunk3372 |
5.6. TRIPLE INTEGRALS AND BEYOND 145 Figure 5.33: Three of the planes that bound W: z = y (left), y = x (middle), and x = 1 (right). The fourth is z = 0, the xy-plane. Example 5.20. Find RRR W (y + z) dV if W is the solid region in R3 bounded by the planes z = y, y = x, x = 1, and z = 0. See Figure 5.33. The planes in ... | Multivariable_Calculus_Shimamoto_Page_157_Chunk3373 |
146 CHAPTER 5. REAL-VALUED FUNCTIONS: INTEGRATION antidifferentiation at a time. Thus: ZZZ W (y + z) dV = Z 1 0 Z x 0 Z y 0 (y + z) dz dy dx = Z 1 0 Z x 0 (yz + 1 2z2) z=y z=0 dy dx = Z 1 0 Z x 0 3 2y2 dy dx = Z 1 0 1 2y3 y=x y=0 dx = Z 1 0 1 2x3 dx = 1 8x4 1 0 = 1 8. Example 5.21. Express the same integ... | Multivariable_Calculus_Shimamoto_Page_158_Chunk3374 |
5.7. EXERCISES FOR CHAPTER 5 147 Figure 5.35: Integrating first with respect to x (left) or y (right) 5.7 Exercises for Chapter 5 Section 1 Volume and iterated integrals In Exercises 1.1–1.4, evaluate the given iterated integral. 1.1. Z 2 1 Z 4 3 x2y dy dx 1.2. Z 1 0 Z x x2 (x + y) dy dx 1.3. Z π/2 0 Z sin y 0 (... | Multivariable_Calculus_Shimamoto_Page_159_Chunk3375 |
148 CHAPTER 5. REAL-VALUED FUNCTIONS: INTEGRATION 1.13. Z 1 −1 Z √ 1−x2 0 f(x, y) dy dx 1.14. Z 6 −3 Z −1+√3+y 1 3 y f(x, y) dx dy 1.15. Consider the iterated integral Z 1 1 4 Z 4 1 x yexy dy dx. (a) Sketch the domain of integration D in the xy-plane. (b) Write an equivalent expression with the order of integr... | Multivariable_Calculus_Shimamoto_Page_160_Chunk3376 |
5.7. EXERCISES FOR CHAPTER 5 149 Figure 5.36: A 3 by 3 subdivision of R = [0, 1] × [0, 1] (d) Let δ be a positive real number. If R is subdivided into a grid of subrectangles of dimensions △xi by △yj, where △xi < δ and △yj < δ for all i, j, show that any Riemann sum based on the subdivision lies in the range 0 ≤P i,j f... | Multivariable_Calculus_Shimamoto_Page_161_Chunk3377 |
150 CHAPTER 5. REAL-VALUED FUNCTIONS: INTEGRATION (b) Let g: U →R be another continuous function on U. If f(a) > g(a), prove that there exists a rectangle R containing a such that: ZZ R f dA > ZZ R g dA. (Hint: Consider f −g.) 2.5. In this exercise, we use the double integral to give another proof of the equality of mi... | Multivariable_Calculus_Shimamoto_Page_162_Chunk3378 |
5.7. EXERCISES FOR CHAPTER 5 151 Figure 5.37: The fudgsicle 3.8. Let R be a rectangle in R2, and let f : R →R be an integrable function. If f is continuous, then it is true that there is a point of R at which f assumes its average value, that is, there exists a point c of R such that f(c) = f. The proof is not hard, bu... | Multivariable_Calculus_Shimamoto_Page_163_Chunk3379 |
152 CHAPTER 5. REAL-VALUED FUNCTIONS: INTEGRATION 5.2. Let S be the cylindrical surface x2 + y2 = 4, 0 ≤z ≤3. Find the integral RR S z2 dS. 5.3. Consider the graph z = f(x, y) of a smooth real-valued function f defined on a bounded subset D of R2. Show that the surface area of the graph is given by the formula: Surface... | Multivariable_Calculus_Shimamoto_Page_164_Chunk3380 |
5.7. EXERCISES FOR CHAPTER 5 153 6.3. Find RRR W xyz dV if W is the solid region in R3 below the surface z = x2 + y2 and above the square 0 ≤x ≤1, 0 ≤y ≤1. 6.4. Find RRR W (xz + y) dV if W is the solid region in R3 above the xy-plane bounded by the surface y = x2 and the planes z = y, y = 1, and z = 0. 6.5. Find RRR W ... | Multivariable_Calculus_Shimamoto_Page_165_Chunk3381 |
154 CHAPTER 5. REAL-VALUED FUNCTIONS: INTEGRATION | Multivariable_Calculus_Shimamoto_Page_166_Chunk3382 |
Part IV Vector-valued functions 155 | Multivariable_Calculus_Shimamoto_Page_167_Chunk3383 |
Chapter 6 Differentiability and the chain rule So far, we have studied functions of which at least one of the domain or codomain is a subset of R, in other words: • vector-valued functions of one variable, also known as paths, α: I →Rn, where I ⊂R, or • real-valued functions of n variables f : U →R, where U ⊂Rn. Now, w... | Multivariable_Calculus_Shimamoto_Page_169_Chunk3384 |
158 CHAPTER 6. DIFFERENTIABILITY AND THE CHAIN RULE Figure 6.1: Continuity of a vector-valued function: given any ball B(f(a), ϵ) about f(a), there is a ball B(a, δ) about a such that f(B(a, δ)) ⊂B(f(a), ϵ). We say that f is a continuous function if it is continuous at every point of its domain. Proposition 6.1. If f(x... | Multivariable_Calculus_Shimamoto_Page_170_Chunk3385 |
6.2. DIFFERENTIABILITY REVISITED 159 is continuous at a. When this happens, we write limx→a f(x) = L. As before, it is immediate that f : U →Rm is continuous at a if and only if limx→a f(x) = f(a). One consequence of this is that Proposition 6.1 then implies that a vector-valued limit can also be viewed as a sequence o... | Multivariable_Calculus_Shimamoto_Page_171_Chunk3386 |
160 CHAPTER 6. DIFFERENTIABILITY AND THE CHAIN RULE To repeat a point made in the real-valued case, the derivative is not a single number, but rather a matrix or, even better, the linear part of a good affine approximation of f near a. We begin by calculating a few simple examples of Df(a). In general, it is an m by n ... | Multivariable_Calculus_Shimamoto_Page_172_Chunk3387 |
6.3. THE CHAIN RULE: A CONCEPTUAL APPROACH 161 According to Proposition 6.6, we may apply the C1 test for real-valued functions to each of the components of a vector-valued function to obtain the following criterion. Theorem 6.8 (The C1 test). If all the partial derivatives ∂fi ∂xj (i.e., the entries of Df(x)) exist an... | Multivariable_Calculus_Shimamoto_Page_173_Chunk3388 |
162 CHAPTER 6. DIFFERENTIABILITY AND THE CHAIN RULE In terms of the sizes of the various parties involved, the left side of the chain rule is a p by n matrix, while the right side is a product (p by m) × (m by n). This need not be memorized. In practice, it usually takes care of itself. The chain rule says that the der... | Multivariable_Calculus_Shimamoto_Page_174_Chunk3389 |
6.4. THE CHAIN RULE: A COMPUTATIONAL APPROACH 163 dot product of the ith row and jth column: ... · · · (i, j) · · · ... = ∗ ∗ · · · ∗ ∗ ∗ ∗ ... ∗ ∗ . There is one such dot product for each entry of D(g ◦f)(a). We shall write out these entries in a few cases in a moment and discover... | Multivariable_Calculus_Shimamoto_Page_175_Chunk3390 |
164 CHAPTER 6. DIFFERENTIABILITY AND THE CHAIN RULE w = g x y z s t Figure 6.3: Dependence diagram for tweak 1 the ordinary derivative terms d ds (or d dt) in equation (6.5) with partial derivatives ∂ ∂s (or ∂ ∂t). For instance, for ∂w ∂s , the Little Chain Rule says: ∂w ∂s = ∂g ∂x ∂x ∂s + ∂g ∂y ∂y ∂s + ∂g ∂z ∂z ∂s. (6... | Multivariable_Calculus_Shimamoto_Page_176_Chunk3391 |
6.4. THE CHAIN RULE: A COMPUTATIONAL APPROACH 165 as highlighted in bold in the figure. Again, the collection of the four individual partial derivatives of the composition computed in this way can be organized into a single matrix equation: " ∂w1 ∂s ∂w1 ∂t ∂w2 ∂s ∂w2 ∂t # = " ∂g1 ∂x ∂g1 ∂y ∂g1 ∂z ∂g2 ∂x ∂g2 ∂y ∂g2 ∂z #... | Multivariable_Calculus_Shimamoto_Page_177_Chunk3392 |
166 CHAPTER 6. DIFFERENTIABILITY AND THE CHAIN RULE Figure 6.6: The level set f(x, y) = c defines y implicitly as a function of x on an interval of x values. w = f x y x Figure 6.7: A dependence diagram for implicit differentiation On the other hand, w has a constant value of c, so dw dx = 0. Hence: ∂f ∂x + ∂f ∂y dy dx... | Multivariable_Calculus_Shimamoto_Page_178_Chunk3393 |
6.5. EXERCISES FOR CHAPTER 6 167 (b) Let U be an open set in Rn, and let f : U →Rm be a function, where f(x) = (f1(x), f2(x), . . . , fm(x)). Show that, if f is continuous at a point a of U, then so are f1, f2, . . . , fm. 1.2. Let U be an open set in Rn, a a point of U, and f : U −{a} →Rm a function defined on U, exce... | Multivariable_Calculus_Shimamoto_Page_179_Chunk3394 |
168 CHAPTER 6. DIFFERENTIABILITY AND THE CHAIN RULE (a) Find Df(x, y) and Dg(x, y). (b) Show that f and g are differentiable at every point (x, y) of R6. Section 3 The chain rule: a conceptual approach 3.1. Let f : R2 →R3 and g: R3 →R2 be given by: f(s, t) = | Multivariable_Calculus_Shimamoto_Page_180_Chunk3395 |
6.5. EXERCISES FOR CHAPTER 6 169 4.3. The spherical substitutions x = ρ sin ϕ cos θ, y = ρ sin ϕ sin θ, and z = ρ cos ϕ convert a smooth real-valued function f(x, y, z) into a function of ρ, ϕ, and θ: w = f(ρ sin ϕ cos θ, ρ sin ϕ sin θ, ρ cos ϕ). (a) Find formulas for ∂w ∂ρ , ∂w ∂ϕ , and ∂w ∂θ in terms of ∂f ∂x, ∂f ∂y ... | Multivariable_Calculus_Shimamoto_Page_181_Chunk3396 |
170 CHAPTER 6. DIFFERENTIABILITY AND THE CHAIN RULE Assume that the condition f(x, y, z) = c defines x and y implicitly as smooth functions of z on some interval of z values in R, that is, x = x(z) and y = y(z) are functions that satisfy f(x(z), y(z), z) = c. Then the corresponding portion of the level set is parametri... | Multivariable_Calculus_Shimamoto_Page_182_Chunk3397 |
6.5. EXERCISES FOR CHAPTER 6 171 (b) Find an analogous expression for the mixed partial derivative ∂2w ∂r ∂θ. (c) Find an analogous expression for ∂2w ∂r2 . (d) If f(x, y) = x2 +y2 +xy, use your answer to part (b) to find ∂2w ∂r ∂θ. Then, verify that it is the same as the result obtained by writing w in terms of r and ... | Multivariable_Calculus_Shimamoto_Page_183_Chunk3398 |
172 CHAPTER 6. DIFFERENTIABILITY AND THE CHAIN RULE | Multivariable_Calculus_Shimamoto_Page_184_Chunk3399 |
Chapter 7 Change of variables We now turn to the counterpart for integrals of the chain rule. It is called the change of variables theorem. For functions of one variable, the corresponding notion is the method of substitution. The one-variable and multivariable versions are expressed in equations that are similar forma... | Multivariable_Calculus_Shimamoto_Page_185_Chunk3400 |
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