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college_math.Corrals_Vector_Calculus | exercise.4.2.1 | Evaluate the given double integral: $\int_{0}^{1} \int_{\sqrt{x}}^{1} 24 x^{2} y d y d x$ | $1 $ | GNU Free Documentation License | college_math.vector_calculus | 1 |
college_math.Corrals_Vector_Calculus | exercise.1.4.5 | Calculate $\mathbf{v} \times \mathbf{w}$:
$\mathbf{v}=-\mathbf{i}+2 \mathbf{j}+\mathbf{k}, \mathbf{w}=-3 \mathbf{i}+6 \mathbf{j}+3 \mathbf{k}$ | 0 | GNU Free Documentation License | college_math.vector_calculus | 0 |
college_math.Corrals_Vector_Calculus | exercise.1.1.3 | For the points $P=(0,0,0), Q=(1,3,2), R=(1,0,1), S=(2,3,4)$, does $\overrightarrow{P Q}=\overrightarrow{R S}$ ? | No | GNU Free Documentation License | college_math.vector_calculus | No |
college_math.Corrals_Vector_Calculus | exercise.3.5.9 | Find all local maxima and minima of the function $f(x, y) = 4x^{2} - 4xy + 2y^{2} + 10x - 6y$. | local min. $(-1,1 / 2) $ | GNU Free Documentation License | college_math.vector_calculus | local min. $(-1,1 / 2) |
college_math.Corrals_Vector_Calculus | exercise.5.6.3 | Find the Laplacian of the function $f(x, y, z)=\left(x^{2}+y^{2}+z^{2}\right)^{3 / 2}$. | $12 \sqrt{x^{2}+y^{2}+z^{2}} $ | GNU Free Documentation License | college_math.vector_calculus | 12 \sqrt{x^{2}+y^{2}+z^{2}} |
college_math.Corrals_Vector_Calculus | exercise.1.4.31 | Describe geometrically the set of points with position vector $\mathbf{x}$ satisfying the equation
$$(\mathbf{v} \times \mathbf{x}) \times \mathbf{x}=\mathbf{v}$$
for given vector $\mathbf{v} \neq \mathbf{0}$ | A circle of radius $\frac{1}{\|\mathbf{v}\|}$ centered at the origin in the normal plane to $\mathbf{v}$. | GNU Free Documentation License | college_math.vector_calculus | A circle of radius $\frac{1}{\|\mathbf{v}\|}$ centered at the origin in the normal plane to $\mathbf{v}$. |
college_math.Corrals_Vector_Calculus | exercise.4.1.3 | Find the volume under the surface $z=f(x, y)$ over the rectangle $R$: $f(x, y)=x^{3}+y^{2}, R=[0,1] \times[0,1]$ | $\frac{7}{12} $ | GNU Free Documentation License | college_math.vector_calculus | \frac{7}{12} |
college_math.Corrals_Vector_Calculus | exercise.4.3.3 | Evaluate the given triple integral: $\int_{0}^{\pi} \int_{0}^{x} \int_{0}^{x y} x^{2} \sin z d z d y d x$ | $\left(2 \cos \left(\pi^{2}\right)+\pi^{4}-2\right) / 4 $ | GNU Free Documentation License | college_math.vector_calculus | \left(2 \cos \left(\pi^{2}\right)+\pi^{4}-2\right) / 4 |
college_math.Corrals_Vector_Calculus | exercise.4.6.1 | Find the center of mass of the region $R$ with the given density function $\delta(x, y)$:
$R=\{(x, y): 0 \leq x \leq 2,0 \leq y \leq 4\}, \delta(x, y)=2 y$ | $(1,8 / 3)$ | GNU Free Documentation License | college_math.vector_calculus | (1,8 / 3) |
college_math.Corrals_Vector_Calculus | exercise.3.1.5 | State the domain and range of the given function: $f(x, y, z)=\sin (x y z)$ | domain: $\mathbb{R}^{3}$, range: $[-1,1] $ | GNU Free Documentation License | college_math.vector_calculus | domain: $\mathbb{R}^{3}$, range: $[-1,1] |
college_math.Corrals_Vector_Calculus | exercise.5.5.7 | Calculate $\int_{C} \mathbf{f} \cdot d \mathbf{r}$ for the given vector field $\mathbf{f}(x, y, z)$ and curve $C$: $\mathbf{f}(x, y, z)=(y-2 z) \mathbf{i}+x y \mathbf{j}+(2 x z+y) \mathbf{k} ; \quad C: x=t, y=2 t, z=t^{2}-1,0 \leq t \leq 1$ | $67 / 15 $ | GNU Free Documentation License | college_math.vector_calculus | 67 / 15 |
college_math.Corrals_Vector_Calculus | exercise.4.3.7 | Evaluate the given triple integral: $\int_{1}^{2} \int_{2}^{4} \int_{0}^{3} 1 d x d y d z$ | 6 | GNU Free Documentation License | college_math.vector_calculus | 6 |
college_math.Corrals_Vector_Calculus | exercise.3.1.13 | Evaluate the limit: $\lim _{(x, y) \rightarrow(1,1)} \frac{x^{2}-y^{2}}{x-y}$ | $2 $ | GNU Free Documentation License | college_math.vector_calculus | 2 |
college_math.Corrals_Vector_Calculus | exercise.3.1.17 | Evaluate the limit: $\lim _{(x, y) \rightarrow(0,0)} \frac{x}{y}$ | does not exist | GNU Free Documentation License | college_math.vector_calculus | does not exist |
college_math.Corrals_Vector_Calculus | exercise.4.2.5 | Evaluate the given double integral: $\int_{0}^{\pi / 2} \int_{0}^{y} \cos x \sin y d x d y$ | $\frac{\pi}{4} $ | GNU Free Documentation License | college_math.vector_calculus | \frac{\pi}{4} |
college_math.Corrals_Vector_Calculus | exercise.4.1.7 | Evaluate the double integral: $\int_{0}^{2} \int_{0}^{1}(x+2) d x d y$ | $5 $ | GNU Free Documentation License | college_math.vector_calculus | 5 |
college_math.Corrals_Vector_Calculus | exercise.4.2.3 | Evaluate the given double integral: $\int_{1}^{2} \int_{0}^{\ln x} 4 x d y d x$ | $8 \ln 2-3$ | GNU Free Documentation License | college_math.vector_calculus | 8 \ln 2-3 |
college_math.Corrals_Vector_Calculus | exercise.5.1.3 | Calculate the line integral $\int_{C} f(x, y) d s$ for the given function $f(x, y)$ and curve $C$.
$f(x, y)=2 x+y ; \quad C$ : polygonal path from $(0,0)$ to $(3,0)$ to $(3,2)$ | 23 | GNU Free Documentation License | college_math.vector_calculus | 23 |
college_math.Corrals_Vector_Calculus | exercise.4.1.1 | Find the volume under the surface $z=f(x, y)$ over the rectangle $R$: $f(x, y)=4 x y, R=[0,1] \times[0,1]$ | $1 $ | GNU Free Documentation License | college_math.vector_calculus | 1 |
college_math.Corrals_Vector_Calculus | exercise.3.5.13 | Find three positive numbers $x, y, z$ whose sum is 10 such that $x^{2}y^{2}z$ is a maximum. | $x=y=4, z=2$ | GNU Free Documentation License | college_math.vector_calculus | x=y=4, z=2 |
college_math.Corrals_Vector_Calculus | exercise.3.5.7 | Find all local maxima and minima of the function $f(x, y) = \sqrt{x^{2} + y^{2}}$. | local min. $(0,0) $ | GNU Free Documentation License | college_math.vector_calculus | local min. $(0,0) |
college_math.Corrals_Vector_Calculus | exercise.3.5.3 | Find all local maxima and minima of the function $f(x, y) = x^{3} - 3x + y^{3} - 3y$. | local min. $(1,1)$; local max. $(-1,-1)$; saddle pts. $(1,-1),(-1,1) $ | GNU Free Documentation License | college_math.vector_calculus | local min. $(1,1)$; local max. $(-1,-1)$; saddle pts. $(1,-1),(-1,1) |
college_math.Corrals_Vector_Calculus | exercise.5.1.7 | Calculate the line integral $\int_{C} \mathbf{f} \cdot d \mathbf{r}$ for the given vector field $\mathbf{f}(x, y)$ and curve $C$.
$\mathbf{f}(x, y)=y \mathbf{i}-x \mathbf{j} ; \quad C: x=\cos t, y=\sin t, 0 \leq t \leq 2 \pi$ | $-2 \pi$ | GNU Free Documentation License | college_math.vector_calculus | -2 \pi |
college_math.Corrals_Vector_Calculus | exercise.4.7.2 | For $\sigma>0$ and $\mu>0$, evaluate
$$
\int_{-\infty}^{\infty} \frac{1}{\sigma \sqrt{2 \pi}} e^{-(x-\mu)^{2} / 2 \sigma^{2}} d x
$$ | 1 | GNU Free Documentation License | college_math.vector_calculus | 1 |
college_math.Corrals_Vector_Calculus | exercise.4.6.5 | Find the center of mass of the region $R$ with the given density function $\delta(x, y)$:
$R=\left\{(x, y): y \geq 0, x^{2}+y^{2} \leq 1\right\}, \delta(x, y)=y$ | $(0,3 \pi / 16) $ | GNU Free Documentation License | college_math.vector_calculus | (0,3 \pi / 16) |
college_math.Corrals_Vector_Calculus | exercise.4.6.9 | Find the center of mass of the solid $S$ with the given density function $\delta(x, y, z)$:
$S=\{(x, y, z): 0 \leq x \leq 1,0 \leq y \leq 1,0 \leq z \leq 1\}, \delta(x, y, z)=x^{2}+y^{2}+z^{2}$ | $(7 / 12,7 / 12,7 / 12)$ | GNU Free Documentation License | college_math.vector_calculus | (7 / 12,7 / 12,7 / 12) |
college_math.Corrals_Vector_Calculus | exercise.2.3.1 | Find the tangent line, the osculating plane, and the curvature at each point of the curve $\mathbf{f}(t)= (\cos t, \sin t, t)$. | $\frac{3 \pi \sqrt{5}}{2} $ | GNU Free Documentation License | college_math.vector_calculus | \frac{3 \pi \sqrt{5}}{2} |
college_math.Corrals_Vector_Calculus | exercise.1.4.3 | Calculate $\mathbf{v} \times \mathbf{w}$:
$\mathbf{v}=(2,1,4), \mathbf{w}=(1,-2,0)$ | $(8,4,-5) $ | GNU Free Documentation License | college_math.vector_calculus | (8,4,-5) |
college_math.Corrals_Vector_Calculus | exercise.2.1.5 | Find the velocity $\mathbf{v}(t)$ and acceleration $\mathbf{a}(t)$ of an object with the given position vector $\mathbf{r}(t)$: $\mathbf{r}(t)=(t, t-\sin t, 1-\cos t)$ | $\mathbf{v}(t)=(1,1-\cos t, \sin t)$, $\mathbf{a}(t)=(0, \sin t, \cos t) $ | GNU Free Documentation License | college_math.vector_calculus | \mathbf{v}(t)=(1,1-\cos t, \sin t)$, $\mathbf{a}(t)=(0, \sin t, \cos t) |
college_math.Corrals_Vector_Calculus | exercise.3.3.9 | Find the equation of the tangent plane to the given surface at the point $P$: $x^{2}+y^{2}-z^{2}=0$, $P=(3,4,5)$. | $3 x+4 y-5 z=0$ | GNU Free Documentation License | college_math.vector_calculus | 3 x+4 y-5 z=0 |
college_math.Corrals_Vector_Calculus | exercise.4.3.1 | Evaluate the given triple integral: $\int_{0}^{3} \int_{0}^{2} \int_{0}^{1} x y z d x d y d z$ | $\frac{9}{2} $ | GNU Free Documentation License | college_math.vector_calculus | \frac{9}{2} |
college_math.Corrals_Vector_Calculus | exercise.3.3.1 | Find the equation of the tangent plane to the surface $z=f(x, y)$ at the point $P$: $f(x, y)=x^{2}+y^{3}$, $P=(1,1,2)$. | $2 x+3 y-z-3=0 $ | GNU Free Documentation License | college_math.vector_calculus | 2 x+3 y-z-3=0 |
college_math.Corrals_Vector_Calculus | exercise.1.4.7 | Calculate the area of the triangle $\triangle P Q R$:
$P=(5,1,-2), Q=(4,-4,3), R=(2,4,0)$ | $16.72 $ | GNU Free Documentation License | college_math.vector_calculus | 16.72 |
college_math.Corrals_Vector_Calculus | exercise.4.2.10 | Evaluate the double integral: $\iint_{R} f(x, y) d A$, where $f(x, y)=x^{2}+y$ and $R$ is the triangle with vertices $(0,0),(2,0)$ and $(0,1)$. | $\frac{6}{5} $ | GNU Free Documentation License | college_math.vector_calculus | \frac{6}{5} |
college_math.Corrals_Vector_Calculus | exercise.4.6.7 | Find the center of mass of the solid $S$ with the given density function $\delta(x, y, z)$:
$S=\left\{(x, y, z): z \geq 0, x^{2}+y^{2}+z^{2} \leq a^{2}\right\}, \delta(x, y, z)=x^{2}+y^{2}+z^{2}$ | $(0,0,5 a / 12) $ | GNU Free Documentation License | college_math.vector_calculus | (0,0,5 a / 12) |
college_math.Corrals_Vector_Calculus | exercise.5.5.13 | State whether or not the vector field $\mathbf{f}(x, y, z)$ has a potential in $\mathbb{R}^{3}$ (you do not need to find the potential itself): $\mathbf{f}(x, y, z)=x y \mathbf{i}-\left(x-y z^{2}\right) \mathbf{j}+y^{2} z \mathbf{k}$ | No | GNU Free Documentation License | college_math.vector_calculus | No |
college_math.Corrals_Vector_Calculus | exercise.4.5.7 | Evaluate $\iint_{R} \sin \left(\frac{x+y}{2}\right) \cos \left(\frac{x-y}{2}\right) d A$, where $R$ is the triangle with vertices $(0,0),(2,0)$ and $(1,1)$. (Hint: Use the change of variables $u=(x+y) / 2, v=(x-y) / 2$.) | $1-\frac{\sin 2}{2}$ | GNU Free Documentation License | college_math.vector_calculus | 1-\frac{\sin 2}{2} |
college_math.Corrals_Vector_Calculus | exercise.1.3.9 | Let $\mathbf{v}=(8,4,3)$ and $\mathbf{w}=(-2,1,4)$. Is $\mathbf{v} \perp \mathbf{w}$ ? Justify your answer. | Yes, since $\mathbf{v} \cdot \mathbf{w}=0 | GNU Free Documentation License | college_math.vector_calculus | Yes, since $\mathbf{v} \cdot \mathbf{w}=0 |
college_math.Corrals_Vector_Calculus | exercise.3.4.7 | Compute the gradient $\nabla f$ for the function $f(x, y, z)=\sin (x y z)$. | $\quad(y z \cos (x y z), x z \cos (x y z), x y \cos (x y z))$ | GNU Free Documentation License | college_math.vector_calculus | \quad(y z \cos (x y z), x z \cos (x y z), x y \cos (x y z)) |
college_math.Corrals_Vector_Calculus | exercise.5.7.5 | For $\mathbf{f}(\rho, \theta, \phi)=\mathbf{e}_{\rho}+\rho \cos \theta \mathbf{e}_{\theta}+\rho \mathbf{e}_{\phi}$ in spherical coordinates, find $\operatorname{div} \mathbf{f}$ and curlf. | $\operatorname{div} \mathbf{f}=\frac{2}{\rho}-\frac{\sin \theta}{\sin \phi}+\cot \phi, \operatorname{curl} \mathbf{f}=\cot \phi \cos \theta \mathbf{e}_{\rho}+$ $2 \mathbf{e}_{\theta}-2 \cos \theta \mathbf{e}_{\phi} \mathbf{6}$ | GNU Free Documentation License | college_math.vector_calculus | \operatorname{div} \mathbf{f}=\frac{2}{\rho}-\frac{\sin \theta}{\sin \phi}+\cot \phi, \operatorname{curl} \mathbf{f}=\cot \phi \cos \theta \mathbf{e}_{\rho}+$ $2 \mathbf{e}_{\theta}-2 \cos \theta \mathbf{e}_{\phi} \mathbf{6} |
college_math.Corrals_Vector_Calculus | exercise.4.1.11 | Evaluate the double integral: $\int_{0}^{2} \int_{1}^{4} x y d x d y$ | $15 $ | GNU Free Documentation License | college_math.vector_calculus | 15 |
college_math.Corrals_Vector_Calculus | exercise.5.7.3 | Let $f(x, y, z)=\frac{z}{x^{2}+y^{2}}$ in Cartesian coordinates. Find $\nabla f$ in cylindrical coordinates. | $-\frac{2 z}{r^{3}} \mathbf{e}_{r}+\frac{1}{r^{2}} \mathbf{e}_{z}$ | GNU Free Documentation License | college_math.vector_calculus | -\frac{2 z}{r^{3}} \mathbf{e}_{r}+\frac{1}{r^{2}} \mathbf{e}_{z} |
college_math.Corrals_Vector_Calculus | exercise.3.4.9 | Compute the gradient $\nabla f$ for the function $f(x, y, z)=x^{2}+y^{2}+z^{2}$. | $\quad(2 x, 2 y, 2 z) $ | GNU Free Documentation License | college_math.vector_calculus | \quad(2 x, 2 y, 2 z) |
college_math.Corrals_Vector_Calculus | exercise.4.1.9 | Evaluate the double integral: $\int_{0}^{\pi / 2} \int_{0}^{1} x y \cos \left(x^{2} y\right) d x d y$ | $\frac{1}{2} $ | GNU Free Documentation License | college_math.vector_calculus | \frac{1}{2} |
college_math.Corrals_Vector_Calculus | exercise.5.3.7 | Is there a potential $F(x, y)$ for $\mathbf{f}(x, y)=(8 x y+3) \mathbf{i}+4\left(x^{2}+y\right) \mathbf{j}$ ? If so, find one. | Yes. $F(x, y)=4 x^{2} y+2 y^{2}+3 x$ | GNU Free Documentation License | college_math.vector_calculus | Yes. $F(x, y)=4 x^{2} y+2 y^{2}+3 x |
college_math.Corrals_Vector_Calculus | exercise.5.2.1 | Evaluate $\oint_{C}\left(x^{2}+y^{2}\right) d x+2 x y d y$ for $C: x=\cos t, y=\sin t, 0 \leq t \leq 2 \pi$. | $0 $ | GNU Free Documentation License | college_math.vector_calculus | 0 |
college_math.Corrals_Vector_Calculus | exercise.4.2.7 | Evaluate the given double integral: $\int_{0}^{2} \int_{0}^{y} 1 d x d y$ | $2 $ | GNU Free Documentation License | college_math.vector_calculus | 2 |
college_math.Corrals_Vector_Calculus | exercise.3.3.7 | Find the equation of the tangent plane to the given surface at the point $P$: $\frac{x^{2}}{4}+\frac{y^{2}}{9}+\frac{z^{2}}{16}=1$, $P=\left(1,2, \frac{2 \sqrt{11}}{3}\right)$. | $\frac{1}{2}(x-1)+\frac{4}{9}(y-2)+\frac{\sqrt{11}}{12}(z-\frac{2 \sqrt{11}}{3})=0 $ | GNU Free Documentation License | college_math.vector_calculus | \frac{1}{2}(x-1)+\frac{4}{9}(y-2)+\frac{\sqrt{11}}{12}(z-\frac{2 \sqrt{11}}{3})=0 |
college_math.Corrals_Vector_Calculus | exercise.2.3.3 | Find the tangent line, the osculating plane, and the curvature at each point of the curve $\mathbf{f}(t)= (t \sin t, t \cos t)$. | $2\left(5^{3 / 2}-8\right) $ | GNU Free Documentation License | college_math.vector_calculus | 2\left(5^{3 / 2}-8\right) |
college_math.Corrals_Vector_Calculus | exercise.3.4.3 | Compute the gradient $\nabla f$ for the function $f(x, y)=\sqrt{x^{2}+y^{2}+4}$. | $\left(\frac{x}{\sqrt{x^{2}+y^{2}+4}}, \frac{y}{\sqrt{x^{2}+y^{2}+4}}\right) $ | GNU Free Documentation License | college_math.vector_calculus | \left(\frac{x}{\sqrt{x^{2}+y^{2}+4}}, \frac{y}{\sqrt{x^{2}+y^{2}+4}}\right) |
college_math.Corrals_Vector_Calculus | exercise.3.4.5 | Compute the gradient $\nabla f$ for the function $f(x, y)=\ln (x y)$. | $\left(\frac{1}{x}, \frac{1}{y}\right)$ | GNU Free Documentation License | college_math.vector_calculus | \left(\frac{1}{x}, \frac{1}{y}\right) |
college_math.Corrals_Vector_Calculus | exercise.1.6.9 | Find the trace of the hyperbolic paraboloid $\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=\frac{z}{c}$ in the $x y$-plane | lines $\frac{x}{a}=\frac{y}{b}, z=0$ and $\frac{x}{a}=-\frac{y}{b}, z=0$ | GNU Free Documentation License | college_math.vector_calculus | lines $\frac{x}{a}=\frac{y}{b}, z=0$ and $\frac{x}{a}=-\frac{y}{b}, z=0 |
college_math.Corrals_Vector_Calculus | exercise.4.5.1 | Find the volume $V$ inside the paraboloid $z=x^{2}+y^{2}$ for $0 \leq z \leq 4$. | $8 \pi$ | GNU Free Documentation License | college_math.vector_calculus | 8 \pi |
college_math.Corrals_Vector_Calculus | exercise.1.3.7 | Find the angle $\theta$ between the vectors $\mathbf{v}=-\mathbf{i}+2 \mathbf{j}+\mathbf{k}$ and $\mathbf{w}=-3 \mathbf{i}+6 \mathbf{j}+3 \mathbf{k}$. | $0^{\circ} $ | GNU Free Documentation License | college_math.vector_calculus | 0^{\circ} |
college_math.Corrals_Vector_Calculus | exercise.3.1.1 | State the domain and range of the given function: $f(x, y)=x^{2}+y^{2}-1$ | domain: $\mathbb{R}^{2}$, range: $[-1, \infty) $ | GNU Free Documentation License | college_math.vector_calculus | domain: $\mathbb{R}^{2}$, range: $[-1, \infty) |
college_math.Corrals_Vector_Calculus | exercise.1.4.1 | Calculate $\mathbf{v} \times \mathbf{w}$:
$\mathbf{v}=(5,1,-2), \mathbf{w}=(4,-4,3)$ | $(-5,-23,-24) $ | GNU Free Documentation License | college_math.vector_calculus | (-5,-23,-24) |
college_math.Corrals_Vector_Calculus | exercise.3.3.3 | Find the equation of the tangent plane to the surface $z=f(x, y)$ at the point $P$: $f(x, y)=x^{2} y$, $P=(-1,1,1)$. | $-2 x+y-z-2=0$ | GNU Free Documentation License | college_math.vector_calculus | -2 x+y-z-2=0 |
college_math.Corrals_Vector_Calculus | exercise.5.2.3 | Is there a potential $F(x, y)$ for $\mathbf{f}(x, y)=y \mathbf{i}-x \mathbf{j}$ ? If so, find one. | No | GNU Free Documentation License | college_math.vector_calculus | No |
college_math.Corrals_Vector_Calculus | exercise.3.5.1 | Find all local maxima and minima of the function $f(x, y) = x^{3} - 3x + y^{2}$. | local min. $(1,0)$; saddle pt. $(-1,0)$ | GNU Free Documentation License | college_math.vector_calculus | local min. $(1,0)$; saddle pt. $(-1,0) |
college_math.Corrals_Vector_Calculus | exercise.1.3.1 | Let $\mathbf{v}=(5,1,-2)$ and $\mathbf{w}=(4,-4,3)$. Calculate $\mathbf{v} \cdot \mathbf{w}$. | $10 $ | GNU Free Documentation License | college_math.vector_calculus | 10 |
college_math.Corrals_Vector_Calculus | exercise.3.5.11 | For a rectangular solid of volume 1000 cubic meters, find the dimensions that will minimize the surface area. (Hint: Use the volume condition to write the surface area as a function of just two variables.) | width $=$ height $=\operatorname{depth}=10$ | GNU Free Documentation License | college_math.vector_calculus | width $=$ height $=\operatorname{depth}=10 |
college_math.Corrals_Vector_Calculus | exercise.1.6.1 | Determine if the given equation describes a sphere. If so, find its radius and center: $x^{2}+y^{2}+z^{2}-4 x-6 y-10 z+37=0$ | radius: 1 , center: $(2,3,5)$ | GNU Free Documentation License | college_math.vector_calculus | radius: 1 , center: $(2,3,5) |
college_math.Corrals_Vector_Calculus | exercise.3.4.1 | Compute the gradient $\nabla f$ for the function $f(x, y)=x^{2}+y^{2}-1$. | $(2 x, 2 y)$ | GNU Free Documentation License | college_math.vector_calculus | (2 x, 2 y) |
college_math.Corrals_Vector_Calculus | exercise.3.4.13 | Find the directional derivative of $f(x, y)=\sqrt{x^{2}+y^{2}+4}$ at the point $P=(1,1)$ in the direction of $\mathbf{v}=\left(\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}}\right)$. | $\quad \frac{1}{\sqrt{3}}$ | GNU Free Documentation License | college_math.vector_calculus | \quad \frac{1}{\sqrt{3}} |
college_math.Corrals_Vector_Calculus | exercise.3.4.15 | Find the directional derivative of $f(x, y, z)=\sin (x y z)$ at the point $P=(1,1,1)$ in the direction of $\mathbf{v}=\left(\frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}\right)$. | $\sqrt{3} \cos (1) ; 1$. increase: $(45,20)$, decrease: $(-45,-20)$ | GNU Free Documentation License | college_math.vector_calculus | \sqrt{3} \cos (1) ; 1$. increase: $(45,20)$, decrease: $(-45,-20) |
college_math.Corrals_Vector_Calculus | exercise.1.4.11 | Find the volume of the parallelepiped with adjacent sides $\mathbf{u}, \mathbf{v}, \mathbf{w}$:
$\mathbf{u}=(1,1,3), \mathbf{v}=(2,1,4), \mathbf{w}=(5,1,-2)$ | $9 $ | GNU Free Documentation License | college_math.vector_calculus | 9 |
college_math.Corrals_Vector_Calculus | exercise.3.5.5 | Find all local maxima and minima of the function $f(x, y) = 2x^{3} + 6xy + 3y^{2}$. | local min. $(1,-1)$, saddle pt. $(0,0) $ | GNU Free Documentation License | college_math.vector_calculus | local min. $(1,-1)$, saddle pt. $(0,0) |
college_math.Corrals_Vector_Calculus | exercise.5.2.5 | Is there a potential $F(x, y)$ for $\mathbf{f}(x, y)=x y^{2} \mathbf{i}+x^{3} y \mathbf{j}$ ? If so, find one. | No | GNU Free Documentation License | college_math.vector_calculus | No |
college_math.Corrals_Vector_Calculus | exercise.5.5.9 | Calculate $\int_{C} \mathbf{f} \cdot d \mathbf{r}$ for the given vector field $\mathbf{f}(x, y, z)$ and curve $C$: $\mathbf{f}(x, y, z)=x y \mathbf{i}+(z-x) \mathbf{j}+2 y z \mathbf{k} ; \quad C$ : the polygonal path from $(0,0,0)$ to $(1,0,0)$ to $(1,2,0)$ to $(1,2,-2)$ | 6 | GNU Free Documentation License | college_math.vector_calculus | 6 |
college_math.Corrals_Vector_Calculus | exercise.4.5.3 | Find the volume $V$ of the solid inside both $x^{2}+y^{2}+z^{2}=4$ and $x^{2}+y^{2}=1$. | $\frac{4 \pi}{3}\left(8-3^{3 / 2}\right)$ | GNU Free Documentation License | college_math.vector_calculus | \frac{4 \pi}{3}\left(8-3^{3 / 2}\right) |
college_math.Corrals_Vector_Calculus | exercise.5.2.4 | Is there a potential $F(x, y)$ for $\mathbf{f}(x, y)=x \mathbf{i}-y \mathbf{j}$ ? If so, find one. | Yes. $F(x, y)=\frac{x^{2}}{2}-\frac{y^{2}}{2} $ | GNU Free Documentation License | college_math.vector_calculus | Yes. $F(x, y)=\frac{x^{2}}{2}-\frac{y^{2}}{2} |
college_math.Corrals_Vector_Calculus | exercise.2.1.1 | Calculate $\mathbf{f}^{\prime}(t)$ and find the tangent line at $\mathbf{f}(0)$ for the following function: $\mathbf{f}(t)=\left(t+1, t^{2}+1, t^{3}+1\right)$ | $\mathbf{f}^{\prime}(t)=\left(1,2 t, 3 t^{2}\right), \quad x=1+t, \quad y=z=$ $1 $ | GNU Free Documentation License | college_math.vector_calculus | \mathbf{f}^{\prime}(t)=\left(1,2 t, 3 t^{2}\right), \quad x=1+t, \quad y=z=$ $1 |
college_math.Corrals_Vector_Calculus | exercise.5.1.11 | Calculate the line integral $\int_{C} \mathbf{f} \cdot d \mathbf{r}$ for the given vector field $\mathbf{f}(x, y)$ and curve $C$.
$\mathbf{f}(x, y)=\left(x^{2}+y^{2}\right) \mathbf{i} ; \quad C: x=2+\cos t, y=\sin t, 0 \leq t \leq 2 \pi$ | 0 | GNU Free Documentation License | college_math.vector_calculus | 0 |
college_math.Corrals_Vector_Calculus | exercise.3.4.11 | Find the directional derivative of $f(x, y)=x^{2}+y^{2}-1$ at the point $P=(1,1)$ in the direction of $\mathbf{v}=\left(\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}}\right)$. | $\quad 2 \sqrt{2} $ | GNU Free Documentation License | college_math.vector_calculus | \quad 2 \sqrt{2} |
college_math.Corrals_Vector_Calculus | exercise.5.5.5 | Calculate $\int_{C} \mathbf{f} \cdot d \mathbf{r}$ for the given vector field $\mathbf{f}(x, y, z)$ and curve $C$: $\mathbf{f}(x, y, z)=y \mathbf{i}-x \mathbf{j}+z \mathbf{k} ; \quad C: x=\cos t, y=\sin t, z=t, 0 \leq t \leq 2 \pi$ | $2 \pi(\pi-1)$ | GNU Free Documentation License | college_math.vector_calculus | 2 \pi(\pi-1) |
college_math.Corrals_Vector_Calculus | exercise.2.1.3 | Calculate $\mathbf{f}^{\prime}(t)$ and find the tangent line at $\mathbf{f}(0)$ for the following function: $\mathbf{f}(t)=(\cos 2 t, \sin 2 t, t)$ | $\mathbf{f}^{\prime}(t)=(-2 \sin 2 t, 2 \cos 2 t, 1) ; \quad x=1$, $y=2 t, \quad z=t $ | GNU Free Documentation License | college_math.vector_calculus | \mathbf{f}^{\prime}(t)=(-2 \sin 2 t, 2 \cos 2 t, 1) ; \quad x=1$, $y=2 t, \quad z=t |
college_math.Corrals_Vector_Calculus | exercise.1.1.2 | For the points $P=(1,-1,1), Q=(2,-2,2), R=(2,0,1), S=(3,-1,2)$, does $\overrightarrow{P Q}=\overrightarrow{R S}$ ? | Yes | GNU Free Documentation License | college_math.vector_calculus | Yes |
college_math.Corrals_Vector_Calculus | exercise.5.6.5 | Find the Laplacian of the function $f(x, y, z)=x^{3}+y^{3}+z^{3}$. | $6(x+y+z) $ | GNU Free Documentation License | college_math.vector_calculus | 6(x+y+z) |
college_math.Corrals_Vector_Calculus | exercise.5.5.2 | Calculate $\int_{C} f(x, y, z) d s$ for the given function $f(x, y, z)$ and curve $C$: $f(x, y, z)=\frac{x}{y}+y+2 y z ; \quad C: x=t^{2}, y=t, z=1,1 \leq t \leq 2$ | $(17 \sqrt{17}-5 \sqrt{5}) / 3 $ | GNU Free Documentation License | college_math.vector_calculus | (17 \sqrt{17}-5 \sqrt{5}) / 3 |
college_math.Corrals_Vector_Calculus | exercise.4.1.5 | Evaluate the double integral: $\int_{0}^{1} \int_{1}^{2}(1-y) x^{2} d x d y$ | $\frac{7}{6} $ | GNU Free Documentation License | college_math.vector_calculus | \frac{7}{6} |
college_math.Corrals_Vector_Calculus | exercise.5.6.1 | Find the Laplacian of the function $f(x, y, z)=x+y+z$. | $0 $ | GNU Free Documentation License | college_math.vector_calculus | 0 |
college_math.Corrals_Vector_Calculus | exercise.3.7.1 | Find the constrained maxima and minima of $f(x, y)=2 x+y$ given that $x^{2}+y^{2}=4$. | $\max .\left(\frac{4}{\sqrt{5}}, \frac{2}{\sqrt{5}}\right) $ | GNU Free Documentation License | college_math.vector_calculus | \max .\left(\frac{4}{\sqrt{5}}, \frac{2}{\sqrt{5}}\right) |
college_math.Corrals_Vector_Calculus | exercise.5.5.4 | Calculate $\int_{C} \mathbf{f} \cdot d \mathbf{r}$ for the given vector field $\mathbf{f}(x, y, z)$ and curve $C$: $\mathbf{f}(x, y, z)=\mathbf{i}-\mathbf{j}+\mathbf{k} ; \quad C: x=3 t, y=2 t, z=t, 0 \leq t \leq 1$ | 2 | GNU Free Documentation License | college_math.vector_calculus | 2 |
college_math.Corrals_Vector_Calculus | exercise.5.1.1 | Calculate the line integral $\int_{C} f(x, y) d s$ for the given function $f(x, y)$ and curve $C$.
$f(x, y)=x y ; \quad C: x=\cos t, y=\sin t, 0 \leq t \leq \pi / 2$ | $1 / 2$ | GNU Free Documentation License | college_math.vector_calculus | 1 / 2 |
college_math.Corrals_Vector_Calculus | exercise.3.1.3 | State the domain and range of the given function: $f(x, y)=\sqrt{x^{2}+y^{2}-4}$ | domain: $\left\{(x, y): x^{2}+y^{2} \geq 4\right\}$, range: $[0, \infty) $ | GNU Free Documentation License | college_math.vector_calculus | domain: $\left\{(x, y): x^{2}+y^{2} \geq 4\right\}$, range: $[0, \infty) |
college_math.Corrals_Vector_Calculus | exercise.4.3.10 | Find the volume $V$ of the solid $S$ bounded by the three coordinate planes, bounded above by the plane $x+y+z=2$, and bounded below by the plane $z=x+y$. | $\frac{1}{3}$ | GNU Free Documentation License | college_math.vector_calculus | \frac{1}{3} |
college_math.Corrals_Vector_Calculus | exercise.5.3.5 | Is there a potential $F(x, y)$ for $\mathbf{f}(x, y)=\left(y^{2}+3 x^{2}\right) \mathbf{i}+2 x y \mathbf{j}$ ? If so, find one. | Yes. $F(x, y)=x y^{2}+x^{3}$ | GNU Free Documentation License | college_math.vector_calculus | Yes. $F(x, y)=x y^{2}+x^{3} |
college_math.Corrals_Vector_Calculus | exercise.4.6.3 | Find the center of mass of the region $R$ with the given density function $\delta(x, y)$:
$R=\left\{(x, y): y \geq 0, x^{2}+y^{2} \leq a^{2}\right\}, \delta(x, y)=1$ | $\left(0, \frac{4 a}{3 \pi}\right) \quad$ | GNU Free Documentation License | college_math.vector_calculus | \left(0, \frac{4 a}{3 \pi}\right) \quad |
college_math.Corrals_Vector_Calculus | exercise.4.2.6 | Evaluate the given double integral: $\int_{0}^{\infty} \int_{0}^{\infty} x y e^{-\left(x^{2}+y^{2}\right)} d x d y$ | $\frac{1}{4} $ | GNU Free Documentation License | college_math.vector_calculus | \frac{1}{4} |
college_math.Corrals_Vector_Calculus | exercise.3.1.11 | Evaluate the limit: $\lim _{(x, y) \rightarrow(1,-1)} \frac{x^{2}-2 x y+y^{2}}{x-y}$ | $2 $ | GNU Free Documentation License | college_math.vector_calculus | 2 |
college_math.Corrals_Vector_Calculus | exercise.3.1.7 | Evaluate the limit: $\lim _{(x, y) \rightarrow(0,0)} \cos (x y)$ | $1 $ | GNU Free Documentation License | college_math.vector_calculus | 1 |
college_math.Corrals_Vector_Calculus | exercise.5.5.3 | Calculate $\int_{C} f(x, y, z) d s$ for the given function $f(x, y, z)$ and curve $C$: $f(x, y, z)=z^{2} ; \quad C: x=t \sin t, y=t \cos t, z=\frac{2 \sqrt{2}}{3} t^{3 / 2}, 0 \leq t \leq 1$ | $2 / 5 $ | GNU Free Documentation License | college_math.vector_calculus | 2 / 5 |
college_math.Corrals_Vector_Calculus | exercise.5.7.2 | Let $f(x, y, z)=e^{-x^{2}-y^{2}-z^{2}}$ in Cartesian coordinates. Find the Laplacian of the function in spherical coordinates. | $\left(4 \rho^{2}-6\right) e^{-\rho^{2}} $ | GNU Free Documentation License | college_math.vector_calculus | \left(4 \rho^{2}-6\right) e^{-\rho^{2}} |
college_math.Corrals_Vector_Calculus | exercise.1.3.5 | Find the angle $\theta$ between the vectors $\mathbf{v}=(2,1,4)$ and $\mathbf{w}=(1,-2,0)$. | $90^{\circ} $ | GNU Free Documentation License | college_math.vector_calculus | 90^{\circ} |
college_math.Corrals_Vector_Calculus | exercise.3.7.5 | Find the constrained maxima and minima of $f(x, y, z)=x+y^{2}+2 z$ given that $4 x^{2}+9 y^{2}-36 z^{2}=36$. | $\frac{8 a b c}{3 \sqrt{3}}$ | GNU Free Documentation License | college_math.vector_calculus | \frac{8 a b c}{3 \sqrt{3}} |
college_math.Corrals_Vector_Calculus | exercise.4.5.9 | Find the volume inside the elliptic cylinder $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}} \leq 1$ for $0 \leq z \leq 2$. | $2 \pi a b$ | GNU Free Documentation License | college_math.vector_calculus | 2 \pi a b |
college_math.Corrals_Vector_Calculus | exercise.1.4.9 | Calculate the area of the parallelogram $P Q R S$:
$P=(2,1,3), Q=(1,4,5), R=(2,5,3), S=(3,2,1)$ | 4 \sqrt{5} | GNU Free Documentation License | college_math.vector_calculus | 4 \sqrt{5} |
college_math.Corrals_Vector_Calculus | exercise.4.3.5 | Evaluate the given triple integral: $\int_{1}^{e} \int_{0}^{y} \int_{0}^{1 / y} x^{2} z d x d z d y$ | $\frac{1}{6} $ | GNU Free Documentation License | college_math.vector_calculus | \frac{1}{6} |
college_math.Corrals_Vector_Calculus | exercise.5.5.11 | State whether or not the vector field $\mathbf{f}(x, y, z)$ has a potential in $\mathbb{R}^{3}$ (you do not need to find the potential itself): $\mathbf{f}(x, y, z)=a \mathbf{i}+b \mathbf{j}+c \mathbf{k}(a, b, c$ constant $)$ | Yes | GNU Free Documentation License | college_math.vector_calculus | Yes |
college_math.Corrals_Vector_Calculus | exercise.5.1.9 | Calculate the line integral $\int_{C} \mathbf{f} \cdot d \mathbf{r}$ for the given vector field $\mathbf{f}(x, y)$ and curve $C$.
$\mathbf{f}(x, y)=\left(x^{2}-y\right) \mathbf{i}+\left(x-y^{2}\right) \mathbf{j} ; \quad C: x=\cos t, y=\sin t, 0 \leq t \leq 2 \pi$ | $2 \pi$ | GNU Free Documentation License | college_math.vector_calculus | 2 \pi |
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