Chapter 9

In Problems \(25-32\), verify the given identity. Assume continuity of all partial derivatives. $$ \nabla \times(f \mathbf{F})=f(\nabla \times \mathbf{F})+(\nabla f) \times \mathbf{F} $$

Find points on the surface \(x^{2}+3 y^{2}+4 z^{2}-2 x y=16\) at which the tangent plane is parallel to (a) the \(x z\) -plane, (b) the \(y z\) -plane, and (c) the \(x y\) -plane.

Suppose \(m\) is the mass of a moving particle. Newton's second law of motion can be written in vector form as $$ \mathbf{F}=m \mathbf{a}=\frac{d}{d t}(m \mathbf{v})=\frac{d \mathbf{p}}{d t} $$ where \(\mathbf{p}=m \mathbf{v}\) is called linear momentum. The angular momentum of the particle with respect to the origin is defined to be \(\mathbf{L}=\mathbf{r} \times \mathbf{p}\), where \(\mathbf{r}\) is its position vector. If the torque of the particle about the origin is \(\boldsymbol{\tau}=\mathbf{r} \times \mathbf{F}=\mathbf{r} \times d \mathbf{p} / d t\) show that \(\tau\) is the time rate of change of angular momentum.

Find the first partial derivatives of the given function. $$ h(r, s)=\frac{\sqrt{r}}{s}-\frac{\sqrt{s}}{r} $$

In Problems, find the indicated derivative. Assume that all vector functions are differentiable. $$ \frac{d}{d t}[\mathbf{r}(t) \cdot(t \mathbf{r}(t))] $$

In Problems, find a vector that gives the direction in which the given function decreases most rapidly at the indicated point. Find the minimum rate. $$ f(x, y)=x^{3}-y^{3} ;(2,-2) $$

Evaluate the given iterated integral by changing to polar coordinates. $$ \int_{-\sqrt{\pi}}^{\sqrt{\pi}} \int_{0}^{\sqrt{\pi-x^{2}}} \sin \left(x^{2}+y^{2}\right) d y d x $$

Find the volume of the solid bounded by the graphs of the given equations. $$ y=x^{2}, y+z=3, z=0 $$

Verify the given identity. Assume continuity of all partial derivatives. $$ \nabla \times(f \mathbf{F})=f(\nabla \times \mathbf{F})+(\nabla f) \times \mathbf{F} $$

Find a vector that gives the direction in which the given function decreases most rapidly at the indicated point. Find the minimum rate. $$ f(x, y)=x^{3}-y^{3} ;(2,-2) $$

Find the indicated derivative. Assume that all vector functions are differentiable. $$ \frac{d}{d t}[\mathbf{r}(t) \cdot(t \mathbf{r}(t))] $$

Evaluate the double integral \(\iint\left(\frac{x^{2}}{25}+\frac{y^{2}}{9}\right) d A\), where \(R\) is the elliptical region whose boundary is the graph of \(x^{2} / 25+y^{2} / 9=1\). Use the substitutions \(u=x / 5, v=y / 3\), and polar coordinates.

In Problems, evaluate the given iterated integral by changing to polar coordinates. $$ \int_{0}^{1} \int_{\sqrt{1-x^{2}}}^{\sqrt{4-x^{2}}} \frac{x^{2}}{x^{2}+y^{2}} d y d x+\int_{1}^{2} \int_{0}^{\sqrt{4-x^{2}}} \frac{x^{2}}{x^{2}+y^{2}} d y d x $$

Let \(\mathbf{F}\) be a vector field. Find the flux of \(\mathbf{F}\) through the given surface. Assume the surface \(S\) is oriented upward. \(\mathbf{F}=x \mathbf{i}+2 z \mathbf{j}+y \mathbf{k} ; S\) that portion of the cylinder \(y^{2}+z^{2}=4\) in the first octant bounded by \(x=0, x=3, y=0, z=0\)

Find the volume of the solid bounded by the graphs of the given equations. $$ z=1+x^{2}+y^{2}, 3 x+y=3, x=0, y=0, z=0, \text { first octant } $$

In Problems \(25-32\), verify the given identity. Assume continuity of all partial derivatives. $$ \operatorname{curl}(\operatorname{grad} f)=\mathbf{0} $$

If \(\mathbf{F}\) is a conservative force field, show that the work done along any simple closed path is zero.

Evaluate \(\int_{C} \mathbf{F} \cdot d \mathbf{r}\). \(\mathbf{F}(x, y)=y^{3} \mathbf{i}-x^{2} y \mathbf{j} ; \mathbf{r}(t)=e^{-2 t} \mathbf{i}+e^{t} \mathbf{j}, 0 \leq t \leq \ln 2\)

Find the first partial derivatives of the given function. $$ w=2 \sqrt{x} y-y e^{y / z} $$

In Problems, find the indicated derivative. Assume that all vector functions are differentiable. $$ \frac{d}{d t}\left[\mathbf{r}(t) \cdot\left(\mathbf{r}^{\prime}(t) \times \mathbf{r}^{\prime \prime}(t)\right)\right] $$

In Problems, find a vector that gives the direction in which the given function decreases most rapidly at the indicated point. Find the minimum rate. $$ F(x, y, z)=\sqrt{x z} e^{y} ;(16,0,9) $$

Evaluate the given iterated integral by changing to polar coordinates. $$ \int_{0}^{1} \int_{\sqrt{1-x^{2}}}^{\sqrt{4}-x^{2}} \frac{x^{2}}{x^{2}+y^{2}} d y d x+\int_{1}^{2} \int_{0}^{\sqrt{4}-x^{2}} \frac{x^{2}}{x^{2}+y^{2}} d y d x $$

Evaluate \(\int_{C} \mathbf{F} \cdot d \mathbf{r}\). If \(\mathbf{F}\) is a conservative force field, show that the work done along any simple closed path is zero.

Verify the given identity. Assume continuity of all partial derivatives. \(\operatorname{curl}(\operatorname{grad} f)=\mathbf{0}\)

Show that the second equation is an equation of the tangent plane to the graph of the first equation at \(\left(x_{0}, y_{0}, z_{0}\right)\). $$ \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}+\frac{z^{2}}{c^{2}}=1 ; \frac{x x_{0}}{a^{2}}+\frac{y y_{0}}{b^{2}}+\frac{z z_{0}}{c^{2}}=1 $$

Find the indicated derivative. Assume that all vector functions are differentiable. $$ \frac{d}{d t}\left[\mathbf{r}(t) \cdot\left(\mathbf{r}^{\prime}(t) \times \mathbf{r}^{\prime \prime}(t)\right)\right] $$

In Problems, evaluate the given iterated integral by changing to polar coordinates. $$ \int_{0}^{1} \int_{0}^{\sqrt{2 y-y^{2}}}\left(1-x^{2}-y^{2}\right) d x d y $$

Let \(\mathbf{F}\) be a vector field. Find the flux of \(\mathbf{F}\) through the given surface. Assume the surface \(S\) is oriented upward. \(\mathbf{F}=z \mathbf{k} ; S\) that part of the paraboloid \(z=5-x^{2}-y^{2}\) inside the cylinder \(x^{2}+y^{2}=4\)

Find the volume of the solid bounded by the graphs of the given equations. $$ z=x+y, x^{2}+y^{2}=9, x=0, y=0, z=0, \text { first octant } $$

In Problems \(25-32\), verify the given identity. Assume continuity of all partial derivatives. $$ \operatorname{div}(\text { curl } \mathbf{F})=0 $$

A particle in the plane is attracted to the origin with a force \(\mathbf{F}=\|\mathbf{r}\|^{n} \mathbf{r}\), where \(n\) is a positive integer and \(\mathbf{r}=x \mathbf{i}+y \mathbf{j}\) is the position vector of the particle. Show that \(\mathbf{F}\) is conservative. Find the work done in moving the particle between \(\left(x_{1}, y_{1}\right)\) and \(\left(x_{2}, y_{2}\right) .\)

Evaluate \(\int_{C} \mathbf{F} \cdot d \mathbf{r}\). \(\mathbf{F}(x, y, z)=e^{x} \mathbf{i}+x e^{x y} \mathbf{j}+x y e^{x y 2} \mathbf{k} ; \mathbf{r}(t)=t \mathbf{i}+t^{2} \mathbf{j}+t^{3} \mathbf{k}\), \(0 \leq t \leq 1\)

Show that the second equation is an equation of the tangent plane to the graph of the first equation at \(\left(x_{0}, y_{0}, z_{0}\right)\). $$ \frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}+\frac{z^{2}}{c^{2}}=1 ; \frac{x x_{0}}{a^{2}}-\frac{y y_{0}}{b^{2}}+\frac{z z_{0}}{c^{2}}=1 $$

In Problems, find the indicated derivative. Assume that all vector functions are differentiable. $$ \frac{d}{d t}\left[\mathbf{r}_{1}(t) \times\left(\mathbf{r}_{2}(t) \times \mathbf{r}_{3}(t)\right)\right] $$

Find the first partial derivatives of the given function. $$ w=x y \ln (x z) $$

In Problems, find a vector that gives the direction in which the given function decreases most rapidly at the indicated point. Find the minimum rate. $$ F(x, y, z)=\ln \frac{x y}{z} ;\left(\frac{1}{2}, \frac{1}{6}, \frac{1}{3}\right) $$

Set up, but do not evaluate, the iterated integrals giving the mass of the solid that has the given shape and density. $$ x^{2}+y^{2}-z^{2}=1, z=-1, z=2 ; p(x, y, z)=z^{2} $$

\(\mathbf{F}=z \mathbf{k} ; S\) that part of the paraboloid \(z=5-x^{2}-y^{2}\) inside the cylinder \(x^{2}+y^{2}=4\)

Evaluate the given iterated integral by changing to polar coordinates. $$ \int_{0}^{1} \int_{0}^{\sqrt{2 y-y^{2}}}\left(1-x^{2}-y^{2}\right) d x d y $$

Evaluate \(\int_{C} \mathbf{F} \cdot d \mathbf{r}\). $$ \begin{aligned} &\mathbf{F}(x, y, z)=e^{x} \mathbf{i}+x e^{x y} \mathbf{j}+x y e^{x y z} \mathbf{k} ; \mathbf{r}(t)=t \mathbf{i}+t^{2} \mathbf{j}+t^{3} \mathbf{k} \\ &0 \leq t \leq 1 \end{aligned} $$

Verify the given identity. Assume continuity of all partial derivatives. \(\operatorname{div}(\operatorname{curl} \mathbf{F})=0\)

Find a vector that gives the direction in which the given function decreases most rapidly at the indicated point. Find the minimum rate. $$ F(x, y, z)=\ln \frac{x y}{z} ;\left(\frac{1}{2}, \frac{1}{6}, \frac{1}{3}\right) $$

Find the indicated derivative. Assume that all vector functions are differentiable. $$ \frac{d}{d t}\left[\mathbf{r}_{1}(t) \times\left(\mathbf{r}_{2}(t) \times \mathbf{r}_{3}(t)\right)\right] $$

In Problems, evaluate the given iterated integral by changing to polar coordinates. $$ \int_{-5}^{5} \int_{0}^{\sqrt{25-x^{2}}}(4 x+3 y) d y d x $$

Let \(P\) and \(Q\) be continuous and have continuous first partial derivatives in a simply connected region of the \(x y\) -plane. If \(\int_{A}^{B} P d x+Q d y\) is independent of the path, show that \&t \(P d x+Q d y=0\) on every piecewise- smooth simple closed curve \(C\) in the region

Find the volume of the solid bounded by the graphs of the given equations. $$ y z=6, x=0, x=5, y=1, y=6, z=0 $$

In Problems \(25-32\), verify the given identity. Assume continuity of all partial derivatives. $$ \operatorname{div}(\mathbf{F} \times \mathbf{G})=\mathbf{G} \cdot \operatorname{curl} \mathbf{F}-\mathbf{F} \cdot \operatorname{curl} \mathbf{G} $$

Find the work done by the force \(\mathbf{F}(x, y)=y \mathbf{i}+x\) j acting along \(y=\ln x\) from \((1,0)\) to \((e, 1)\)

Show that every tangent plane to the graph of \(z^{2}=x^{2}+y^{2}\) passes through the origin.

In Problems, find the indicated derivative. Assume that all vector functions are differentiable. $$ \frac{d}{d t}\left[\mathbf{r}_{1}(2 t)+\mathbf{r}_{2}\left(\frac{1}{t}\right)\right] $$