Chapter 16

Find a parametrization of the surface. (There are many correct ways to do these, so your answers may not be the same as those in the back of the book.) Parabolic cylinder between planes The surface cut from the parabolic cylinder \(z=4-y^{2}\) by the planes \(x=0, x=2,\) and \(z=0\)

Evaluate \(\int_{C}(x+y) d s\) where \(C\) is the straight-line segment \(x=t, y=(1-t), z=0,\) from \((0,1,0)\) to \((1,0,0)\)

Let \(S\) be the cylinder \(x^{2}+y^{2}=a^{2}, 0 \leq z \leq h,\) together with its top, \(x^{2}+y^{2} \leq a^{2}, z=h .\) Let \(\mathbf{F}=-y \mathbf{i}+x \mathbf{j}+x^{2} \mathbf{k} .\) Use Stokes' Theorem to find the flux of \(\nabla \times \mathbf{F}\) outward through \(S\) .

Find a potential function \(f\) for the field \(\mathbf{F}.\) \(\mathbf{F}=e^{y+2 z}(\mathbf{i}+x \mathbf{j}+2 x \mathbf{k})\)

Integrate \(G(x, y, z)=y+z\) over the surface of the wedge in the first octant bounded by the coordinate planes and the planes \(x=2\) and \(y+z=1 .\)

Find a parametrization of the surface. (There are many correct ways to do these, so your answers may not be the same as those in the back of the book.) Parabolic cylinder between planes The surface cut from the parabolic cylinder \(y=x^{2}\) by the planes \(z=0, z=3,\) and \(y=2\)

Evaluate $$\iint_{S} \nabla \times(y \mathbf{i}) \cdot \mathbf{n} d \sigma,$$ where \(S\) is the hemisphere \(x^{2}+y^{2}+z^{2}=1, z \geq 0.\)

Evaluate \(\int_{C}(x-y+z-2) d s\) where \(C\) is the straight-line segment \(x=t, y=(1-t), z=1,\) from \((0,1,1)\) to \((1,0,1)\)

Find the line integrals of \(\mathbf{F}\) from \((0,0,0)\) to \((1,1,1)\) over each of the following paths in the accompanying figure. $$ \begin{array}{l}{\text { a. The straight-line path } C_{1} : \mathbf{r}(t)=t \mathbf{i}+t \mathbf{j}+t \mathbf{k}, \quad 0 \leq t \leq 1} \\ {\text { b. The curved path } C_{2} : \mathbf{r}(t)=t \mathbf{i}+t^{2} \mathbf{j}+t^{4} \mathbf{k}, \quad 0 \leq t \leq 1} \\ {\text { c. The path } C_{3} \cup C_{4} \text { consisting of the line segment from }(0,0,0)} \\ {\text { to }(1,1,0) \text { followed by the segment from }(1,1,0) \text { to }(1,1,1)}\end{array} $$ $$ \mathbf{F}=x y \mathbf{i}+y z \mathbf{j}+x z \mathbf{k} $$

Find a potential function \(f\) for the field \(\mathbf{F}.\) \(\mathbf{F}=(y \sin z) \mathbf{i}+(x \sin z) \mathbf{j}+(x y \cos z) \mathbf{k}\)

Find a parametrization of the surface. (There are many correct ways to do these, so your answers may not be the same as those in the back of the book.) Circular cylinder band The portion of the cylinder \(y^{2}+z^{2}=9\) between the planes \(x=0\) and \(x=3\)

Integrate \(G(x, y, z)=x y z\) over the surface of the rectangular solid cut from the first octant by the planes \(x=a, y=b,\) and \(z=c.\)

Suppose \(\mathbf{F}=\nabla \times \mathbf{A},\) where $$\mathbf{A}=(y+\sqrt{z}) \mathbf{i}+e^{x y z} \mathbf{j}+\cos (x z) \mathbf{k}.$$ Determine the flux of \(\mathbf{F}\) outward through the hemisphere \(x^{2}+y^{2}+z^{2}=1, z \geq 0.\)

\(\begin{array}{l}{\text { Evaluate } \int_{C}(x y+y+z) d s \text { along the curve } \mathbf{r}(t)=2 \mathbf{i}+} \\ {t \mathbf{j}+(2-2 t) \mathbf{k}, 0 \leq t \leq 1}\end{array}\)

Find the line integrals of \(\mathbf{F}\) from \((0,0,0)\) to \((1,1,1)\) over each of the following paths in the accompanying figure. $$ \begin{array}{l}{\text { a. The straight-line path } C_{1} : \mathbf{r}(t)=t \mathbf{i}+t \mathbf{j}+t \mathbf{k}, \quad 0 \leq t \leq 1} \\ {\text { b. The curved path } C_{2} : \mathbf{r}(t)=t \mathbf{i}+t^{2} \mathbf{j}+t^{4} \mathbf{k}, \quad 0 \leq t \leq 1} \\ {\text { c. The path } C_{3} \cup C_{4} \text { consisting of the line segment from }(0,0,0)} \\ {\text { to }(1,1,0) \text { followed by the segment from }(1,1,0) \text { to }(1,1,1)}\end{array} $$ $$ \mathbf{F}=\left(3 x^{2}-3 x\right) \mathbf{i}+3 z \mathbf{j}+\mathbf{k} $$

Find a potential function \(f\) for the field \(\mathbf{F}.\) $$\begin{aligned} \mathbf{F}=\left(\ln x+\sec ^{2}(x+y)\right) \mathbf{i}+& \\\ &\left(\sec ^{2}(x+y)+\frac{y}{y^{2}+z^{2}}\right) \mathbf{j}+\frac{z}{y^{2}+z^{2}} \mathbf{k} \end{aligned}$$

Wedge \(\quad \mathbf{F}=2 x z \mathbf{i}-x y \mathbf{j}-z^{2} \mathbf{k}\) \(D :\) The wedge cut from the first octant by the plane \(y+z=4\) and the elliptical cylinder \(4 x^{2}+y^{2}=16\)

Find a parametrization of the surface. (There are many correct ways to do these, so your answers may not be the same as those in the back of the book.) Circular cylinder band The portion of the cylinder \(x^{2}+z^{2}=4\) above the \(x y\) -plane between the planes \(y=-2\) and \(y=2\)

Integrate \(G(x, y, z)=x y z\) over the surface of the rectangular solid bounded by the planes \(x=\pm a, y=\pm b,\) and \(z=\pm c.\)

\(\begin{array}{l}{\text { Evaluate } \int_{C} \sqrt{x^{2}+y^{2}} d s \text { along the curve } \mathbf{r}(t)=(4 \cos t) \mathbf{i}+} \\ {(4 \sin t) \mathbf{j}+3 t \mathbf{k},-2 \pi \leq t \leq 2 \pi}\end{array} \)

Find the line integrals of \(\mathbf{F}\) from \((0,0,0)\) to \((1,1,1)\) over each of the following paths in the accompanying figure. $$ \begin{array}{l}{\text { a. The straight-line path } C_{1} : \mathbf{r}(t)=t \mathbf{i}+t \mathbf{j}+t \mathbf{k}, \quad 0 \leq t \leq 1} \\ {\text { b. The curved path } C_{2} : \mathbf{r}(t)=t \mathbf{i}+t^{2} \mathbf{j}+t^{4} \mathbf{k}, \quad 0 \leq t \leq 1} \\ {\text { c. The path } C_{3} \cup C_{4} \text { consisting of the line segment from }(0,0,0)} \\ {\text { to }(1,1,0) \text { followed by the segment from }(1,1,0) \text { to }(1,1,1)}\end{array} $$ $$ \mathbf{F}=(y+z) \mathbf{i}+(z+x) \mathbf{j}+(x+y) \mathbf{k} $$

Find a potential function \(f\) for the field \(\mathbf{F}.\) $$\begin{array}{r}{\mathbf{F}=\frac{y}{1+x^{2} y^{2}} \mathbf{i}+\left(\frac{x}{1+x^{2} y^{2}}+\frac{z}{\sqrt{1-y^{2} z^{2}}}\right) \mathbf{j}+} \\ {\left(\frac{y}{\sqrt{1-y^{2} z^{2}}}+\frac{1}{z}\right) \mathbf{k}}\end{array}$$

Integrate \(G(x, y, z)=x+y+z\) over the portion of the plane \(2 x+2 y+z=2\) that lies in the first octant.

In Exercises \(13-18,\) use the surface integral in Stokes' Theorem to calculate the flux of the curl of the field \(\mathbf{F}\) across the surface \(S\) in the direction of the outward unit normal \(\mathbf{n}\) . $$\begin{array}{l}{\mathbf{F}=2 \mathbf{z} \mathbf{i}+3 \mathbf{x} \mathbf{j}+5 y \mathbf{k}} \\ {S : \mathbf{r}(r, \theta)=(r \cos \theta) \mathbf{i}+(r \sin \theta) \mathbf{j}+\left(4-r^{2}\right) \mathbf{k}}, \\ {0 \leq r \leq 2, \quad 0 \leq \theta \leq 2 \pi}\end{array}$$

In Exercises \(5-14,\) use Green's Theorem to find the counterclockwise circulation and outward flux field \(\mathbf{F}\) and curve \(C .\) $$ \begin{array}{l}{\mathbf{F}=\left(x+e^{x} \sin y\right) \mathbf{i}+\left(x+e^{x} \cos y\right) \mathbf{j}} \\ {C : \text { The right- hand loop of the lemniscate } r^{2}=\cos 2 \theta}\end{array}$$

Find the line integral of \(f(x, y, z)=x+y+z\) over the straight-line segment from \((1,2,3)\) to \((0,-1,1)\) .

Find the line integrals along the given path \(C .\) $$ \int_{C}(x-y) d x, \text { where } C : x=t, y=2 t+1, \text { for } 0 \leq t \leq 3 $$

Show that the differential forms in the integrals are exact. Then evaluate the integrals. \(\int_{(0,0,0)}^{(2,3,-6)} 2 x d x+2 y d y+2 z d z\)

Integrate \(G(x, y, z)=x \sqrt{y^{2}+4}\) over the surface cut from the parabolic cylinder \(y^{2}+4 z=16\) by the planes \(x=0, x=1,\) and \(z=0 .\)

In Exercises \(13-18,\) use the surface integral in Stokes' Theorem to calculate the flux of the curl of the field \(\mathbf{F}\) across the surface \(S\) in the direction of the outward unit normal $$\begin{aligned} \mathbf{F} &=(y-z) \mathbf{i}+(z-x) \mathbf{j}+(x+z) \mathbf{k} \\ S : & \mathbf{r}(r, \theta)=(r \cos \theta) \mathbf{i}+(r \sin \theta) \mathbf{j}+\left(9-r^{2}\right) \mathbf{k}, \\ 0 & \leq r \leq 3, \quad 0 \leq \theta \leq 2 \pi \end{aligned}$$

Find the line integrals along the given path \(C .\) $$ \int_{C} \frac{x}{y} d y, \text { where } C : x=t, y=t^{2}, \text { for } 1 \leq t \leq 2 $$

Show that the differential forms in the integrals are exact. Then evaluate the integrals. \(\int_{(1,1,2)}^{(3,5,0)} y z d x+x z d y+x y d z\)

Find a parametrization of the surface. (There are many correct ways to do these, so your answers may not be the same as those in the back of the book.) Tilted plane inside cylinder The portion of the plane \(x-y+2 z=2\) $$\begin{array}{l}{\text { a. Inside the cylinder } x^{2}+z^{2}=3} \\ {\text { b. Inside the cylinder } y^{2}+z^{2}=2}\end{array}$$

Find a parametrization of the surface. (There are many correct ways to do these, so your answers may not be the same as those in the back of the book.) Circular cylinder band The portion of the cylinder \((x-2)^{2}+\) \(z^{2}=4\) between the planes \(y=0\) and \(y=3\)

In Exercises \(13-18,\) use the surface integral in Stokes' Theorem to calculate the flux of the curl of the field \(\mathbf{F}\) across the surface \(S\) in the direction of the outward unit normal $$\begin{array}{l}{\mathbf{F}=x^{2} y \mathbf{i}+2 y^{3} \mathbf{z} \mathbf{j}+3 z \mathbf{k}} \\ {S : \quad \mathbf{r}(r, \theta)=(r \cos \theta) \mathbf{i}+(r \sin \theta) \mathbf{j}+r \mathbf{k}}, \\ {0 \leq r \leq 1, \quad 0 \leq \theta \leq 2 \pi}\end{array}$$

Find the counterclockwise circulation and outward flux of the field \(\mathbf{F}=x y \mathbf{i}+y^{2} \mathbf{j}\) around and over the boundary of the region enclosed by the curves \(y=x^{2}\) and \(y=x\) in the first quadrant.

Integrate \(G(x, y, z)=z-x\) over the portion of the graph of \(z=x+y^{2}\) above the triangle in the \(x y-\) plane having vertices \((0,\) \(0,0 ),(1,1,0),\) and \((0,1,0) .\) (See accompanying figure.)

Integrate \(f(x, y, z)=x+\sqrt{y}-z^{2}\) over the path from \((0,0,0)\) to \((1,1,1)(\) see accompanying figure) given by $$ \begin{array}{ll}{C_{1} :} & {\mathbf{r}(t)=t \mathbf{i}+t^{2} \mathbf{j}, \quad 0 \leq t \leq 1} \\ {C_{2} :} & {\mathbf{r}(t)=\mathbf{i}+\mathbf{j}+t \mathbf{k}, \quad 0 \leq t \leq 1}\end{array} $$ The paths of integration for Exercises 15 and 16

Find a parametrization of the surface. (There are many correct ways to do these, so your answers may not be the same as those in the back of the book.) Circular cylinder band The portion of the cylinder \(y^{2}+\) \((z-5)^{2}=25\) between the planes \(x=0\) and \(x=10\)

Integrate \(G(x, y, z)=x\) over the surface given by \begin{equation}z=x^{2}+y \text { for } 0 \leq x \leq 1, \quad-1 \leq y \leq 1.\end{equation}

In Exercises \(13-18,\) use the surface integral in Stokes' Theorem to calculate the flux of the curl of the field \(\mathbf{F}\) across the surface \(S\) in the direction of the outward unit normal $$\begin{array}{l}{\mathbf{F}=(x-y) \mathbf{i}+(y-z) \mathbf{j}+(z-x) \mathbf{k}} \\ {S : \quad \mathbf{r}(r, \theta)=(r \cos \theta) \mathbf{i}+(r \sin \theta) \mathbf{j}+(5-r) \mathbf{k}}, \\ {0 \leq r \leq 5, \quad 0 \leq \theta \leq 2 \pi}\end{array}$$

Find the counterclockwise circulation and the outward flux of the field \(\mathbf{F}=(-\sin y) \mathbf{i}+(x \cos y) \mathbf{j}\) around and over the square cut from the first quadrant by the lines \(x=\pi / 2\) and \(y=\pi / 2\) .

Integrate \(f(x, y, z)=x+\sqrt{y}-z^{2}\) over the path from \((0,0,0)\) to \((1,1,1)\) (see accompanying figure) given by $$ \begin{array}{ll}{C_{1} :} & {\mathbf{r}(t)=t \mathbf{k}, \quad 0 \leq t \leq 1} \\ {C_{2} :} & {\mathbf{r}(t)=t \mathbf{j}+\mathbf{k}, \quad 0 \leq t \leq 1} \\ {C_{3} :} & {\mathbf{r}(t)=t \mathbf{i}+\mathbf{j}+\mathbf{k}, \quad 0 \leq t \leq 1}\end{array} $$ The paths of integration for Exercises 15 and 16

Show that the differential forms in the integrals are exact. Then evaluate the integrals. \(\int_{(0,0,0)}^{(3,3,1)} 2 x d x-y^{2} d y-\frac{4}{1+z^{2}} d z\)

Use a parametrization to express the area of the surface as a double integral. Then evaluate the integral. (There are many correct ways to set up the integrals, so your integrals may not be the same as those in the back of the book. They should have the same values, however.) Tilted plane inside cylinder The portion of the plane \(y+2 z=2\) inside the cylinder \(x^{2}+y^{2}=1\)

Integrate \(G(x, y, z)=x y z\) over the triangular surface with vertices (1,0,0),(0,2,0), and (0,1,1).

In Exercises \(13-18,\) use the surface integral in Stokes' Theorem to calculate the flux of the curl of the field \(\mathbf{F}\) across the surface \(S\) in the direction of the outward unit normal $$\begin{array}{l}{\mathbf{F}=3 y \mathbf{i}+(5-2 x) \mathbf{j}+\left(z^{2}-2\right) \mathbf{k}} \\ {S : \quad \mathbf{r}(\phi, \theta)=(\sqrt{3} \sin \phi \cos \theta) \mathbf{i}+(\sqrt{3} \sin \phi \sin \theta) \mathbf{j}+} \\ {(\sqrt{3} \cos \phi) \mathbf{k}, \quad 0 \leq \phi \leq \pi / 2, \quad 0 \leq \theta \leq 2 \pi}\end{array}$$

a. Show that the outward flux of the position vector field \(\mathbf{F}=\) \(x \mathbf{i}+y \mathbf{j}+z \mathbf{k}\) through a smooth closed surface \(S\) is three times the volume of the region enclosed by the surface. b. Let n be the outward unit normal vector field on \(S .\) Show that it is not possible for \(\mathbf{F}\) be orthogonal to \(\mathbf{n}\) at every point of \(S\) .

Along the curve \(\mathbf{r}(t)=t \mathbf{i}-\mathbf{j}+t^{2} \mathbf{k}, 0 \leq t \leq 1,\) evaluate each of the following integrals. $$ \text { a. } \int_{C}(x+y-z) d x \quad \text { b. } \int_{C}(x+y-z) d y \\\ \text { c. } \int_{C}(x+y-z) d z $$

\(\begin{array}{l}{\text { Integrate } f(x, y, z)=(x+y+z) /\left(x^{2}+y^{2}+z^{2}\right) \text { over the path }} \\ {\mathbf{r}(t)=t \mathbf{i}+t \mathbf{j}+t \mathbf{k}, 0 < a \leq t \leq b}\end{array}\)