Chapter 15

Use the Divergence Theorem to find the flux of F across the surface ? with outward orientation. $$ \begin{array}{l}{\mathbf{F}(x, y, z)=\left(x^{2}+y\right) \mathbf{i}+z^{2} \mathbf{j}+\left(e^{y}-z\right) \mathbf{k} ; \sigma \text { is the surface }} \\\ {\text { of the rectangular solid bounded by the coordinate planes }} \\\ {\text { and the planes } x=3, y=1, \text { and } z=2}\end{array} $$

Use a graphing utility to generate a plot of the vector field. $$ \mathbf{F}(x, y)=\mathbf{i}+\cos y \mathbf{j} $$

In each part, evaluate the integral $$ \int_{C}(3 x+2 y) d x+(2 x-y) d y $$ along the stated curve. (a) The line segment from \((0,0)\) to \((1,1) .\) (b) The parabolic arc \(y=x^{2}\) from \((0,0)\) to \((1,1) .\) (c) The curve \(y=\sin (\pi x / 2)\) from \((0,0 \text { to }(1,1)\). (d) The curve \(x=y^{3}\) from \((0,0)\) to \((1,1) .\)

Find the flux of the vector field \(\mathbf{F}\) across \(\sigma\) \(\mathbf{F}(x, y, z)=y \mathbf{j}+\mathbf{k} ; \sigma\) is the portion of the paraboloid \(z=x^{2}+y^{2}\) below the plane \(z=4,\) oriented by downward unit normals.

Determine whether the statement is true or false. Explain your answer. If \(\sigma\) has surface area \(S,\) and if $$ \iint_{\sigma} f(x, y, z) d S=S $$ then \(f(x, y, z)\) is equal to 1 identically on \(\sigma .\)

Use Stokes' Theorem to evaluate \(\oint_{C} \mathbf{F} \cdot d \mathbf{r}\) $$ \begin{array}{l}{\mathbf{F}(x, y, z)=x y \mathbf{i}+y z \mathbf{j}+z x \mathbf{k} ; C \text { is the triangle in the plane }} \\ {x+y+z=1 \text { with vertices }(1,0,0),(0,1,0), \text { and }(0,0,1)} \\ {\text { with a counterclockwise orientation looking from the first }} \\ {\text { octant toward the origin. }}\end{array} $$

Show that the integral is independent of the path, and use Theorem 15.3.1 to find its value. $$ \int_{(0,0)}^{(1, \pi / 2)} e^{x} \sin y d x+e^{x} \cos y d y $$

Use Green’s Theorem to evaluate the integral. In each exercise, assume that the curve C is oriented counterclockwise. \(\oint_{C} x^{2} y d x-y^{2} x d y,\) where \(C\) is the boundary of the region in the first quadrant, enclosed between the coordinate axes and the circle \(x^{2}+y^{2}=16\)

Use the Divergence Theorem to find the flux of F across the surface ? with outward orientation. $$ \begin{array}{l}{\mathbf{F}(x, y, z)=z^{3} \mathbf{i}-x^{3} \mathbf{j}+y^{3} \mathbf{k}, \quad \text { where } \sigma \text { is the sphere }} \\\ {x^{2}+y^{2}+z^{2}=a^{2}}\end{array} $$

Use a graphing utility to generate a plot of the vector field. $$ \mathbf{F}(x, y)=y \mathbf{i}-x \mathbf{j} $$

In each part, evaluate the integral $$ \int_{C} y d x+z d y-x d z $$ along the stated curve. (a) The line segment from \((0,0,0)\) to \((1,1,1) .\) (b) The twisted cubic \(x=t, y=t^{2}, z=t^{3}\) from \((0,0,0)\) to \((1,1,1) .\) (c) The helix \(x=\cos \pi t, y=\sin \pi t, z=t\) from \((1,0,0)\) to \((-1,0,1)\)

Find the flux of the vector field \(\mathbf{F}\) across \(\sigma\) \(\mathbf{F}(x, y, z)=x \mathbf{k} ;\) the surface \(\sigma\) is the portion of the paraboloid \(z=x^{2}+y^{2}\) below the plane \(z=y,\) oriented by downward unit normals.

Use Green’s Theorem to evaluate the integral. In each exercise, assume that the curve C is oriented counterclockwise. \(\oint_{C} \tan ^{-1} y d x-\frac{y^{2} x}{1+y^{2}} d y,\) where \(C\) is the square with vertices \((0,0),(1,0),(1,1),\) and \((0,1)\)

Use the Divergence Theorem to find the flux of F across the surface ? with outward orientation. $$ \begin{array}{l}{\mathbf{F}(x, y, z)=(x-z) \mathbf{i}+(y-x) \mathbf{j}+(z-y) \mathbf{k} ; \sigma \text { is the sur- }} \\ {\text { face of the cylindrical solid bounded by } x^{2}+y^{2}=a^{2}} \\ {z=0, \text { and } z=1}\end{array} $$

Determine whether the statement is true or false. Explain your answer. The vector-valued function $$ \mathbf{F}(x, y)=y \mathbf{i}+x^{2} \mathbf{j}+x y \mathbf{k} $$ is an example of a vector field in the \(x y\) -plane.

Find the flux of the vector field \(\mathbf{F}\) across \(\sigma\) \(\mathbf{F}(x, y, z)=x^{2} \mathbf{i}+y x \mathbf{j}+z x \mathbf{k} ; \sigma\) is the portion of the plane \(6 x+3 y+2 z=6\) in the first octant, oriented by unit normals with positive components.

Determine whether the statement is true or false. Explain your answer. If \(\sigma\) is the portion of a plane \(z=c\) over a region \(R\) in the \(x y\) -plane, then $$ \iint_{\sigma} f(x, y, z) d S=\iint_{R} f(x, y, c) d A $$ for every continuous function \(f\) on \(\sigma .\)

Use Stokes' Theorem to evaluate \(\oint_{C} \mathbf{F} \cdot d \mathbf{r}\) $$ \begin{array}{l}{\mathbf{F}(x, y, z)=(z+\sin x) \mathbf{i}+\left(x+y^{2}\right) \mathbf{j}+\left(y+e^{z}\right) \mathbf{k} ; \quad C \text { is }} \\ {\text { the intersection of the sphere } x^{2}+y^{2}+z^{2}=1 \text { and the cone }} \\ {z=\sqrt{x^{2}+y^{2}} \text { with counterclockwise orientation looking }} \\ {\text { down the positive } z \text { -axis. }}\end{array} $$

Use Green’s Theorem to evaluate the integral. In each exercise, assume that the curve C is oriented counterclockwise. \(\oint_{C} \cos x \sin y d x+\sin x \cos y d y,\) where \(C\) is the triangle with vertices \((0,0),(3,3),\) and \((0,3)\)

Show that the integral is independent of the path, and use Theorem 15.3.1 to find its value. $$ \int_{(-1,2)}^{(0,1)}(3 x-y+1) d x-(x+4 y+2) d y $$

Use the Divergence Theorem to find the flux of F across the surface ? with outward orientation. $$ \begin{array}{l}{\mathbf{F}(x, y, z)=x \mathbf{i}+y \mathbf{j}+z \mathbf{k} ; \sigma \text { is the surface of the solid }} \\ {\text { bounded by the paraboloid } z=1-x^{2}-y^{2} \text { and the } x y-} \\ {\text { plane. }}\end{array} $$

Find the flux of the vector field \(\mathbf{F}\) across \(\sigma\) in the direction of positive orientation. \(\mathbf{F}(x, y, z)=x \mathbf{i}+y \mathbf{j}+\mathbf{k} ; \sigma\) is the portion of the paraboloid $$ \mathbf{r}(u, v)=u \cos v \mathbf{i}+u \sin v \mathbf{j}+\left(1-u^{2}\right) \mathbf{k} $$ with \(1 \leq u \leq 2,0 \leq v \leq 2 \pi\)

Determine whether the statement is true or false. Explain your answer. $$ \begin{array}{l}{\text { Stokes' Theorem equates a line integral and a surface inte- }} \\ {\text { gral. }}\end{array} $$

Use Green’s Theorem to evaluate the integral. In each exercise, assume that the curve C is oriented counterclockwise. \(\oint_{C} x^{2} y d x+\left(y+x y^{2}\right) d y,\) where \(C\) is the boundary of the region enclosed by \(y=x^{2}\) and \(x=y^{2}\)

Use the Divergence Theorem to find the flux of F across the surface ? with outward orientation. $$ \begin{array}{l}{\mathbf{F}(x, y, z)=x^{3} \mathbf{i}+y^{3} \mathbf{j}+z^{3} \mathbf{k} ; \sigma \text { is the surface of the cylin- }} \\ {\text { drical solid bounded by } x^{2}+y^{2}=4, z=0, \text { and } z=3}\end{array} $$

Find the flux of the vector field \(\mathbf{F}\) across \(\sigma\) in the direction of positive orientation. \(\mathbf{F}(x, y, z)=e^{-y} \mathbf{i}-y \mathbf{j}+x \sin z \mathbf{k} ; \sigma\) is the portion of the elliptic cylinder $$ \mathbf{r}(u, v)=2 \cos v \mathbf{i}+\sin v \mathbf{j}+u \mathbf{k} $$ with \(0 \leq u \leq 5,0 \leq v \leq 2 \pi\)

Let \(C\) be the boundary of the region enclosed between \(y=x^{2}\) and \(y=2 x\). Assuming that \(C\) is oriented counterclockwise, evaluate the following integrals by Green's Theorem: $$ \text { (a) } \oint_{C}\left(6 x y-y^{2}\right) d x \quad \text { (b) } \oint_{C}\left(6 x y-y^{2}\right) d y $$

Determine whether the statement is true or false. Explain your answer. If a smooth oriented curve \(C\) in the \(x y\) -plane is a contour for a differentiable function \(f(x, y),\) then $$ \int_{C} \nabla f \cdot d \mathbf{r}=0 $$

Find the flux of the vector field \(\mathbf{F}\) across \(\sigma\) in the direction of positive orientation. \(\mathbf{F}(x, y, z)=\sqrt{x^{2}+y^{2}} \mathbf{k} ; \sigma\) is the portion of the cone $$ \mathbf{r}(u, v)=u \cos v \mathbf{i}+u \sin v \mathbf{j}+2 u \mathbf{k} $$ with \(0 \leq u \leq \sin v, 0 \leq v \leq \pi\)

Determine whether the statement is true or false. Explain your answer. (In Exercises 16–18, assume that C is a simple, smooth, closed curve, oriented counterclockwise.) Green's Theorem allows us to replace any line integral by a double integral.

Confirm that the force field \(\mathbf{F}\) is conservative in some open connected region containing the points \(P\) and \(Q,\) and then find the work done by the force field on a particle moving along an arbitrary smooth curve in the region from \(P\) to \(Q .\) $$ \mathbf{F}(x, y)=x y^{2} \mathbf{i}+x^{2} y \mathbf{j} ; P(1,1), Q(0,0) $$

Use the Divergence Theorem to find the flux of F across the surface ? with outward orientation. $$ \begin{array}{l}{\mathbf{F}(x, y, z)=\left(x^{3}-e^{y}\right) \mathbf{i}+\left(y^{3}+\sin z\right) \mathbf{j}+\left(z^{3}-x y\right) \mathbf{k}} \\ {\text { where } \sigma \text { is the surface of the solid bounded above by }} \\ {z=\sqrt{4-x^{2}-y^{2}} \text { and below by the } x y \text { -plane. [Hint: Use }} \\ { \text { spherical coordinates. } ]}\end{array} $$

Confirm that \(\phi\) is a potential function for \(\mathbf{F}(\mathbf{r})\) on some region, and state the region. $$ \begin{array}{l}{\text { (a) } \phi(x, y)=\tan ^{-1} x y} \\ {\quad \mathbf{F}(x, y)=\frac{y}{1+x^{2} y^{2}} \mathbf{i}+\frac{x}{1+x^{2} y^{2}} \mathbf{j}} \\ {\text { (b) } \quad \phi(x, y, z)=x^{2}-3 y^{2}+4 z^{2}} \\\ {\quad \mathbf{F}(x, y, z)=2 x \mathbf{i}-6 y \mathbf{j}+8 z \mathbf{k}}\end{array} $$

Evaluate the line integral with respect to \(s\) along the curve \(C .\) $$ \begin{array}{l}{\int_{C} \frac{1}{1+x} d s} \\ {C: \mathbf{r}(t)=t \mathbf{i}+\frac{2}{3} t^{3 / 2} \mathbf{j} \quad(0 \leq t \leq 3)}\end{array} $$

Find the flux of the vector field \(\mathbf{F}\) across \(\sigma\) in the direction of positive orientation. \(\mathbf{F}(x, y, z)=x \mathbf{i}+y \mathbf{j}+z \mathbf{k} ; \sigma\) is the portion of the sphere $$ \mathbf{r}(u, v)=2 \sin u \cos v \mathbf{i}+2 \sin u \sin v \mathbf{j}+2 \cos u \mathbf{k} $$ with \(0 \leq u \leq \pi / 3,0 \leq v \leq 2 \pi\)

Confirm that the force field \(\mathbf{F}\) is conservative in some open connected region containing the points \(P\) and \(Q,\) and then find the work done by the force field on a particle moving along an arbitrary smooth curve in the region from \(P\) to \(Q .\) $$ \mathbf{F}(x, y)=2 x y^{3} \mathbf{i}+3 x^{2} y^{2} \mathbf{j} ; P(-3,0), Q(4,1) $$

Determine whether the statement is true or false. Explain your answer. (In Exercises 16–18, assume that C is a simple, smooth, closed curve, oriented counterclockwise.) If $$ \int_{C} f(x, y) d x+g(x, y) d y=0 $$ then \(\partial g / \partial x=\partial f / \partial y\) at all points in the region bounded by \(C .\)

Use the Divergence Theorem to find the flux of F across the surface ? with outward orientation. $$ \begin{array}{l}{\mathbf{F}(x, y, z)=2 x z \mathbf{i}+y z \mathbf{j}+z^{2} \mathbf{k}, \quad \text { where } \sigma \text { is the surface }} \\ {\text { of the solid bounded above by } z=\sqrt{a^{2}-x^{2}-y^{2}} \text { and be- }} \\ {\text { low by the } x y \text { -plane. }}\end{array} $$

Confirm that \(\phi\) is a potential function for \(\mathbf{F}(\mathbf{r})\) on some region, and state the region. $$ \begin{array}{l}{\text { (a) } \phi(x, y)=2 y^{2}+3 x^{2} y-x y^{3}} \\\ {\quad \mathbf{F}(x, y)=\left(6 x y-y^{3}\right) \mathbf{i}+\left(4 y+3 x^{2}-3 x y^{2}\right) \mathbf{j}}\end{array} $$ $$ \begin{aligned} \text { (b) } \phi(x, y, z)=x \sin z+y \sin x+z \sin y & \\\ \mathbf{F}(x, y, z)=(\sin z+y \cos x) \mathbf{i}+(\sin x+z \cos y) \mathbf{j} \\\\+(\sin y+x \cos z) \mathbf{k} \end{aligned} $$

Evaluate the line integral with respect to \(s\) along the curve \(C .\) $$ \begin{array}{l}{\int_{C} \frac{x}{1+y^{2}} d s} \\ {C: x=1+2 t, \quad y=t \quad(0 \leq t \leq 1)}\end{array} $$

Let \(\sigma\) be the surface of the cube bounded by the planes \(x=\pm 1, y=\pm 1, z=\pm 1,\) oriented by outward unit normals. In each part, find the flux of \(\mathbf{F}\) across \(\sigma .\) $$ \begin{array}{l}{\text { (a) } \mathbf{F}(x, y, z)=x \mathbf{i}} \\ {\text { (b) } \mathbf{F}(x, y, z)=x \mathbf{i}+y \mathbf{j}+z \mathbf{k}} \\ {\text { (c) } \mathbf{F}(x, y, z)=x^{2} \mathbf{i}+y^{2} \mathbf{j}+z^{2} \mathbf{k}}\end{array} $$

Consider the vector field given by the formula $$ \mathbf{F}(x, y, z)=(x-z) \mathbf{i}+(y-x) \mathbf{j}+(z-x y) \mathbf{k} $$ (a) Use Stokes' Theorem to find the circulation around the triangle with vertices \(A(1,0,0), B(0,2,0),\) and \(C(0,0,1)\) oriented counterclockwise looking from the origin toward the first octant. (b) Find the circulation density of \(\mathbf{F}\) at the origin in the direction of \(\mathbf{k}\). (c) Find the unit vector \(\mathbf{n}\) such that the circulation density of \(\mathbf{F}\) at the origin is maximum in the direction of \(\mathbf{n} .\)

Confirm that the force field \(\mathbf{F}\) is conservative in some open connected region containing the points \(P\) and \(Q,\) and then find the work done by the force field on a particle moving along an arbitrary smooth curve in the region from \(P\) to \(Q .\) $$ \mathbf{F}(x, y)=y e^{x y} \mathbf{i}+x e^{x y} \mathbf{j} ; P(-1,1), Q(2,0) $$

Use the Divergence Theorem to find the flux of F across the surface ? with outward orientation. $$ \begin{array}{l}{\mathbf{F}(x, y, z)=x^{2} \mathbf{i}+y^{2} \mathbf{j}+z^{2} \mathbf{k} ; \sigma \text { is the surface of the coni- }} \\ {\text { cal solid bounded by } z=\sqrt{x^{2}+y^{2}} \text { and } z=1}\end{array} $$

Find div F and curl F. $$ \mathbf{F}(x, y, z)=x z^{3} \mathbf{i}+2 y^{4} x^{2} \mathbf{j}+5 z^{2} y \mathbf{k} $$ $$ \mathbf{F}(x, y, z)=x^{2} \mathbf{i}-2 \mathbf{j}+y z \mathbf{k} $$

Evaluate the line integral with respect to \(s\) along the curve \(C .\) $$ \begin{array}{l}{\int_{C} 3 x^{2} y z d s} \\ {C: x=t, y=t^{2}, z=\frac{2}{3} t^{3} \quad(0 \leq t \leq 1)}\end{array} $$

Let \(\sigma\) be the closed surface consisting of the portion of the paraboloid \(z=x^{2}+y^{2}\) for which \(0 \leq z \leq 1\) and capped by the disk \(x^{2}+y^{2} \leq 1\) in the plane \(z=1 .\) Find the flux of the vector field \(\mathbf{F}(x, y, z)=z \mathbf{j}-y \mathbf{k}\) in the outward direction across \(\sigma .\)

(a) Let \(\sigma\) denote the surface of a solid \(G\) with \(\mathbf{n}\) the outward unit normal vector field to \(\sigma\). Assume that \(\mathbf{F}\) is a vector field with continuous first-order partial derivatives on \(\sigma .\) Prove that $$ \iint_{\sigma}(\operatorname{curl} \mathbf{F}) \cdot \mathbf{n} d S=0 $$ [Hint: Let \(C\) denote a simple closed curve on \(\sigma\) that separates the surface into two subsurfaces \(\sigma_{1}\) and \(\sigma_{2}\) that share \(C\) as their common boundary. Apply Stokes 'Theorem to \(\sigma_{1}\) and to \(\sigma_{2}\) and add the results. (b) The vector field curl(F) is called the curl field of F. In words, interpret the formula in part (a) as a statement about the flux of the curl field.

Confirm that the force field \(\mathbf{F}\) is conservative in some open connected region containing the points \(P\) and \(Q,\) and then find the work done by the force field on a particle moving along an arbitrary smooth curve in the region from \(P\) to \(Q .\) $$ \mathbf{F}(x, y)=e^{-y} \cos x \mathbf{i}-e^{-y} \sin x \mathbf{j} ; \quad P(\pi / 2,1), Q(-\pi / 2,0) $$

Determine whether the statement is true or false. Explain your answer. (In Exercises 16–18, assume that C is a simple, smooth, closed curve, oriented counterclockwise.) It must be the case that $$ \int_{C} e^{x^{2}} d x+\sin y^{3} d y=0 $$