Chapter 15

Evaluate the surface integral $$ \iint_{\sigma} f(x, y, z) d S $$ \(f(x, y, z)=z^{2} ; \sigma\) is the portion of the cone \(z=\sqrt{x^{2}+y^{2}}\) between the planes \(z=1\) and \(z=2\)

Verify Formula (2) in Stokes’ Theorem by evaluating the line integral and the surface integral. Assume that the surface has an upward orientation.. $$ \begin{array}{l}{\mathbf{F}(x, y, z)=(x-y) \mathbf{i}+(y-z) \mathbf{j}+(z-x) \mathbf{k} ; \sigma \text { is the por- }} \\ {\text { tion of the plane } x+y+z=1 \text { in the first octant. }}\end{array} $$

Determine whether \(\mathbf{F}\) is a conservative vector field. If so, find a potential function for it. $$ \mathbf{F}(x, y)=x \mathbf{i}+y \mathbf{j} $$

Verify Formula (1) in the Divergence Theorem by evaluating the surface integral and the triple integral. $$ \begin{array}{l}{\mathbf{F}(x, y, z)=x \mathbf{i}+y \mathbf{j}+z \mathbf{k} ; \sigma \text { is the surface of the cube }} \\ {\text { bounded by the planes } x=0, x=1, y=0, y=1, z=0} \\ {z=1}\end{array} $$

Evaluate the line integral using Green’s Theorem and check the answer by evaluating it directly. \(\oint_{C} y^{2} d x+x^{2} d y,\) where \(C\) is the square with vertices \((0,0)\) \((1,0),(1,1),\) and \((0,1)\) oriented counterclockwise.

Let \(C\) be the line segment from \((0,0)\) to \((0,1) .\) In each part, evaluate the line integral along \(C\) by inspection, and explain your reasoning. $$ \begin{array}{lll}{\text { (a) } \int_{C} d s} & {\text { (b) } \int_{C} \sin x y d y} & {}\end{array} $$

Evaluate the surface integral $$ \iint_{\sigma} f(x, y, z) d S $$ \(f(x, y, z)=x y ; \sigma\) is the portion of the plane \(x+y+z=1\) lying in the first octant.

Determine whether \(\mathbf{F}\) is a conservative vector field. If so, find a potential function for it. $$ \mathbf{F}(x, y)=3 y^{2} \mathbf{i}+6 x y \mathbf{j} $$

Verify Formula (1) in the Divergence Theorem by evaluating the surface integral and the triple integral. $$ \begin{array}{l}{\mathbf{F}(x, y, z)=x \mathbf{i}+y \mathbf{j}+z \mathbf{k} ; \sigma \text { is the spherical surface }} \\\ {x^{2}+y^{2}+z^{2}=1}\end{array} $$

Evaluate the line integral using Green’s Theorem and check the answer by evaluating it directly. \(\oint_{C} y d x+x d y,\) where \(C\) is the unit circle oriented counterclockwise.

Evaluate the surface integral $$ \iint_{\sigma} f(x, y, z) d S $$ \(f(x, y, z)=x^{2} y ; \sigma\) is the portion of the cylinder \(x^{2}+z^{2}=1\) between the planes \(y=0, y=1,\) and above the \(x y\) -plane.

Determine whether \(\mathbf{F}\) is a conservative vector field. If so, find a potential function for it. $$ \mathbf{F}(x, y)=x^{2} y \mathbf{i}+5 x y^{2} \mathbf{j} $$

Verify Formula (1) in the Divergence Theorem by evaluating the surface integral and the triple integral. $$ \begin{array}{l}{\mathbf{F}(x, y, z)=2 x \mathbf{i}-y z \mathbf{j}+z^{2} \mathbf{k} ; \text { the surface } \sigma \text { is the parab- }} \\ {\text { oloid } z=x^{2}+y^{2} \text { capped by the disk } x^{2}+y^{2} \leq 1 \text { in the }} \\ {\text { plane } z=1}\end{array} $$

Use Green’s Theorem to evaluate the integral. In each exercise, assume that the curve C is oriented counterclockwise. \(\oint_{C} 3 x y d x+2 x y d y,\) where \(C\) is the rectangle bounded by \(x=-2, x=4, y=1,\) and \(y=2\)

Determine whether the statement about the vector field \(\mathbf{F}(x, y)\) is true or false. If false, explain why. $$ \mathbf{F}(x, y)=x^{2} \mathbf{i}-y \mathbf{j} $$ $$ \begin{array}{l}{\text { (a) }\|\mathbf{F}(x, y)\| \rightarrow 0 \text { as }(x, y) \rightarrow(0,0)} \\ {\text { (b) If }(x, y) \text { is on the positive } y \text { -axis, then the vector }} \\ {\text { points in the negative } y \text { -direction. }} \\ {\text { (c) If }(x, y) \text { is in the first quadrant, then the vector points }} \\ {\text { down and to the right. }}\end{array} $$

Evaluate the surface integral $$ \iint_{\sigma} f(x, y, z) d S $$ \(f(x, y, z)=\left(x^{2}+y^{2}\right) z ; \sigma\) is the portion of the sphere \(x^{2}+y^{2}+z^{2}=4\) above the plane \(z=1\)

Verify Formula (2) in Stokes’ Theorem by evaluating the line integral and the surface integral. Assume that the surface has an upward orientation.. $$ \begin{array}{l}{\mathbf{F}(x, y, z)=(z-y) \mathbf{i}+(z+x) \mathbf{j}-(x+y) \mathbf{k} ; \sigma \text { is the por- }} \\ {\text { tion of the paraboloid } z=9-x^{2}-y^{2} \text { above the } x y \text { -plane. }}\end{array} $$

Determine whether \(\mathbf{F}\) is a conservative vector field. If so, find a potential function for it. $$ \mathbf{F}(x, y)=e^{x} \cos y \mathbf{i}-e^{x} \sin y \mathbf{j} $$

Verify Formula (1) in the Divergence Theorem by evaluating the surface integral and the triple integral. $$ \begin{array}{l}{\mathbf{F}(x, y, z)=x y \mathbf{i}+y z \mathbf{j}+x z \mathbf{k} ; \sigma \text { is the surface of the cube }} \\ {\text { bounded by the planes } x=0, x=2, y=0, y=2, z=0} \\ {z=2}\end{array} $$

Use Green’s Theorem to evaluate the integral. In each exercise, assume that the curve C is oriented counterclockwise. \(\oint_{C}\left(x^{2}-y^{2}\right) d x+x d y,\) where \(C\) is the circle \(x^{2}+y^{2}=9\)

Determine whether the statement about the vector field \(\mathbf{F}(x, y)\) is true or false. If false, explain why. $$ \mathbf{F}(x, y)=\frac{x}{\sqrt{x^{2}+y^{2}}} \mathbf{i}-\frac{y}{\sqrt{x^{2}+y^{2}}} \mathbf{j} $$ $$ \begin{array}{l}{\text { (a) As }(x, y) \text { moves away from the origin, the lengths of }} \\ {\text { the vectors decrease. }} \\ {\text { (b) If }(x, y) \text { is a point on the positive } x \text { -axis, then the }} \\\ {\text { vector points up. }} \\ {\text { (c) If }(x, y) \text { is a point on the positive } y \text { -axis, the vector }} \\ {\text { points to the right. }}\end{array} $$

Let \(\sigma\) be the cylindrical surface that is represented by the vector-valued function \(\mathbf{r}(u, v)=\cos v \mathbf{i}+\sin v \mathbf{j}+\) \(u \mathbf{k}\) with \(0 \leq u \leq 1\) and \(0 \leq v \leq 2 \pi\) (a) Find the unit normal \(\mathbf{n}=\mathbf{n}(u, v)\) that defines the positive orientation of \(\sigma .\) (b) Is the positive orientation inward or outward? Justify your answer.

Evaluate the surface integral $$ \iint_{\sigma} f(x, y, z) d S $$ \(f(x, y, z)=x-y-z ; \sigma\) is the portion of the plane \(x+y=1\) in the first octant between \(z=0\) and \(z=1\)

Use Stokes' Theorem to evaluate \(\oint_{C} \mathbf{F} \cdot d \mathbf{r}\) $$ \begin{array}{l}{\mathbf{F}(x, y, z)=z^{2} \mathbf{i}+2 x \mathbf{j}-y^{3} \mathbf{k} ; C \text { is the circle } x^{2}+y^{2}=1} \\ {\text { in the } x y \text { -plane with counterclockwise orientation looking }} \\ {\text { down the positive } z \text { -axis. }}\end{array} $$

Determine whether \(\mathbf{F}\) is a conservative vector field. If so, find a potential function for it. $$ \mathbf{F}(x, y)=(\cos y+y \cos x) \mathbf{i}+(\sin x-x \sin y) \mathbf{j} $$

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

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

Sketch the vector field by drawing some representative non intersecting vectors. The vectors need not be drawn to scale, but they should be in reasonably correct proportion relative to each other. $$ \mathbf{F}(x, y)=2 \mathbf{i}-\mathbf{j} $$

Let \(\sigma\) be the conical surface that is represented by the parametric equations \(x=r \cos \theta, y=r \sin \theta, z=r\) with \(1 \leq r \leq 2\) and \(0 \leq \theta \leq 2 \pi\) (a) Find the unit normal \(\mathbf{n}=\mathbf{n}(r, \theta)\) that defines the positive orientation of \(\sigma .\) (b) Is the positive orientation upward or downward? Justify your answer.

Evaluate the surface integral $$ \iint_{\sigma} f(x, y, z) d S $$ \(f(x, y, z)=x+y ; \sigma\) is the portion of the plane \(z=6-2 x-3 y\) in the first octant.

Determine whether \(\mathbf{F}\) is a conservative vector field. If so, find a potential function for it. $$ \mathbf{F}(x, y)=x \ln y \mathbf{i}+y \ln x \mathbf{j} $$

Determine whether the statement is true or false. Explain your answer. $$ \begin{array}{l}{\text { If } G \text { is a solid whose surface } \sigma \text { is oriented outward, and if }} \\ {\text { div } \mathbf{F}>0 \text { at all points of } G, \text { then the flux of } \mathbf{F} \text { across } \sigma \text { is }} \\ {\text { positive. }}\end{array} $$

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

Sketch the vector field by drawing some representative non intersecting vectors. The vectors need not be drawn to scale, but they should be in reasonably correct proportion relative to each other. $$ \mathbf{F}(x, y)=y \mathbf{j}, \quad y>0 $$

Find the flux of the vector field \(\mathbf{F}\) across \(\sigma\) \(\mathbf{F}(x, y, z)=x \mathbf{i}+y \mathbf{j}+2 z \mathbf{k} ; \sigma\) is the portion of the surface \(z=1-x^{2}-y^{2}\) above the \(x y-\) plane, oriented by upward normals.

Evaluate the surface integral $$ \iint_{\sigma} f(x, y, z) d S $$ \(f(x, y, z)=x+y+z ; \sigma\) is the surface of the cube defined by the inequalities \(0 \leq x \leq 1,0 \leq y \leq 1,0 \leq z \leq 1\) [Hint: Integrate over each face separately.]

Use Green’s Theorem to evaluate the integral. In each exercise, assume that the curve C is oriented counterclockwise. \(\oint_{C}\left(x^{2}-y\right) d x+x d y,\) where \(C\) is the circle \(x^{2}+y^{2}=4\)

Determine whether the statement is true or false. Explain your answer. $$ \begin{array}{l}{\text { The continuity equation for incompressible fluids states that }} \\ {\text { the divergence of the velocity vector field of the fluid is zero. }}\end{array} $$

Sketch the vector field by drawing some representative non intersecting vectors. The vectors need not be drawn to scale, but they should be in reasonably correct proportion relative to each other. $$ \begin{array}{l}{\mathbf{F}(x, y)=y \mathbf{i}-x \mathbf{j} . \text { [Note: Each vector in the field is per- }} \\ { \text { pendicular to the position vector }\mathbf{r}=x \mathbf{i}+y \mathbf{j} \cdot]}\end{array} $$

Let \(C\) be the curve represented by the equations $$ x=2 t, \quad y=3 t^{2} \quad(0 \leq t \leq 1) $$ In each part, evaluate the line integral along \(C .\) \(\begin{array}{ll}{\text { (a) } \int_{C}(x-y) d s} & {\text { (b) } \int_{C}(x-y) d x} \\ {\text { (c) } \int_{C}(x-y) d y}\end{array}\)

Find the flux of the vector field \(\mathbf{F}\) across \(\sigma\) \(\mathbf{F}(x, y, z)=(x+y) \mathbf{i}+(y+z) \mathbf{j}+(z+x) \mathbf{k} ; \sigma\) is the portion of the plane \(x+y+z=1\) in the first octant, oriented by unit normals with positive components.

Evaluate the surface integral $$ \iint_{\sigma} f(x, y, z) d S $$ \(f(x, y, z)=x^{2}+y^{2} ; \sigma\) is the surface of the sphere \(x^{2}+y^{2}+z^{2}=a^{2}\)

Use Stokes' Theorem to evaluate \(\oint_{C} \mathbf{F} \cdot d \mathbf{r}\) $$ \begin{array}{l}{\mathbf{F}(x, y, z)=-3 y^{2} \mathbf{i}+4 z \mathbf{j}+6 x \mathbf{k} ; C \text { is the triangle in the }} \\ {\text { plane } z=\frac{1}{2} y \text { with vertices }(2,0,0),(0,2,1), \text { and }(0,0,0)} \\\ {\text { with a counterclockwise orientation looking down the pos- }} \\\ {\text { itive } z \text { -axis. }}\end{array} $$

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

(a) Show that the line integral \(\int_{C} y \sin x d x-\cos x d y\) is independent of the path. (b) Evaluate the integral in part (a) along the line segment from \((0,1)\) to \((\pi,-1) .\) (c) Evaluate the integral \(\int_{(0,1)}^{(\pi,-1)} y \sin x d x-\cos x d y\) using Theorem \(15.3 .1,\) and confirm that the value is the same as that obtained in part (b).

Sketch the vector field by drawing some representative non intersecting vectors. The vectors need not be drawn to scale, but they should be in reasonably correct proportion relative to each other. $$ \begin{array}{l}{\mathbf{F}(x, y)=\frac{x \mathbf{i}+y \mathbf{j}}{\sqrt{x^{2}+y^{2}}} \text { . [Note: Each vector in the field is }} \\ {\text { a unit vector in the same direction as the position vector }} \\\ {\mathbf{r}=x \mathbf{i}+y \mathbf{j} .]}\end{array} $$

Let \(C\) be the curve represented by the equations $$ x=t, \quad y=3 t^{2}, \quad z=6 t^{3} \quad(0 \leq t \leq 1) $$ In each part, evaluate the line integral along \(C .\) $$ \begin{array}{ll}{\text { (a) } \int_{C} x y z^{2} d s} & {\text { (b) } \int_{C} x y z^{2} d x} \\ {\text { (c) } \int_{C} x y z^{2} d y} & {\text { (d) } \int_{C} x y z^{2} d z}\end{array} $$

Find the flux of the vector field \(\mathbf{F}\) across \(\sigma\) \(\mathbf{F}(x, y, z)=x \mathbf{i}+y \mathbf{j}+2 z \mathbf{k} ; \sigma\) is the portion of the cone \(z^{2}=x^{2}+y^{2}\) between the planes \(z=1\) and \(z=2,\) oriented by upward unit normals.

Use Stokes' Theorem to evaluate \(\oint_{C} \mathbf{F} \cdot d \mathbf{r}\) $$ \begin{array}{l}{\mathbf{F}(x, y, z)=x y \mathbf{i}+x^{2} \mathbf{j}+z^{2} \mathbf{k} ; C \text { is the intersection of the }} \\ {\text { paraboloid } z=x^{2}+y^{2} \text { and the plane } z=y \text { with a counter- }} \\\ {\text { clockwise orientation looking down the positive } z \text { -axis. }}\end{array} $$

Use Green’s Theorem to evaluate the integral. In each exercise, assume that the curve C is oriented counterclockwise. \(\oint_{C} \ln (1+y) d x-\frac{x y}{1+y} d y,\) where \(C\) is the triangle with vertices \((0,0),(2,0),\) and \((0,4)\)