Chapter 16

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

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.) Plane inside cylinder The portion of the plane \(z=-x\) inside the cylinder \(x^{2}+y^{2}=4\)

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 $$\begin{aligned} \mathbf{F} &=y^{2} \mathbf{i}+z^{2} \mathbf{j}+x \mathbf{k} \\ S : & \mathbf{r}(\phi, \theta)=(2 \sin \phi \cos \theta) \mathbf{i}+(2 \sin \phi \sin \theta) \mathbf{j}+(2 \cos \phi) \mathbf{k}, \\ 0 & \leq \phi \leq \pi / 2, \quad 0 \leq \theta \leq 2 \pi \end{aligned}$$

Find the counterclockwise circulation of \(\mathbf{F}=\left(y+e^{x} \ln y\right) \mathbf{i}+\) \(\left(e^{x} / y\right) \mathbf{j}\) around the boundary of the region that is bounded above by the curve \(y=3-x^{2}\) and below by the curve \(y=x^{4}+1\)

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.) Cone frustum The portion of the cone \(z=2 \sqrt{x^{2}+y^{2}}\) between the planes \(z=2\) and \(z=6\)

Use a parametrization to find the flux \(\iint_{S} \mathbf{F} \cdot \mathbf{n} d \sigma\) a cross the surface in the specified direction. \(\begin{array}{l}{\text { Parabolic cylinder } \mathbf{F}=z^{2} \mathbf{i}+x \mathbf{j}-3 z \mathbf{k} \text { outward (normal }} \\ {\text { away from the } x \text { -axis) through the surface cut from the parabolic }} \\\ {\text { cylinder } z=4-y^{2} \text { by the planes } x=0, x=1, \text { and } z=0}\end{array}\)

Let \(C\) be the smooth curve \(\mathbf{r}(t)=(2 \cos t) \mathbf{i}+(2 \sin t) \mathbf{j}+\) \(\left(3-2 \cos ^{3} t\right) \mathbf{k},\) oriented to be traversed counterclockwise around the \(z\) -axis when viewed from above. Let \(S\) be the piecewise smooth cylindrical surface \(x^{2}+y^{2}=4,\) below the curve for \(z \geq 0,\) together with the base disk in the \(x y\) -plane. Note that \(C\) lies on the cylinder \(S\) and above the \(x y\) -plane (see the accompanying figure). Verify Equation \((4)\) in Stokes' Theorem for the vector field \(\mathbf{F}=y \mathbf{i}-x \mathbf{j}+x^{2} \mathbf{k}.\)

Let \(\mathbf{F}=(y \cos 2 x) \mathbf{i}+\left(y^{2} \sin 2 x\right) \mathbf{j}+\left(x^{2} y+z\right) \mathbf{k} .\) Is there a vector field A such that \(\mathbf{F}=\nabla \times \mathbf{A} ?\) Explain your answer.

In Exercises 19 and 20 , find the work done by \(F\) in moving a particle once counterclockwise around the given curve. $$\mathbf{F}=2 x y^{3} \mathbf{i}+4 x^{2} y^{2} \mathbf{j}$$ \(C :\) The boundary of the "triangular" region in the first quadrant enclosed by the \(x\) -axis, the line \(x=1,\) and the curve \(y=x^{3}\)

Find the work done by \(\mathbf{F}\) over the curve in the direction of increasing \(t .\) $$ \begin{array}{l}{\mathbf{F}=x y \mathbf{i}+y \mathbf{j}-y z \mathbf{k}} \\\ {\mathbf{r}(t)=t \mathbf{i}+t^{2} \mathbf{j}+t \mathbf{k}, \quad 0 \leq t \leq 1}\end{array} $$

Evaluate \(\int_{c} x d s,\) where \(C\) is $$ \begin{array}{l}{\text { a. the straight-line segment } x=t, y=t / 2, \text { from }(0,0) \text { to }(4,2) \text { . }} \\ {\text { b. the parabolic curve } x=t, y=t^{2}, \text { from }(0,0) \text { to }(2,4) .}\end{array} $$

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.) Cone frustum The portion of the cone \(z=\sqrt{x^{2}+y^{2}} / 3\) between the planes \(z=1\) and \(z=4 / 3\)

Use a parametrization to find the flux \(\iint_{S} \mathbf{F} \cdot \mathbf{n} d \sigma\) across the surface in the specified direction. \(\begin{array}{l}{\text { Parabolic cylinder } \quad \mathbf{F}=x^{2} \mathbf{j}-x z \mathbf{k} \text { outward (normal away }} \\ {\text { from the } y z-\text { plane through the surface cut from the parabolic cylinder }} \\\ {\text y=x^{2},-1 \leq x \leq 1, \text { by the planes } z=0 \text { and } z=2}\end{array} \)

Verify Stokes' Theorem for the vector field \(\mathbf{F}=2 x y \mathbf{i}+x \mathbf{j}+\) \((y+z) \mathbf{k}\) and surface \(z=4-x^{2}-y^{2}, z \geq 0,\) oriented with unit normal n pointing upward.

Outward flux of a gradient field Let \(S\) be the surface of the portion of the solid sphere \(x^{2}+y^{2}+z^{2} \leq a^{2}\) that lies in the first octant and let \(f(x, y, z)=\ln \sqrt{x^{2}+y^{2}+z^{2}} .\) Calculate $$\iint_{S} \nabla f \cdot \mathbf{n} d \sigma$$ \((\nabla f \cdot \mathbf{n}\) is the derivative of \(f\) in the direction of outward normal \(\mathbf{n} .)\)

In Exercises 19 and 20 , find the work done by \(F\) in moving a particle once counterclockwise around the given curve. $$\begin{array}{l}{\mathbf{F}=(4 x-2 y) \mathbf{i}+(2 x-4 y) \mathbf{j}} \\\ {C : \text { The circle }(x-2)^{2}+(y-2)^{2}=4}\end{array}$$

Find the work done by \(\mathbf{F}\) over the curve in the direction of increasing \(t .\) $$ \begin{array}{l}{\mathbf{F}=2 y \mathbf{i}+3 x \mathbf{j}+(x+y) \mathbf{k}} \\\ {\mathbf{r}(t)=(\cos t) \mathbf{i}+(\sin t) \mathbf{j}+(t / 6) \mathbf{k}, \quad 0 \leq t \leq 2 \pi}\end{array} $$

\(\text{Evaluate}\int_{C} \sqrt{x+2 y} d s, \text { where } C \text{ is }\) $$ \begin{array}{l}{\text { a. the straight-line segment } x=t, y=4 t, \text { from }(0,0) \text { to }(1,4) \text { . }} \\ {\text { b. } C_{1} \cup C_{2} ; C_{1} \text { is the line segment from }(0,0) \text { to }(1,0) \text { and } C_{2} \text { is }} \\ {\text { the line segment from }(1,0) \text { to }(1,2) \text { . }}\end{array} $$

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.) Circular cylinder band The portion of the cylinder \(x^{2}+y^{2}=1\) between the planes \(z=1\) and \(z=4\)

Use a parametrization to find the flux \(\iint_{S} \mathbf{F} \cdot \mathbf{n} d \sigma\) across the surface in the specified direction. \(\begin{array}{l}{\text { Sphere } \quad \mathbf{F}=z \mathbf{k} \text { across the portion of the sphere } x^{2}+y^{2}+} \\ {z^{2}=a^{2} \text { in the first octant in the direction away from the origin }}\end{array}\)

Let \(\mathbf{F}\) be a field whose components have continuous first partial derivatives throughout a portion of space containing a region \(D\) bounded by a smooth closed surface \(S .\) If \(|\mathbf{F}| \leq 1,\) can any bound be placed on the size of $$\iiint_{D} \nabla \cdot \mathbf{F} d V ?$$ Give reasons for your answer.

Apply Green's Theorem to evaluate the integrals in Exercises \(21-24\) $$\oint\left(y^{2} d x+x^{2} d y\right)$$ $$C : The triangle bounded by x=0, x+y=1, y=0$$

Find the line integral of \(f(x, y)=y e^{x^{2}}\) along the curve \(\mathbf{r}(t)=4 t \mathbf{i}-3 t \mathbf{j},-1 \leq t \leq 2\)

Find the work done by \(\mathbf{F}\) over the curve in the direction of increasing \(t .\) $$ \begin{array}{l}{\mathbf{F}=z \mathbf{i}+x \mathbf{j}+y \mathbf{k}} \\\ {\mathbf{r}(t)=(\sin t) \mathbf{i}+(\cos t) \mathbf{j}+t \mathbf{k}, \quad 0 \leq t \leq 2 \pi}\end{array} $$

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.) Circular cylinder band The portion of the cylinder \(x^{2}+z^{2}=\) 10 between the planes \(y=-1\) and \(y=1\)

Use a parametrization to find the flux \(\iint_{S} \mathbf{F} \cdot \mathbf{n} d \sigma\) across the surface in the specified direction. \(\begin{array}{l}{\text { Sphere } \quad \mathbf{F}=x \mathbf{i}+y \mathbf{j}+z \mathbf{k} \text { across the sphere } x^{2}+y^{2}+} \\\ {z^{2}=a^{2} \text { in the direction away from the origin }}\end{array}\)

Zero circulation Let \(f(x, y, z)=\left(x^{2}+y^{2}+z^{2}\right)^{-1 / 2}\) . Show that the clockwise circulation of the field \(\mathbf{F}=\nabla f\) around the circle \(x^{2}+y^{2}=a^{2}\) in the \(x y\) -plane is zero \(\begin{array}{l}{\text { a. by taking } \mathbf{r}=(a \cos t) \mathbf{i}+(a \sin t) \mathbf{j}, 0 \leq t \leq 2 \pi, \text { and }} \\ {\text { integrating } \mathbf{F} \cdot d \mathbf{r} \text { over the circle. }} \\\ {\text { b. by applying Stokes' Theorem. }}\end{array}\)

Apply Green's Theorem to evaluate the integrals in Exercises \(21-24\) $$\begin{array}{l}{\oint_{C}(3 y d x+2 x d y)} \\ {C : \text { The boundary of } 0 \leq x \leq \pi, 0 \leq y \leq \sin x}\end{array}$$

\(\begin{array}{l}{\text { Find the line integral of } f(x, y)=x-y+3 \text { along the curve }} \\ {\mathbf{r}(t)=(\cos t) \mathbf{i}+(\sin t) \mathbf{j}, 0 \leq t \leq 2 \pi}\end{array}\)

Find the work done by \(\mathbf{F}\) over the curve in the direction of increasing \(t .\) $$ \begin{array}{l}{\mathbf{F}=6 z \mathbf{i}+y^{2} \mathbf{j}+12 x \mathbf{k}} \\\ {\mathbf{r}(t)=(\sin t) \mathbf{i}+(\cos t) \mathbf{j}+(t / 6) \mathbf{k}, \quad 0 \leq t \leq 2 \pi}\end{array} $$

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.) Parabolic cap The cap cut from the paraboloid \(z=2-x^{2}-y^{2}\) by the cone \(z=\sqrt{x^{2}+y^{2}}\)

Let \(C\) be a simple closed smooth curve in the plane \(2 x+2 y+z=2,\) oriented as shown here. Show that $$\oint_{C} 2 y d x+3 z d y-x d z$$ depends only on the area of the region enclosed by \(C\) and not on the position or shape of \(C .\)

Use a parametrization to find the flux \(\iint_{S} \mathbf{F} \cdot \mathbf{n} d \sigma\) across the surface in the specified direction. \(\begin{array}{l}{\text { Plane } \mathbf{F}=2 x y \mathbf{i}+2 y z \mathbf{j}+2 x z \mathbf{k} \text { upward across the portion of }} \\ {\text { the plane } x+y+z=2 a \text { that lies above the square } 0 \leq x \leq a,} \\\ {0 \leq y \leq a, \text { in the } x y-\text { plane }}\end{array}\)

Calculate the net outward flux of the vector field $$\mathbf{F}=x y \mathbf{i}+\left(\sin x z+y^{2}\right) \mathbf{j}+\left(e^{r y^{2}}+x\right) \mathbf{k}$$ over the surface \(S\) surrounding the region \(D\) bounded by the planes \(y=0, z=0, z=2-y\) and the parabolic cylinder \(z=1-x^{2}\) .

Apply Green's Theorem to evaluate the integrals in Exercises \(21-24\) $$\begin{array}{l}{\oint(6 y+x) d x+(y+2 x) d y} \\ {C : \text { The circle }(x-2)^{2}+(y-3)^{2}=4}\end{array}$$

Evaluate \(\int_{C} \frac{x^{2}}{y^{4 / 3}} d s,\) where \(C\) is the curve \(x=t^{2}, y=t^{3},\) for \(1 \leq t \leq 2\)

Evaluate \(\int_{C} x y d x+(x+y) d y\) along the curve \(y=x^{2}\) from \((-1,1)\) to \((2,4) .\)

Revisiting Example 6 Evaluate the integral $$\int_{(1,1,1)}^{(2,3,-1)} y d x+x d y+4 d z$$ from Example 6 by finding parametric equations for the line segment from \((1,1,1)\) to \((2,3,-1)\) and evaluating the line integral of \(\mathbf{F}=y \mathbf{i}+x \mathbf{j}+4 \mathbf{k}\) along the segment. Since \(\mathbf{F}\) is conservative, the integral is independent of the path.

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.) Parabolic band The portion of the paraboloid \(z=x^{2}+y^{2}\) between the planes \(z=1\) and \(z=4\)

Show that if \(\mathbf{F}=x \mathbf{i}+y \mathbf{j}+z \mathbf{k},\) then \(\nabla \times \mathbf{F}=\mathbf{0}.\)

Use a parametrization to find the flux \(\iint_{S} \mathbf{F} \cdot \mathbf{n} d \sigma\) across the surface in the specified direction. \(\begin{array}{l}{\text { Cylinder } \mathbf{F}=x \mathbf{i}+y \mathbf{j}+z \mathbf{k} \text { outward through the portion of }} \\ {\text { the cylinder } x^{2}+y^{2}=1 \text { cut by the planes } z=0 \text { and } z=a}\end{array} \)

Compute the net outward flux of the vector field \(\mathbf{F}=(x \mathbf{i}+y \mathbf{j}+z \mathbf{k}) /\left(x^{2}+y^{2}+z^{2}\right)^{3 / 2}\) across the ellipsoid \(9 x^{2}+4 y^{2}+6 z^{2}=36\)

Find the line integral of \(f(x, y)=\sqrt{y} / x\) along the curve \(\mathbf{r}(t)=t^{3} \mathbf{i}+t^{4} \mathbf{j}, 1 / 2 \leq t \leq 1\)

Evaluate \(\int_{C}(x-y) d x+(x+y) d y\) counterclockwise around the triangle with vertices \((0,0),(1,0),\) and \((0,1)\) .

Evaluate $$\int_{C} x^{2} d x+y z d y+\left(y^{2} / 2\right) d z$$ along the line segment \(C\) joining \((0,0,0)\) to \((0,3,4).\)

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.) Sawed-off sphere The lower portion cut from the sphere \(x^{2}+y^{2}+z^{2}=2\) by the cone \(z=\sqrt{x^{2}+y^{2}}\)

Find a vector field with twice-differentiable components whose curl is \(x \mathbf{i}+y \mathbf{j}+z \mathbf{k}\) or prove that no such field exists.

Let \(\mathbf{F}\) be a differentiable vector field and let \(g(x, y, z)\) be a differentiable scalar function. Verify the following identities. $$ \begin{array}{l}{\text { a. } \nabla \cdot(g \mathbf{F})=g \nabla \cdot \mathbf{F}+\nabla g \cdot \mathbf{F}} \\ {\text { b. } \nabla \times(g \mathbf{F})=g \nabla \times \mathbf{F}+\nabla g \times \mathbf{F}}\end{array} $$

The reason is that by Equation \((4),\) run backward, $$\begin{aligned} \text { Area of } R &=\iint_{R} d y d x=\iint_{K}\left(\frac{1}{2}+\frac{1}{2}\right) d y d x \\ &=\oint_{C} \frac{1}{2} x d y-\frac{1}{2} y d x \end{aligned}$$ Use the Green's Theorem area formula given above to find the areas bof the regions enclosed by the curves in Exercises \(25-28 .\) $$\mathbf{r}(t)=(a \cos t) \mathbf{i}+(a \sin t) \mathbf{j}, \quad 0 \leq t \leq 2 \pi$$

Evaluate \(\int_{C} \mathbf{F} \cdot \mathbf{T} d s\) for the vector field \(\mathbf{F}=x^{2} \mathbf{i}-y \mathbf{j}\) along the curve \(x=y^{2}\) from \((4,2)\) to \((1,-1)\) .