Chapter 14

Calculate \(\iint_{\partial S} \mathbf{F} \cdot \mathbf{n} d S\). In each case, \(\mathbf{r}=x \mathbf{i}+y \mathbf{j}+z \mathbf{k}\). (a) \(\mathbf{F}=\mathbf{r} /\|\mathbf{r}\|^{3} ; S\) is the solid sphere \((x-2)^{2}+y^{2}+z^{2} \leq 1\). (b) \(\mathbf{F}=\mathbf{r} /\|\mathbf{r}\|^{3} ; S\) is the solid sphere \(x^{2}+y^{2}+z^{2} \leq a^{2}\). (c) \(\mathbf{F}=\mathbf{r} /\|\mathbf{r}\|^{2} ; S\) as in part (b). (d) \(\mathbf{F}=f(\|\mathbf{r}\|) \mathbf{r}, f\) any scalar function; \(S\) as in part (b). (e) \(\mathbf{F}=\|\mathbf{r}\|^{n} \mathbf{r}, n \geq 0 ; S\) is the solid sphere \(x^{2}+y^{2}+z^{2} \leq a z\) ( \(\rho \leq a \cos \phi\) in spherical coordinates).

Plot the parametric surface over the indicated domain. \(\mathbf{r}(u, v)=u \mathbf{i}+3 \sin v \mathbf{j}+5 \cos v \mathbf{k} ;-6 \leq u \leq 6\), \(0 \leq v \leq 2 \pi\)

Find the work done by the force field \(\mathbf{F}\) in moving a particle along the curve \(C\). \(\mathbf{F}(x, y)=e^{x} \mathbf{i}-e^{-y} \mathbf{j} ; C\) is the curve \(x=3 \ln t, y=\ln 2 t\), \(1 \leq t \leq 5\).

Let the piecewise smooth, simple closed curve \(C\) be the boundary of a region \(S\) in the \(x y\)-plane. Modify the argument in Example 2 to show that $$ A(S)=\oint_{C}(-y) d x=\oint_{C} x d y $$

Use a CAS to plot the parametric surface over the indicated domain and find the surface area of the resulting surface. \(\mathbf{r}(u, v)=u \sin v \mathbf{i}+u \cos v \mathbf{j}+v \mathbf{k} ;-6 \leq u \leq 6\), \(0 \leq v \leq \pi\)

Find the work done by the force field \(\mathbf{F}\) in moving a particle along the curve \(C\). \(\mathbf{F}(x, y)=(x+y) \mathbf{i}+(x-y) \mathbf{j} ; C\) is the quarter- ellipse, \(x=a \cos t, y=b \sin t, 0 \leq t \leq \pi / 2 .\)

Suppose that \(\nabla^{2} f\) is identically zero in a region \(S\). Show that $$ \iint_{\partial S} f D_{\mathbf{n}} f d S=\iiint_{S}\|\nabla f\|^{2} d V $$

Use a CAS to plot the parametric surface over the indicated domain and find the surface area of the resulting surface. \(\mathbf{r}(u, v)=\sin u \sin v \mathbf{i}+\cos u \sin v \mathbf{j}+\sin v \mathbf{k}\); \(0 \leq u \leq 2 \pi, 0 \leq v \leq 2 \pi\)

Find the work done by the force field \(\mathbf{F}\) in moving a particle along the curve \(C\). \(\mathbf{F}(x, y, z)=(2 x-y) \mathbf{i}+2 z \mathbf{j}+(y-z) \mathbf{k} ; C\) is the line segment from \((0,0,0)\) to \((1,1,1)\).

Establish Green's First Identity $$ \iint_{\partial S} f D_{\mathbf{n}} g d S=\iiint_{S}\left(f \nabla^{2} g+\nabla f \cdot \nabla g\right) d V $$ by applying Gauss's Divergence Theorem to \(\mathbf{F}=f \nabla g\).

Calculate the area of the asteroid \(x^{2 / 3}+y^{2 / 3}=a^{2 / 3}\).

Use a CAS to plot the parametric surface over the indicated domain and find the surface area of the resulting surface. \(\mathbf{r}(u, v)=u^{2} \cos v \mathbf{i}+u^{2} \sin v \mathbf{j}+5 u \mathbf{k} ; 0 \leq u \leq 2 \pi\), \(0 \leq v \leq 2 \pi\)

Find the work done by the force field \(\mathbf{F}\) in moving a particle along the curve \(C\). \(\mathbf{F}(x, y, z)=y \mathbf{i}+z \mathbf{j}+x \mathbf{k} ; C\) is the curve \(x=t, y=t^{2}\), \(z=t^{3}, 0 \leq t \leq 2\).

Let \(\mathbf{F}(\mathbf{r})=\mathbf{r} /\|\mathbf{r}\|^{2}=(x \mathbf{i}+y \mathbf{j}) /\left(x^{2}+y^{2}\right)\). (a) Show that \(\int_{C} \mathbf{F} \cdot \mathbf{n} d s=2 \pi\), where \(C\) is the circle centered at the origin of radius \(a\) and \(\mathbf{n}=(x \mathbf{i}+y \mathbf{j}) / \sqrt{x^{2}+y^{2}}\) is the exterior unit normal to \(C\). (b) Show that div \(\mathbf{F}=0\). (c) Explain why the results of parts (a) and (b) do not contradict the vector form of Green's Theorem. (d) Show that if \(C\) is a smooth simple closed curve then \(\int_{C} \mathbf{F} \cdot \mathbf{n} d s\) equals \(2 \pi\) or 0 accordingly as the origin is inside or outside \(C\).

Sketch a plot of the vector field \(\mathbf{F}=y \mathbf{i}\) for \((x, y)\) in the rectangle \(1 \leq x \leq 2,0 \leq y \leq 2\). From the plot, use the marginal box that describes the interpretation of div and curl to determine whether div is positive, negative, or zero at the point \((1,1)\), and whether a paddle wheel placed at \((1,1)\) would rotate clockwise, counterclockwise, or not at all.

Sketch a plot of the vector field $$ \mathbf{F}=-\frac{x}{\left(1+x^{2}+y^{2}\right)^{3 / 2}} \mathbf{i}-\frac{y}{\left(1+x^{2}+y^{2}\right)^{3 / 2}} \mathbf{j} $$ for \((x, y)\) in the rectangle \(-1 \leq x \leq 1,-1 \leq y \leq 1\). From the plot, use the marginal box that describes the interpretation of div and curl to determine whether div is positive, negative, or zero at the origin, and whether a paddle wheel placed at the origin would rotate clockwise, counterclockwise, or not at all. (For the curl, think of \(\mathbf{F}\) as being a vector field in 3-space with \(z\)-component equal to 0.)

Christy plans to paint both sides of a fence whose base is in the \(x y\)-plane with shape \(x=30 \cos ^{3} t, y=30 \sin ^{3} t\), \(0 \leq t \leq \pi / 2\), and whose height at \((x, y)\) is \(1+\frac{1}{3} y\), all measured in feet. Sketch a picture of the fence and decide how much paint she will need if a gallon covers 200 square feet.

Consider the velocity field \(\mathbf{v}(x, y, z)=-\omega y \mathbf{i}+\omega x \mathbf{j}\), \(\omega>0\) (see Example 2 and Figure 1). Note that \(\mathbf{v}\) is perpendicular to \(x \mathbf{i}+y \mathbf{j}\) and that \(\|\mathbf{v}\|=\omega \sqrt{x^{2}+y^{2}}\). Thus, \(\mathbf{v}\) describes a fluid that is rotating (like a solid) about the \(z\)-axis with constant angular velocity \(\omega\). Show that \(\operatorname{div} \mathbf{v}=0\) and curl \(\mathbf{v}=2 \omega \mathbf{k}\).

A squirrel weighing \(1.2\) pounds climbed a cylindrical tree by following the helical path \(x=\cos t, y=\sin t, z=4 t\), \(0 \leq t \leq 8 \pi\) (distance measured in feet). How much work did it do? Use a line integral, but then think of a trivial way to answer this question.

An object of mass \(m\), which is revolving in a circular orbit with constant angular velocity \(\omega\), is subject to the centrifugal force given by $$ \mathbf{F}(x, y, z)=m \omega^{2} \mathbf{r}=m \omega^{2}(x \mathbf{i}+y \mathbf{j}+z \mathbf{k}) $$ Show that \(f(x, y, z)=\frac{1}{2} m \omega^{2}\left(x^{2}+y^{2}+z^{2}\right)\) is a potential function for \(\mathbf{F}\).

Let \(G\) be the sphere \(x^{2}+y^{2}+z^{2}=a^{2}\). Evaluate each of the following: (a) \(\iint_{G} z d S\) (b) \(\iint_{G} \frac{x+y^{3}+\sin z}{1+z^{4}} d S\)

The scalar function \(\operatorname{div}(\operatorname{grad} f)=\nabla \cdot \nabla f\) (also written \(\left.\nabla^{2} f\right)\) is called the Laplacian, and a function \(f\) satisfying \(\nabla^{2} f=0\) is said to be harmonic, concepts important in physics. Show that \(\nabla^{2} f=f_{x x}+f_{y y}+f_{z z}\). Then find \(\nabla^{2} f\) for each of the following functions and decide which are harmonic. (a) \(f(x, y, z)=2 x^{2}-y^{2}-z^{2}\) (b) \(f(x, y, z)=x y z\) (c) \(f(x, y, z)=x^{3}-3 x y^{2}+3 z\) (d) \(f(x, y, z)=\left(x^{2}+y^{2}+z^{2}\right)^{-1 / 2}\)

The sphere \(x^{2}+y^{2}+z^{2}=a^{2}\) has constant area density \(k\). Find each moment of inertia. (a) About a diameter (b) About a tangent line (assume the Parallel Axis Theorem from Problem 28 of Section 13.5).

Show that (a) \(\operatorname{div}(\mathbf{F} \times \mathbf{G})=\mathbf{G} \cdot \operatorname{curl} \mathbf{F}-\mathbf{F} \cdot \operatorname{curl} \mathbf{G}\) (b) \(\operatorname{div}(\nabla f \times \nabla g)=0\)