Question 18.113

\( W = [0,1] \times [0,1] \times [0,1] \textbf{F} = x \textbf{i} + y \textbf{j} + z \textbf{k}\)

Question 18.114

\(W\) as in Exercise 1, and \(\textbf{F} = zy \textbf{i} + xz \textbf{j} + xy \textbf{k}\)

Question 18.115

\( W = \{ (x,y,z) : x^2 + y^2 + z^2 \leq 1 \}\) \quad (the unit ball), \(\textbf{F} = x \textbf{i} + y \textbf{j} + z \textbf{k}\)

Question 18.116

\(W\) as in Exercise 3, and \(\textbf{F} = -y \textbf{i} + x \textbf{j} + z \textbf{k}\)

Question 18.117

Use the divergence theorem to calculate the flux of \({\bf F} = (x-y) {\bf i} + (y -z) {\bf j} + (z-x){\bf k}\) out of the unit sphere.

Question 18.118

Let \({\bf F}=x^3{\bf i}+y^3{\bf j}+z^3{\bf k}\). Evaluate the surface integral of \({\bf F}\) over the unit sphere.

Question 18.119

Evaluate \({\intop\!\!\!\intop}_{\partial\! W}{\bf F}\,{\cdot}\,{\,d} {\bf S}\), where \({\bf F}=x{\bf i}+y{\bf j}+z{\bf k}\) and \(W\) is the unit cube (in the first octant). Perform the calculation directly and check by using the divergence theorem.

Question 18.120

Repeat the previous exercise for

• (a) \({\bf F}={\bf i}+{\bf j}+{\bf k}\)
• (b) \({\bf F}=x^2{\bf i} +x^2{\bf j} +z^2{\bf k}\)

Question 18.121

Let \({\bf F}=y{\bf i} +z{\bf j} +xz{\bf k}\). Evaluate \({\intop\!\!\!\intop}_{\partial\! W}{\bf F}\,{\cdot}\,{\,d} {\bf S}\) for each of the following regions \(W\):

• (a) \(x^2+y^2\leq z\leq 1\)
• (b) \(x^2+y^2\leq z\leq 1\) and \(x\geq 0\)
• (c) \(x^2+y^2\leq z\leq 1\) and \(x\leq 0\)

Question 18.122

Repeat the previous exercise for \({\bf F}=(x-y){\bf i}+(y-z){\bf j}+ (z-x){\bf k}\). [The solution to part (b) only is in the Study Guide to this text.]

Question 18.123

Find the flux of the vector field \({\bf F}= (x-y^2){\bf i} + y{\bf j} + x^3 {\bf k}\) out of the rectangular solid \([0,1] \times [1,2] \times [1,4]\).

Question 18.124

Evaluate \({\intop\!\!\!\intop}_S{\bf F}\,{\cdot}\,{\,d} {\bf S}\), where \({\bf F}=3xy^2{\bf i}+3x^2y{\bf j}+z^3{\bf k}\) and \(S\) is the surface of the unit sphere.

Question 18.125

Let \(W\) be the pyramid with top vertex \((0,0,1)\), and base vertices at \((0,0,0)\), \((1,0,0)\), \((0,1,0)\), and \((1,1,0)\). Let \(S\) be the two-dimensional closed surface bounding \(W\), oriented outward from \(W\). Use Gauss’ theorem to calculate \({\intop\!\!\!\intop}_{S} \textbf{F} \cdot d \textbf{S}\), where: \[ \textbf{F}(x,y,z) = (x^2 y, 3y^2 z, 9z^2 x). \]

Question 18.126

Let \(W\) be the three-dimensional solid enclosed by the surfaces \(x = y^2\), \(x=9\), \(z=0\), and \(x=z\). Let \(S\) be the boundary of \(W\). Use Gauss’ theorem to find the flux of \(\textbf{F}(x,y,z) = (3x - 5y)i + (4z - 2y)j + (8yz)k\) across \(S\): \({\intop\!\!\!\intop}_{S} \textbf{F} \cdot d \textbf{S}\).

Question 18.127

Evaluate \({\intop\!\!\!\intop}_{\partial\! W} {\bf F} {\,{\cdot}\,} {\bf n} {\,d} A\), where \({\bf F}(x,y,z) = x{\bf i} + y{\bf j} - z{\bf k}\) and \(W\) is the unit cube in the first octant. Perform the calculation directly and check by using the divergence theorem.

Question 18.128

Evaluate the surface integral \({\intop\!\!\!\intop}_{\partial S} {\bf F} {\,{\cdot}\,} {\bf n} {\,d} A\), where \({\bf F}(x,y,z) = {\bf i} + {\bf j} + z(x^2 + y^2)^2{\bf k}\) and \(\partial\! S\) is the surface of the cylinder \(x^2 + y^2 \le 1, 0\le z \le 1\).

Question 18.129

Prove that \begin{eqnarray*} \intop\!\!\!\intop\!\!\!\intop\nolimits_{W} ({\nabla}\!f) \,{\cdot}\, {\bf F} {\it {\,d} x}\, {\it dy} {\,d} z &=& \intop\!\!\!\intop\nolimits_{\partial\! W} f{\bf F} {\,{\cdot}\,} {\bf n} {\,d} S\\[6pt] &&-\ \intop\!\!\!\intop\!\!\!\intop\nolimits_{W} f{\nabla} {\,{\cdot}\,} {\bf F} {\it {\,d} x}\, {\it dy}\, {\,d} z. \end{eqnarray*}

475

Question 18.130

Prove the identity \[ {\nabla}{\,{\cdot}\,} ({\bf F} \times {\bf G}) = {\bf G}{\,{\cdot}\,} ({\nabla}\times {\bf F}) - {\bf F} {\,{\cdot}\,} ({\nabla}\times {\bf G}). \]

Question 18.131

Show that \({\intop\!\!\!\intop\!\!\!\intop}_{W}(1/r^2)\, {\it {\,d} x}\,{\it dy}\,{\,d} z = {\intop\!\!\!\intop}_{\partial\! W}({\bf r}{\,{\cdot}\,} {\bf n}/r^2){\,d} S\), where \({\bf r} = x{\bf i} + y{\bf j} + z{\bf k}\).

Question 18.132

Fix vectors \({\bf v}_1,\ldots, {\bf v}_k \in {\mathbb R}^3\) and numbers (“charges”) \(q_1,\ldots, q_k\). Define the function \(\phi\) by \(\phi(x,y,z) = \sum_{i=1}^k q_i/(4\pi \|{\bf r} - {\bf v}_i\|)\), where \({\bf r} =(x,y,z)\). Show that for a closed surface \(S\) and \({\bf E} = - {\nabla}\phi\), \[ \intop\!\!\!\intop\nolimits_{S}{\bf E}{\,{\cdot}\,} {\,d}{\bf S}= Q, \] where \(Q\) is the total charge inside \(S\). (Assume that Gauss’ law from Theorem 10 applies and that none of the charges are on \(S\).)

Question 18.133

Prove Green’s identities \[ \intop\!\!\!\intop\nolimits_{\partial\! W} f{\nabla}g {\,{\cdot}\,} {\bf n}{\,d} S = \intop\!\!\!\intop\!\!\!\intop\nolimits_{W}(f\nabla^2 g + {\nabla}\!f{\,{\cdot}\,} {\nabla}g) {\,d} V \] and \[ \intop\!\!\!\intop\nolimits_{\partial\! W} (f{\nabla}g - g{\nabla} f) {\,{\cdot}\,} {\bf n} {\,d} S = \intop\!\!\!\intop\!\!\!\intop\nolimits_{W} (f \nabla^2 g - g\nabla^2\! f) {\,d} V. \]

Question 18.134

Suppose \({\bf F}\) satisfies div \({\bf F}=0\) and curl \({\bf F} = {\bf 0}\) on all of \({\mathbb R}^3\). Show that we can write \({\bf F} = {\nabla}\! f\), where \(\nabla^2 f = 0\).

Question 18.135

Let \(\rho\) be a continuous function on \({\mathbb R}^3\) such that \(\rho({\bf q})=0\) except for \({\bf q}\) in some region \(W\). Let \({\bf q}\in W\) be denoted by \({\bf q} = (x,y,z)\). The potential of \(\rho\) is defined to be the function \[ \phi({\bf p}) = \intop\!\!\!\intop\!\!\!\intop\nolimits_{W} \frac{\rho({\bf q})}{4\pi\|{\bf p}-{\bf q}\|} {\,d} V({\bf q}), \] where \(\|{\bf p}-{\bf q}\|\) is the distance between \({\bf p}\) and \({\bf q}\).

• (a) Using the method of Theorem 10, show that \[ \intop\!\!\!\intop\nolimits_{\partial\! W} {\nabla}\phi {\,{\cdot}\,} {\bf n} {\,d} S= -\intop\!\!\!\intop\!\!\!\intop\nolimits_W \rho{\,d} V \] for those regions \(W\) that can be partitioned into a finite union of symmetric elementary regions.
• (b) Show that \(\phi\) satisfies Poisson’s equation \[ \nabla^2\phi = -\rho. \]

[HINT: Use part (a).] (Notice that if \(\rho\) is a charge density, then the integral defining \(\phi\) may be thought of as the sum of the potential at \({\bf p}\) caused by point charges distributed over \(W\) according to the density \(\rho\).)

Question 18.136

Suppose \({\bf F}\) is tangent to the closed surface \(S= \partial\! W\) of a region \(W\). Prove that \[ \intop\!\!\!\intop\!\!\!\intop\nolimits_{W} ({\hbox{div }} {\bf F}){\,d} V= 0. \]

Question 18.137

Use Gauss’ law and symmetry to prove that the electric field due to a charge \(Q\) evenly spread over the surface of a sphere is the same outside the surface as the field from a point charge \(Q\) located at the center of the sphere. What is the field inside the sphere?

Question 18.138

Reformulate the previous exercise in terms of gravitational fields.

Question 18.139

Show how Gauss’ law can be used to solve part (b) of Exercise 29 in Section 18.3.

Question 18.140

Let \(S\) be a closed surface. Use Gauss’ theorem to show that if \({\bf F}\) is a \(C^2\) vector field, then we have \({\intop\!\!\!\intop}_s({\nabla} \times {\bf F})\, {\cdot}\, d{\bf S} =0\).

Question 18.141

Let \(S\) be the surface of region \(W\). Show that \[ \intop\!\!\!\intop\nolimits_S {\bf r}\,{\cdot}\, {\bf n}\, dS =3 \hbox{ volume } (W). \]

Explain this geometrically.

Question 18.142

For a steady-state charge distribution and divergence-free current distribution, the electric and magnetic fields \({\bf E}(x,y,z)\) and \({\bf H}(x,y,z)\) satisfy \begin{eqnarray*} &&{\nabla}\times{\bf E}= {\bf 0},\qquad {\nabla}\,{\cdot}\, {\bf H}=0,\qquad {\nabla}\,{\cdot}\,{\bf J}=0, \qquad {\nabla}\,{\cdot}\, {\bf E}=\rho,\\ &&\qquad\hbox{and}\qquad {\nabla}\times {\bf H}={\bf J}. \end{eqnarray*}

Here \(\rho=\rho(x,y,z)\) and \({\bf J}(x,y,z)\) are assumed to be known. The radiation that the fields produce through a surface \(S\) is determined by a radiation flux density vector field, called the Poynting vector field, \[ {\bf P}={\bf E}\times{\bf H}. \]

• (a) If \(S\) is a closed surface, show that the radiation flux—that is, the flux of \({\bf P}\) through \(S\)—is given by \[ \intop\!\!\!\intop\nolimits_{S} {\bf P}\,{\cdot}\,{\,d} {\bf S}=-\intop\!\!\!\intop\!\!\!\intop\nolimits_{V}{\bf E}\,{\cdot}\, {\bf J}{\it {\,d} V}, \] where \(V\) is the region enclosed by \(S\).
• (b) Examples of such fields are \begin{eqnarray*} &&{\bf E}(x,y,z) = z{\bf j}+y{\bf k} \qquad {\rm and}\\ && \qquad {\bf H}(x,y,z) = -xy{\bf i}+x{\bf j}+yz{\bf k}. \end{eqnarray*} In this case, find the flux of the Poynting vector through the hemispherical shell shown in Figure 18.40. (Notice that it is an open surface.)
• (c) The fields of part (b) produce a Poynting vector field passing through the toroidal surface shown in Figure 18.41. What is the flux through this torus?