Vector Analysis

Express the cylindrical unit vectors $\hat{\mathbf{s}}\mathbf{,}\mathbf{ }\hat{\mathbf{\varphi }}\mathbf{,}\mathbf{ }\hat{\mathbf{z}}$   in terms of $\hat{\mathbf{x}}\mathbf{,}\mathbf{ }\hat{\mathbf{y}}\mathbf{,}\mathbf{ }\hat{\mathbf{z}}$  (that is, derive Eq. 1.75). "Invert" your formulas to get  $\hat{\mathbf{x}}\mathbf{,}\mathbf{ }\hat{\mathbf{y}}\mathbf{,}\mathbf{ }\hat{\mathbf{z}}$ in terms of  $\hat{\mathbf{s}}\mathbf{,}\mathbf{ }\hat{\mathbf{\varphi }}\mathbf{,}\mathbf{ }\hat{\mathbf{z}}$

Question: Evaluate the following integrals:

(a) ${\mathbf{\int }}_{\mathbf{2}}^{\mathbf{6}}\mathbf{(}\mathbf{3}{\mathit{x}}^{\mathbf{2}}\mathbf{-}\mathbf{2}\mathit{x}\mathbf{-}\mathbf{1}\mathbf{)}\mathit{\delta }\mathbf{(}\mathit{x}\mathbf{-}\mathbf{3}\mathbf{)}\mathit{d}\mathit{x}$ 

(b)  ${\mathbf{\int }}_{\mathbf{0}}^{\mathbf{5}}\mathbf{(}\mathit{c}\mathit{o}\mathit{s}\mathbf{ }\mathit{x}\mathbf{ }\mathit{\delta }\mathbf{(}\mathit{x}\mathbf{-}\mathit{\pi }\mathbf{)}\mathit{d}\mathit{x}$

(c)  ${\mathbf{\int }}_{\mathbf{0}}^{\mathbf{3}}\mathbf{ }{\mathit{x}}^{\mathbf{2}}\mathbf{ }\mathit{\delta }\mathbf{(}\mathit{x}\mathbf{+}\mathbf{1}\mathbf{)}\mathit{d}\mathit{x}$

(d) ${\mathbf{\int }}_{\mathbf{-}\mathbf{\infty }\mathbf{ }}^{\mathbf{\infty }}\mathbf{In}\mathbf{(}\mathbf{x}\mathbf{+}\mathbf{3}\mathbf{)}\mathbf{\delta }\mathbf{(}\mathbf{x}\mathbf{+}\mathbf{2}\mathbf{)}\mathbf{dx}$

Question:   Evaluate the following integrals:

(a) ${\mathbf{\int }}_{\mathbf{-}\mathbf{2}}^{\mathbf{2}}\mathbf{(}\mathbf{2}\mathit{x}\mathbf{+}\mathbf{3}\mathbf{)}\mathit{\delta }\mathbf{\left(}\mathbf{3}\mathit{x}\mathbf{\right)}\mathit{d}\mathit{x}$

(b) ${\mathbf{\int }}_{\mathbf{0}}^{\mathbf{2}}\mathbf{(}{\mathit{x}}^{\mathbf{3}}\mathbf{+}\mathbf{3}\mathit{x}\mathbf{+}\mathbf{2}\mathbf{)}\mathit{\delta }\mathbf{(}\mathbf{1}\mathbf{-}\mathit{x}\mathbf{)}\mathit{d}\mathit{x}$

(c)  ${\mathbf{\int }}_{\mathbf{-}\mathbf{4}}^{\mathbf{1}}\mathbf{9}{\mathit{x}}^{\mathbf{2}}\mathit{\delta }\mathbf{(}\mathbf{3}\mathit{x}\mathbf{+}\mathbf{1}\mathbf{)}\mathit{d}\mathit{x}$

(d)${\mathbf{\int }}_{\mathbf{-}\mathbf{\infty }}^{\mathbf{a}}\mathit{\delta }\mathbf{(}\mathit{x}\mathbf{-}\mathit{b}\mathbf{)}\mathit{d}\mathit{x}$

 (a) Show that $\mathit{x}\frac{\mathbf{d}}{\mathbf{d}\mathbf{x}}\left(\delta \left(x\right)\right)\mathbf{=}\mathbf{-}\mathit{\delta }\left(x\right)$ 

[Hint: Use integration by parts.] 

(b) Let $\mathit{\theta }\left(x\right)$ be the step function:

 $\mathit{\theta }\left(x\right)\mathbf{=}\{\begin{array}{ll}1& \mathrm{if}\mathit{ }\mathit{x}>0\\ 0,& \mathrm{if}\mathit{ }\mathit{x}\le 0\end{array}$

Show that $\frac{\mathbf{d\theta }}{\mathbf{dx}}\mathbf{=}\mathit{\delta }\mathbf{\left(}\mathbf{x}\mathbf{\right)}$

(a) Write an expression for the volume charge density p(r) of a point charge q at r'. Make sure that the volume integral of p equals q.

(b) What is the volume charge density of an electric dipole, consisting of a point? charge -q at the origin and a point charge +q at a?

(c) What is the volume charge density (in spherical coordinates) of a uniform, in-finitesimally thin spherical shell of radius Rand total charge Q, centered at the origin? [Beware: the integral over all space must equal Q.]

Evaluate the following integrals:

(a) $\mathbf{\int }\mathbf{(}{\mathbf{r}}^{\mathbf{2}}\mathbf{+}\mathbf{r}\mathbf{·}\mathbf{a}\mathbf{+}{\mathbf{a}}^{\mathbf{2}}\mathbf{)}{\mathbf{\delta }}^{\mathbf{3}}\mathbf{(}\mathbf{r}\mathbf{-}\mathbf{a}\mathbf{)}\mathbf{d\tau }$ , where a is a fixed vector, a is its magnitude.

(b) ${\mathbf{\int }}_{\mathbf{v}}{\mathbf{|}\mathbf{r}\mathbf{-}\mathbf{d}\mathbf{|}}^{\mathbf{2}}{\mathbf{\delta }}^{\mathbf{3}}\mathbf{\left(}\mathbf{5}\mathbf{r}\mathbf{\right)}\mathbf{d\tau }$ , where V is a cube of side 2, centered at origin and $\mathbf{b}\mathbf{=}\mathbf{4}\hat{\mathbf{y}}\mathbf{+}\mathbf{3}\hat{\mathbf{z}}$. 

(c) ${\mathbf{\int }}_{\mathbf{v}}{\mathbf{r}}^{\mathbf{4}}\mathbf{+}{\mathbf{r}}^{\mathbf{2}}\mathbf{(}\mathbf{r}\mathbf{·}\mathbf{c}\mathbf{)}\mathbf{+}{\mathbf{c}}^{\mathbf{4}}{\mathbf{\delta }}^{\mathbf{3}}\mathbf{(}\mathbf{r}\mathbf{-}\mathbf{c}\mathbf{)}\mathbf{d\tau }$ , where is a cube of side 6, about the origin, $\mathbf{c}\mathbf{=}\mathbf{5}\hat{\mathbf{x}}\mathbf{+}\mathbf{3}\hat{\mathbf{y}}\mathbf{+}\mathbf{2}\hat{\mathbf{z}}$ and c is its magnitude.

(d) ${\mathbf{\int }}_{\mathbf{v}}\mathbf{r}\mathbf{·}\mathbf{(}\mathbf{d}\mathbf{-}\mathbf{r}\mathbf{)}{\mathbf{\delta }}^{\mathbf{3}}\mathbf{(}\mathbf{e}\mathbf{-}\mathbf{r}\mathbf{)}\mathbf{d\tau }$, where $\mathbf{d}\mathbf{=}\mathbf{(}\mathbf{1}\mathbf{,}\mathbf{2}\mathbf{,}\mathbf{3}\mathbf{)}\mathbf{,}\mathbf{ }\mathbf{e}\mathbf{=}\mathbf{(}\mathbf{3}\mathbf{,}\mathbf{2}\mathbf{,}\mathbf{1}\mathbf{)}$ , and where v is a sphere of radius  1.5 centered at $(2,2,2)$.

Evaluate the integral

                                             $\mathit{J}\mathbf{=}{\mathbf{\int }}_{\mathbf{v}}{\mathit{e}}^{\mathbf{-}\mathbf{r}}\mathbf{(}\mathbf{\nabla }\mathbf{·}\frac{\hat{\mathbf{r}}}{{\mathbf{r}}^{\mathbf{2}}}\mathbf{)}\mathit{d}\mathit{\tau }$,

where V is a sphere of radius R centered at origin by two different methods as in Ex. 1.16. .

(a) Let${\mathit{F}}_{\mathbf{1}}\mathbf{=}{\mathit{x}}^{\mathbf{2}}\mathit{i}$and${\mathit{F}}_{\mathbf{2}}\mathbf{=}\mathit{x}\mathit{i}\mathbf{+}\mathit{y}\mathit{j}\mathbf{+}\mathit{z}\mathit{k}$Calculate the divergence and curl of${\mathit{F}}_{\mathbf{1}}$and${\mathit{F}}_{\mathbf{2}}$which one can be written as the gradient of a scalar? Find a scalar potential that does the job. Which one can be written as the curl of a vector? Find a suitable vector potential.

(b) Show that${\mathit{F}}_{\mathbf{3}}\mathbf{=}\mathit{y}\mathit{z}\mathbf{ }\mathit{i}\mathbf{+}\mathit{z}\mathit{x}\mathbf{ }\mathit{j}\mathbf{+}\mathit{x}\mathit{y}\mathbf{ }\mathit{k}$can be written both as the gradient of a scalar and as the curl of a vector. Find scalar and vector potentials for this function.

For Theorem 1, show that $\mathbf{\left(}\mathbf{d}\mathbf{\right)}\mathbf{\Rightarrow }\mathbf{\left(}\mathbf{a}\mathbf{\right)}\mathbf{,}\mathbf{ }\mathbf{\left(}\mathbf{a}\mathbf{\right)}\mathbf{\Rightarrow }\mathbf{\left(}\mathbf{c}\mathbf{\right)}\mathbf{,}\mathbf{ }\mathbf{\left(}\mathbf{c}\mathbf{\right)}\mathbf{\Rightarrow }\mathbf{\left(}\mathbf{b}\mathbf{\right)}\mathbf{,}\mathbf{ }\mathbf{\left(}\mathbf{b}\mathbf{\right)}\mathbf{\Rightarrow }\mathbf{\left(}\mathbf{c}\mathbf{\right)}$ and  $\left(c\right)\mathbf{\Rightarrow }\left(a\right)$

For Theorem 2, show that $\left(d\right)\Rightarrow \left(a\right),\left(a\right)\Rightarrow \left(c\right),\left(c\right)\Rightarrow \left(b\right),\left(b\right)\Rightarrow \left(c\right)\mathrm{and} \left(c\right)\Rightarrow \left(a\right)$

For Theorem 2, show that $\left(d\right)\Rightarrow \left(a\right)$, $\left(a\right)\Rightarrow \left(c\right)$ , $\left(c\right)\Rightarrow \left(b\right)$, $\left(b\right)\Rightarrow \left(c\right)$and $\left(c\right)\Rightarrow \left(a\right)$

(a) Which of the vectors in Problem 1.15 can be expressed as the gradient of a scalar? Find a scalar function that does the job.

(b) Which can be expressed as the curl of a vector? Find such a vector.

Check the divergence theorem for the function

 $\mathit{v}\mathbf{=}\left(r^{2}cos\theta \right)\stackrel{\mathbf{\wedge }}{\mathbf{r}}\mathbf{+}\left(r^{2}cos\varphi \right)\stackrel{\mathbf{\wedge }}{\mathbf{\theta }}\mathbf{+}{\mathit{r}}^{\mathbf{2}}\mathit{c}\mathit{o}\mathit{s}\mathit{\theta }\mathit{s}\mathit{i}\mathit{n}\mathit{\varphi }\stackrel{\mathbf{\wedge }}{\mathbf{\varphi }}$

using as your volume one octant of the sphere of radius R (Fig. 1.48). Make sure you include the entire surface. [Answer: $\mathit{\pi }{\mathit{R}}^{\mathbf{4}}\mathbf{/}\mathbf{4}$ ]

Check Stokes' theorem using the function $\mathit{v}\mathbf{=}\mathit{a}\mathit{y}\mathit{i}\mathbf{ }\mathbf{+}\mathit{b}\mathit{x}\mathbf{ }\mathit{j}$ (a and b are constants) and the circular path of radius R, centered at the origin in the xyplane. [Answer: $\mathit{\pi }{\mathit{R}}^{\mathbf{2}}\left(b-a\right)$ ], 

Compute the line integral of

 $v=6i+y^{2}j+\left(3y+z\right)k$

along the triangular path shown in Fig. 1.49. Check your answer using Stokes' theorem. [Answer: 8/3]

Compute the line integral of

  $\mathit{v}\mathbf{=}\mathbf{(}\mathit{r}\mathbf{ }\mathit{c}\mathit{o}{\mathit{s}}^{\mathbf{2}}\mathit{\theta }\mathbf{)}\hat{\mathbf{r}}\mathbf{-}\mathbf{(}\mathit{r}\mathbf{ }\mathit{c}\mathit{o}\mathit{s}\mathit{\theta }\mathit{s}\mathit{i}\mathit{n}\mathit{\theta }\mathbf{)}\hat{\mathbf{\theta }}\mathbf{+}\mathbf{3}\mathit{r}\hat{\mathbf{\varphi }}$

around the path shown in Fig. 1.50 (the points are labeled by their Cartesian coordinates).Do it either in cylindrical or in spherical coordinates. Check your answer, using Stokes' theorem. [Answer: 3rr /2]

Check Stokes' theorem for the function $v=yi$, using the triangular surface shown in Fig. 1.51. [Answer:  $a^2$],

Check the divergence theorem for the function

$\mathit{v}\mathbf{=}{\mathit{r}}^{\mathbf{2}}\mathit{s}\mathit{i}\mathit{n}\mathit{\varphi }\mathbf{ }\widehat{\mathbf{r}\mathbf{ }}\mathbf{+}\mathbf{4}{\mathit{r}}^{\mathbf{2}}\mathbf{ }\mathit{c}\mathit{o}\mathit{s}\mathbf{ }\mathit{\theta }\hat{\mathbf{\theta }}\mathbf{+}{\mathit{r}}^{\mathbf{2}}\mathbf{ }\mathit{t}\mathit{a}\mathit{n}\mathbf{ }\mathit{\theta }\hat{\mathbf{\varphi }}$

using the volume of the "ice-cream cone" shown in Fig. 1.52 (the top surface is spherical, with radius R and centered at the origin). [Answer: $\mathit{\pi }{\mathit{R}}^{\mathbf{4}}\mathbf{/}\mathbf{12}\mathbf{ }\mathbf{(}\mathbf{2}\mathit{\pi }\mathbf{+}\mathbf{3}\sqrt{\mathbf{3}}\mathbf{)}$]

Here are two cute checks of the fundamental theorems:

(a) Combine Corollary 2 to the gradient theorem with Stokes' theorem ( $v=\nabla T$, in this case). Show that the result is consistent with what you already knew about second derivatives.

(b) Combine Corollary 2 to Stokes' theorem with the divergence theorem. Show that the result is consistent with what you already knew.

Although the gradient, divergence, and curl theorems are the fundamental integral theorems of vector calculus, it is possible to derive a number of corollaries from them. Show that:

(a) $\underset{v}{\int }\left(\nabla T\right)d\tau =\underset{s}{\oint }Tda$. [Hint: Let v = cT, where c is a constant, in the divergence theorem; use the product rules.]

(b) $\underset{v}{\int }\left(\nabla \times v\right)d\tau =\underset{s}{\oint }v\times da$. [Hint: Replace v by (v x c) in the divergence

theorem.]

(c) $\underset{v}{\int }\left(T{\nabla }^{2}U+\nabla T\cdot \nabla U\right)d\tau =\underset{s}{\oint }T\nabla U\cdot da$ . [Hint: Let  in the

divergence theorem.]

(d) $\underset{v}{\int }\left(U{\nabla }^{2}T+\nabla U\cdot \nabla V\right)d\tau =\underset{s}{\oint }U\nabla T\cdot da$. [Comment: This is sometimes

called Green's second identity; it follows from (c), which is known as

Green's identity.]

(e)  $\underset{S}{\int }\nabla T\times da=\underset{P}{\oint }T\cdot dl$ [Hint: Let v = cT in Stokes' theorem.]

The integral

                                       $\mathit{a}\mathbf{=}{\mathbf{\int }}_{\mathbf{s}}\mathit{d}\mathit{a}$

is sometimes called the vector area of the surface S. If S happens to be flat, then lal is the ordinary (scalar) area, obviously.

(a) Find the vector area of a hemispherical bowl of radius R.

(b) Show that a= 0 for any closed surface. [Hint: Use Prob. 1.6la.]

(c) Show that a is the same for all surfaces sharing the same boundary.

(d) Show that

where the integral is around the boundary line. [Hint: One way to do it is to draw the cone subtended by the loop at the origin. Divide the conical surface up into infinitesimal triangular wedges, each with vertex at the origin and opposite side dl, and exploit the geometrical interpretation of the cross product (Fig. 1.8).]

(e) Show that 

                                    $\mathbf{\oint }\mathbf{c}\mathbf{\cdot }\mathbf{r}\mathbf{=}\mathbf{a}\mathbf{\times }\mathbf{c}$

for any constant vector c. [Hint: Let T = c · r in Prob. 1.61e.] (

(a) Find the divergence of the function

v=r^r$\mathit{v}\mathbf{=}\frac{\hat{\mathbf{r}}}{\mathbf{r}}$

v=r^rFirst compute it directly, as in Eq. 1.84. Test your result using the divergence theorem, as in Eq. 1.85. Is there a delta function at the origin, as there was for $\frac{\hat{\mathbf{r}}}{{\mathbf{r}}^{\mathbf{2}}}$? What is the general formula for the divergence of ${\mathbf{r}}^{\mathbf{n}} \hat{\mathbf{r}}$ ? [Answer: $\mathbf{\nabla }\mathbf{.}\mathbf{(}{\mathbf{r}}^{\mathbf{n}}\mathbf{ }\hat{\mathbf{r}}\mathbf{)}\mathbf{=}\mathbf{(}\mathbf{n}\mathbf{+}\mathbf{2}\mathbf{)}{\mathbf{r}}^{\mathbf{n}\mathbf{-}\mathbf{1}}$] unless $\mathbf{n}\mathbf{=}\mathbf{-}\mathbf{2}$ , in which case it is $\mathbf{4}{\mathbf{\pi \delta }}^{\mathbf{3}}$for$\mathbf{n}\mathbf{<}\mathbf{2}$ the divergence is ill-defined at the origin.]

(b) Find the curl of ${\mathbf{r}}^{\mathbf{n}}\mathbf{ }\hat{\mathbf{r}}$ . Test your conclusion using Prob. 1.61b. [Answer:$\mathbf{\nabla }\mathbf{\times }\mathbf{(}{\mathbf{r}}^{\mathbf{n}}\mathbf{ }\hat{\mathbf{r}}\mathbf{ }\mathbf{)}\mathbf{=}\mathbf{0}$]

In case you're not persuaded that ${\mathbf{\nabla }}^{\mathbf{2}}\left(\frac{1}{r}\right)\mathbf{=}\mathbf{-}\mathbf{4}\mathit{\pi }{\mathit{\delta }}^{\mathbf{3}}\mathbf{\left(}\mathbf{r}\mathbf{\right)}$ (Eq. 1.102) with ${\mathbf{r}}^{\mathbf{\text{'}}}\mathbf{=}\mathbf{0}$ for simplicity), try replacing r  by  $\sqrt{{\mathbf{r}}^{\mathbf{2}}\mathbf{+}{\mathbf{\epsilon }}^{\mathbf{2}}}$ , and watching what happens as $\mathit{\epsilon }\mathbf{\to }{\mathbf{0}}^{\mathbf{16}}$ Specifically, let                                              $\mathbf{D}\mathbf{(}\mathbf{r}\mathbf{,}\mathbf{\epsilon }\mathbf{)}\mathbf{=}\frac{\mathbf{1}}{\mathbf{4}\mathbf{\pi }}{\mathbf{\nabla }}^{\mathbf{2}}\frac{\mathbf{1}}{\sqrt{{\mathbf{r}}^{\mathbf{2}}\mathbf{+}{\mathbf{\epsilon }}^{\mathbf{2}}}}$

To demonstrate that this goes to ${\mathit{\delta }}^{\mathbf{3}}\mathbf{\left(}\mathit{r}\mathbf{\right)}$ as $\mathit{\epsilon }\mathbf{\to }\mathbf{0}$:

(a) Show that $\mathbf{D}\mathbf{=}\mathbf{(}\mathbf{r}\mathbf{,}\mathbf{\epsilon }\mathbf{)}\mathbf{=}\mathbf{(}\mathbf{3}{\mathbf{\epsilon }}^{\mathbf{2}}\mathbf{/}\mathbf{4}\mathbf{\pi }\mathbf{)}{\mathbf{(}{\mathbf{r}}^{\mathbf{2}}\mathbf{+}{\mathbf{\epsilon }}^{\mathbf{2}}\mathbf{)}}^{\mathbf{-}\mathbf{5}\mathbf{/}\mathbf{2}}$

(b) Check that$\mathbf{D}\mathbf{(}\mathbf{0}\mathbf{,}\mathbf{\epsilon }\mathbf{)}\mathbf{\to }\mathbf{\infty }$ , as $\mathit{\epsilon }\mathbf{\to }\mathbf{0}$

(c)Check that $\mathbf{D}\mathbf{(}\mathbf{r}\mathbf{,}\mathbf{\epsilon }\mathbf{)}\mathbf{\to }\mathbf{0}$ , as $\mathbf{\epsilon }\mathbf{\to }\mathbf{0}$, for all $\mathbf{r}\mathbf{\ne }\mathbf{0}$

(d) Check that the integral of $\mathbf{D}\mathbf{(}\mathbf{r}\mathbf{,}\mathbf{\epsilon }\mathbf{)}$ over all space is 1.