Vector Analysis

A uniform current density $\mathbf{J}\mathbf{=}{\mathbf{J}}_{\mathbf{0}}\hat{\mathbf{z}}$ fills a slab straddling the $\mathbf{yz}$  plane, from $\mathbf{x}\mathbf{=}\mathbf{-}\mathbf{a}\mathbf{ }\mathbf{to}\mathbf{ }\mathbf{x}\mathbf{=}\mathbf{+}\mathbf{a}$ to . A magnetic dipole $\mathbf{m}\mathbf{=}{\mathbf{m}}_{\mathbf{0}}\mathbf{x}$ is situated at the origin. 

(a) Find the force on the dipole, using Eq. 6.3. 

(b) Do the same for a dipole pointing in the direction: $\mathbf{m}\mathbf{=}{\mathbf{m}}_{\mathbf{0}}\mathbf{y}$ . 

(c) In the electrostatic case, the expressions $\mathbf{F}\mathbf{=}\mathbf{\nabla }\left(p.E\right)$ and $\mathbf{F}\mathbf{=}\left(p.\nabla \right)\mathbf{E}$are equivalent (prove it), but this is not the case for the magnetic analogs (explain why). As an example, calculate $\left(m.\nabla \right)$ for the configurations in (a) and (b).

Using the definitions in Eqs. 1.1 and 1.4, and appropriate diagrams, show that the dot product and cross product are distributive, 

a) when the three vectors are coplanar;

b) in the general case.

Is the cross product associative? 

       $\mathbf{\text{(A×B)×C=A×(B×C)}}$

If so, prove it; if not, provide a counterexample (the simpler the better).

Find the angle between the body diagonals of a cube. 

Use the cross product to find the components of the unit vector $\hat{\mathit{n}}$ perpendicular to the shaded plane in Fig. 1.11.

Prove the BAC-CAB rule by writing out both sides in component form.

Prove that. $\mathbf{[}\mathbf{A}\mathbf{\times }\left(B\times C\right)\mathbf{]}\mathbf{+}\mathbf{ }\mathbf{[}\mathbf{B}\mathbf{\times }\left(C\times A\right)\mathbf{]}\mathbf{+}\mathbf{[}\mathbf{C}\mathbf{\times }\left(A\times B\right)\mathbf{]}\mathbf{=}\mathbf{0}$ Under what conditions does $\mathit{A}\mathbf{\times }\left(B\times C\right)\mathbf{=}\left(A\times B\right)\mathbf{\times }\mathit{C}$ ?

Find the separation vector r from the source point (2,8,7) to the field point ( 4,6,8). Determine its magnitude ( r ), and construct the unit vector  $\hat{\mathit{r}}$

(a) Prove that the two-dimensional rotation matrix (Eq.1.29) preserves dot products.
 (That is, show that $\overline{{\mathbf{A}}_{\mathbf{y}}{\mathbf{B}}_{\mathbf{y}}}\mathbf{+}\overline{{\mathbf{A}}_{\mathbf{z}}{\mathbf{B}}_{\mathbf{z}}}\mathbf{=}{\mathit{A}}_{\mathbf{y}}{\mathit{B}}_{\mathbf{y}}\mathbf{+}{\mathit{A}}_{\mathbf{z}}{\mathit{B}}_{\mathbf{z}}$.)
(b) What constraints must the elements (Rij ) of the three-dimensional rotation matrix
(Eq.1.30) satisfy, in order to preserve the length of A (for all vectors $\overrightarrow{\mathbf{A}}$ )?

Find the transformation matrix R that describes a rotation by 120° about an axis from the origin through the point (1, 1, 1). The rotation is clockwise as you look down the axis toward the origin.

(a) How do the components of a vectoii transform under a translation of coordinates (X= x, y = y- a, z = z, Fig. 1.16a)?

(b) How do the components of a vector transform under an inversion of coordinates (X= -x, y = -y, z = -z, Fig. 1.16b)?

(c) How do the components of a cross product (Eq. 1.13) transform under inversion? [The cross-product of two vectors is properly called a pseudovector because of this "anomalous" behavior.] Is the cross product of two pseudovectors a vector, or a pseudovector? Name two pseudovector quantities in classical mechanics.

(d) How does the scalar triple product of three vectors transform under inversions? (Such an object is called a pseudoscalar.)

Find the gradients of the following functions: 

(a)   $f(x,y,z) =x$⁴ +$y$³   + $z$⁴

(b) $f(x,y,z)=$$x$² y³ z⁴

(c) $f(x,y,z)=e$^x $\sin \left(y\right) In \left(z\right)$

The height of a certain hill (in feet) is given by $h(x,y) = 10(2xy-3x$² $-4y$² $-18x+28y+12)$

Where y is the distance (in miles) north, x the distance east of South Hadley. 

(a) Where is the top of hill located? 

(b) How high is the hill?

 (c) How steep is the slope (in feet per mile) at a point 1 mile north and one mile east of South Hadley? In what direction is the slope steepest, at that point?

The height of a certain hill (in feet) is given by

 $\mathit{h}\left(x,y\right)\mathbf{=}\mathbf{10}\left(2xy-3x^2-4y^2-18x+28y+12\right)$

Where y is the distance (in miles) north, x the distance east of South Hadley.

 (a) Where is the top of hill located?

(b) How high is the hill?

(c) How steep is the slope (in feet per mile) at a point 1 mile north and one mileeast of South Hadley? In what direction is the slope steepest, at that point?

Find the gradients of the following functions:

(a)  $\mathit{f}\left(x,y,z\right)\mathbf{=}{\mathit{x}}^{\mathbf{4}}\mathbf{+}{\mathit{y}}^{\mathbf{3}}\mathbf{+}{\mathit{z}}^{\mathbf{4}}$

(b)  $\mathit{f}\left(x,y,z\right)\mathbf{=}{\mathit{x}}^{\mathbf{2}}{\mathit{y}}^{\mathbf{3}}{\mathit{z}}^{\mathbf{4}}$

(c) $\mathit{f}\left(x,y,z\right)\mathbf{=}{\mathit{e}}^{\mathbf{x}}\mathit{s}\mathit{i}\mathit{n}\left(y\right)\mathit{l}\mathit{n}\left(z\right)$ 

Let ${}^{\mathbf{r}}$ be the separation vector from a fixed point ${}^{\mathbf{(}\mathbf{x}\mathbf{\text{'}}\mathbf{,}\mathbf{y}\mathbf{\text{'}}\mathbf{,}\mathbf{ }\mathbf{z}\mathbf{\text{'}}\mathbf{)}}$ to the point $\mathbf{(}\mathbf{x}\mathbf{,}\mathbf{y}\mathbf{,}\mathbf{z}\mathbf{)}$, and let r be its length. Show that

(a) ${}^{\mathbf{\nabla }\mathbf{ }\mathbf{\left(}{\mathbf{r}}^{\mathbf{2}}\mathbf{\right)}\mathbf{=}\mathbf{2}\mathbf{r}}$

(b) ${}^{\mathbf{\nabla }\mathbf{ }\mathbf{(}\mathbf{1}\mathbf{/}\mathbf{r}\mathbf{)}\mathbf{=}\mathbf{-}\mathbf{r}\mathbf{/}{\mathbf{r}}^{\mathbf{2}}}$

(c) What is the general formula for ${}^{\mathbf{\nabla }\mathbf{ }\mathbf{(}\mathbf{ }{\mathbf{r}}^{\mathbf{n}}\mathbf{ }\mathbf{)}}$

Suppose that f is a function of two variables (y and z) only. Show that the gradient ${}^{\mathbf{\nabla }\mathbf{f}\mathbf{ }\mathbf{=}\mathbf{(}\mathbf{\partial }\mathbf{f}\mathbf{/}\mathbf{\partial }\mathbf{y}\mathbf{)}\widehat{\mathbf{ }\mathbf{y}}\mathbf{ }\mathbf{(}\mathbf{\partial }\mathbf{f}\mathbf{/}\mathbf{\partial }\mathbf{z}\mathbf{)}\mathbf{ }\hat{\mathbf{z}}}$ transforms as a vector under rotations, Eq 1.29. [Hint: ${}^{\mathbf{(}\mathbf{\partial }\mathbf{f}\mathbf{/}\mathbf{\partial }\overline{\mathbf{y}}\mathbf{)}\mathbf{=}\mathbf{(}\mathbf{\partial }\mathbf{f}\mathbf{/}\mathbf{\partial }\mathbf{y}\mathbf{)}\mathbf{(}\mathbf{\partial }\mathbf{f}\mathbf{/}\mathbf{\partial }\overline{\mathbf{y}}\mathbf{)}\mathbf{+}\mathbf{(}\mathbf{\partial }\mathbf{f}\mathbf{/}\mathbf{\partial }\mathbf{z}\mathbf{)}\mathbf{(}\mathbf{\partial }\mathbf{z}\mathbf{/}\mathbf{\partial }\overline{\mathbf{y}}\mathbf{)}\mathbf{,}}$and the analogous formula for ${}^{\mathbf{\partial }\mathbf{f}\mathbf{/}\mathbf{\partial }\overline{\mathbf{z}}}$. We know that ${}^{\overline{\mathbf{y}}\mathbf{ }\mathbf{=}\mathbf{ycos\varphi }\mathbf{ }\mathbf{+}\mathbf{zsin\varphi }}$ and ${}^{\overline{\mathbf{z}}\mathbf{ }\mathbf{=}\mathbf{-}\mathbf{ycos}\mathbf{\varnothing }\mathbf{ }\mathbf{+}\mathbf{ }\mathbf{zcos}\mathbf{\varnothing }\mathbf{;}}$”solve” these equations for y and z (as functions of ${}^{\overline{\mathbf{y}}}$ and ${}^{\overline{\mathbf{z}}}$ (as functions of ${}^{\overline{\mathbf{y}}}$ and ${}^{\overline{\mathbf{z}}}$), and compute the needed derivatives ${}^{\mathbf{\partial }\mathbf{f}\mathbf{/}\mathbf{\partial }{\overline{\mathbf{y}}}_{\mathbf{,}}\mathbf{ }\mathbf{\partial }\mathbf{z}\mathbf{/}\mathbf{\partial }\overline{\mathbf{y}}\mathbf{ }}$, etc]

Calculate the divergence of the following vector functions:

$$\begin{gathered} \mathbf{\left(}\mathit{a}\mathbf{\right)}\mathbf{ }{\mathit{v}}_{\mathbf{a}}\mathbf{=}{\mathit{x}}^{\mathbf{2}}\hat{\mathbf{x}}\mathbf{+}\mathbf{3}\mathit{x}{\mathit{z}}^{\mathbf{2}}\hat{\mathbf{y}}\mathbf{ }\mathbf{-}\mathbf{2}\mathit{x}\mathit{z}\hat{\mathbf{z}} \\ \mathbf{\left(}\mathit{b}\mathbf{\right)}\mathbf{ }{\mathit{v}}_{\mathbf{b}}\mathbf{=}\mathit{x}\mathit{y}\hat{\mathbf{x}}\mathbf{+}\mathbf{2}\mathit{y}\mathit{z}\hat{\mathbf{y}}\mathbf{+}\mathbf{3}\mathit{z}\mathit{x}\hat{\mathbf{z}} \\ \mathbf{\left(}\mathit{c}\mathbf{\right)}\mathbf{ }{\mathit{v}}_{\mathbf{c}}\mathbf{=}{\mathit{y}}^{\mathbf{2}}\hat{\mathbf{x}}\mathbf{+}\mathbf{(}\mathbf{2}\mathit{x}\mathit{y}\mathbf{+}{\mathit{z}}^{\mathbf{2}}\mathbf{)}\mathbf{+}\mathbf{2}\mathit{y}\mathit{z}\hat{\mathbf{z}} \end{gathered}$$

Sketch the vector function

$\mathbf{v}\mathbf{=}\frac{\hat{\mathbf{r}}}{{\mathbf{r}}^{\mathbf{2}}}$

and compute its divergence. The answer may surprise you ... can you explain it?

In two dimensions, show that the divergence transforms as a scalar under rotations. [Hint: Use Eq. 1.29 to determine ${}^{\overline{{\mathbf{v}}_{\mathbf{y}}}}$ and${}^{{\overline{\mathbf{v}}}_{\mathbf{z}}}$ and the method of Prob. 1.14 to calculate the derivatives. Your aim is to show that  

$\frac{\mathbf{\partial }\overline{{\mathbf{v}}_{\mathbf{y}}}}{\mathbf{\partial }\mathbf{y}}\mathbf{+}\frac{\mathbf{\partial }\overline{{\mathbf{v}}_{\mathbf{y}}}}{\mathbf{\partial }\mathbf{z}}\mathbf{=}\frac{\mathbf{\partial }{\mathbf{v}}_{\mathbf{ }\mathbf{y}}}{\mathbf{\partial }\mathbf{y}}\mathbf{+}\frac{\mathbf{\partial }{\mathbf{v}}_{\mathbf{z}}}{\mathbf{\partial }\mathbf{z}}$

Calculate the curls of the vector functions in Prob. 1.15.

Draw a circle in the xy plane. At a few representative points draw the vector v tangent to the circle, pointing in the clockwise direction. By comparing adjacent vectors, determine the sign of $\mathbf{\partial }{\mathit{v}}_{\mathbf{x}}\mathbf{ }\mathit{l}\mathbf{\partial }\mathit{y}$and $\mathbf{\partial }{\mathit{v}}_{\mathbf{y}}\mathbf{ }\mathit{l}\mathbf{ }\mathbf{\partial }\mathit{x}$According to Eq. 1.41, then, what is the direction of  $\mathbf{\nabla }\mathbf{\times }\mathit{v}$? Explain how this example illustrates the geometrical interpretation of the curl.

Construct a vector function that has zero divergence and zero curl everywhere. (A constant will do the job, of course, but make it something a little more interesting than that!)

(a) If A and B are two vector functions, what does the expression $\left(\mathbf{A}·\nabla \right)\mathbf{B}$  mean?(That is, what are its x, y, and z components, in terms of the Cartesian componentsof A, B, and V?)

(b) Compute $\left(\hat{\mathbf{r}}·\nabla \right)\hat{\mathbf{r}}$ , where $\hat{\mathbf{r}}$  is the unit vector defined in Eq. 1.21.

(c) For the functions in Prob. 1.15, evaluate  $\left({\mathbf{v}}_{\mathbf{a}}·\nabla \right){\mathbf{v}}_{\mathbf{b}}$.

Prove product rules (i), (iv), and (v)

(a) If A and B are two vector functions, what does the expression $\left(\mathbf{A}·\nabla \right)\mathbf{B}$ mean?(That is, what are its x, y, and z components, in terms of the Cartesian componentsof A, B, and V?)

(b) Compute  $\left(\hat{\mathbf{r}}·\nabla \right)\hat{\mathbf{r}}$, where r  is the unit vector defined in Eq. 1.21.

(c) For the functions in Prob. 1.15, evaluate $\left({\mathbf{v}}_{\mathbf{a}}·\nabla \right){\mathbf{v}}_{\mathbf{b}}$ .

(For masochists only.) Prove product rules (ii) and (vi). Refer to Prob. 1.22 for the definition of $\left(A . \overline{V}\right)\mathbf{ }\mathit{B}$.

Derive the three quotient rules.

(a) Check product rule (iv) (by calculating each term separately) for the functions

 $\mathit{A}\mathbf{=}\mathit{x}\hat{\mathbf{x}}\mathbf{+}\mathbf{2}\mathit{y}\hat{\mathbf{y}}\mathbf{+}\mathbf{3}\mathit{z}\mathbf{ }\hat{\mathbf{z}}\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathit{B}\mathbf{=}\mathbf{3}\mathit{x}\hat{\mathbf{x}}\mathbf{-}\mathbf{2}\mathit{x}\mathbf{ }\hat{\mathbf{y}}$                         

(b) Do the same for product rule (ii).

(c) Do the same for rule (vi).

Calculate the Laplacian of the following functions:

(a) 

(b)

(c) .

(d)

Calculate the Laplacian of the following functions:

$$\begin{gathered} \left(a\right){\mathbf{T}}_{\mathbf{a}}\mathbf{=}{\mathbf{x}}^{\mathbf{2}}\mathbf{+}\mathbf{2}\mathbf{xy}\mathbf{+}\mathbf{3}\mathbf{z}\mathbf{+}\mathbf{4} \\ \left(b\right)\mathbf{ }{\mathbf{T}}_{\mathbf{b}}\mathbf{=}\mathbf{sinx}\mathbf{ }\mathbf{siny}\mathbf{ }\mathbf{sinz} \\ \left(c\right)\mathbf{ }{\mathbf{T}}_{\mathbf{c}}\mathbf{=}{\mathbf{e}}^{\mathbf{-}\mathbf{5}}\mathbf{ }\mathbf{sin}\mathbf{4}\mathbf{y}\mathbf{ }\mathbf{cos}\mathbf{ }\mathbf{3}\mathbf{z}\mathbf{.} \\ \left(d\right)\mathbf{ }\mathbf{v}\mathbf{=}{\mathbf{x}}^{\mathbf{2}}\mathbf{ }\hat{\mathbf{x}}\mathbf{+}\mathbf{3}{\mathbf{xz}}^{{\mathbf{2}}^{\mathbf{ }}}\hat{\mathbf{y}}\mathbf{+}\mathbf{-}\mathbf{2}\mathbf{xz}\hat{\mathbf{z}} \end{gathered}$$ 

Prove that the divergence of a curl is always zero. Check it for function  ${\mathbf{V}}_{\mathbf{a}}$ in Prob. 1.15.

Prove that the curl of a gradient is always zero. Check it for function(b) in Pro b. 1.11.

Calculate the volume integral of the function $T=z$² over the tetrahedron with comers at (0,0,0), (1,0,0), (0,1,0), and (0,0,1). 

Calculate the volume integral of the function $\mathit{T}\mathbf{=}{\mathit{z}}^{\mathbf{2}}$ over the tetrahedron with comers at (0,0,0), (1,0,0), (0,1,0), and (0,0,1).

Calculate the line integral of the function $\mathbf{v}\mathbf{=}{\mathbf{x}}^{\mathbf{2}}\mathbf{i}\mathbf{+}\mathbf{2}\mathbf{yx}\mathbf{ }\mathbf{j}\mathbf{+}{\mathbf{y}}^{\mathbf{2}}\mathbf{k}$ from the origin to the point (1,1,1) by three different routes:

(a)   $\mathbf{(}\mathbf{0}\mathbf{,}\mathbf{0}\mathbf{,}\mathbf{0}\mathbf{)}\mathbf{\to }\mathbf{(}\mathbf{1}\mathbf{,}\mathbf{0}\mathbf{,}\mathbf{0}\mathbf{)}\mathbf{\to }\mathbf{(}\mathbf{1}\mathbf{,}\mathbf{1}\mathbf{,}\mathbf{0}\mathbf{)}\mathbf{\to }\mathbf{(}\mathbf{1}\mathbf{,}\mathbf{1}\mathbf{,}\mathbf{1}\mathbf{)}\mathbf{.}$

(b)  $\mathbf{(}\mathbf{0}\mathbf{,}\mathbf{0}\mathbf{,}\mathbf{0}\mathbf{)}\mathbf{\to }\mathbf{(}\mathbf{0}\mathbf{,}\mathbf{0}\mathbf{,}\mathbf{1}\mathbf{)}\mathbf{\to }\mathbf{(}\mathbf{0}\mathbf{,}\mathbf{1}\mathbf{,}\mathbf{1}\mathbf{)}\mathbf{\to }\mathbf{(}\mathbf{1}\mathbf{,}\mathbf{1}\mathbf{,}\mathbf{1}\mathbf{)}\mathbf{.}$

(c) The direct straight line.

(d) What is the line integral around the closed loop that goes out along path (a) and back along path (b)?

Calculate the surface integral of the function in Ex. 1.7, over the bottom of the box. For consistency, let "upward" be the positive direction. Does the surface integral depend only on the boundary line for this function? What is the total flux over the closed surface of the box (including the bottom)? [Note: For the closed surface, the positive direction is "outward," and hence "down," for the bottom face.]

Check the fundamental theorem for gradients, using $\mathbf{T}\mathbf{=}{\mathbf{x}}^{\mathbf{2}}\mathbf{+}\mathbf{4}\mathbf{xy}\mathbf{+}\mathbf{2}{\mathbf{yz}}^{\mathbf{3}}$ the points $\mathbf{a}\mathbf{=}\mathbf{(}\mathbf{0}\mathbf{,}\mathbf{0}\mathbf{,}\mathbf{0}\mathbf{)}\mathbf{,}\mathbf{b}\mathbf{=}\mathbf{(}\mathbf{1}\mathbf{,}\mathbf{1}\mathbf{,}\mathbf{1}\mathbf{)}$ and the three paths in Fig. 1.28.

 $$\begin{gathered} \mathbf{\left(}\mathbf{a}\mathbf{\right)}\mathbf{=}\mathbf{(}\mathbf{0}\mathbf{.}\mathbf{0}\mathbf{.}\mathbf{0}\mathbf{)}\mathbf{\to }\mathbf{(}\mathbf{1}\mathbf{,}\mathbf{0}\mathbf{,}\mathbf{0}\mathbf{)}\mathbf{\to }\mathbf{(}\mathbf{1}\mathbf{,}\mathbf{1}\mathbf{,}\mathbf{0}\mathbf{)}\mathbf{\to }\mathbf{(}\mathbf{1}\mathbf{,}\mathbf{1}\mathbf{,}\mathbf{1}\mathbf{)}\mathbf{.} \\ \mathbf{\left(}\mathbf{b}\mathbf{\right)}\mathbf{=}\mathbf{(}\mathbf{0}\mathbf{.}\mathbf{0}\mathbf{.}\mathbf{0}\mathbf{)}\mathbf{\to }\mathbf{(}\mathbf{0}\mathbf{,}\mathbf{0}\mathbf{,}\mathbf{1}\mathbf{)}\mathbf{\to }\mathbf{(}\mathbf{0}\mathbf{,}\mathbf{1}\mathbf{,}\mathbf{1}\mathbf{)}\mathbf{\to }\mathbf{(}\mathbf{1}\mathbf{,}\mathbf{1}\mathbf{,}\mathbf{1}\mathbf{)}\mathbf{.} \end{gathered}$$ 

  (c) The parabolic path $\mathbf{z}\mathbf{=}{\mathbf{x}}^{\mathbf{2}}\mathbf{,}\mathbf{y}\mathbf{=}\mathbf{x}$

Test the divergence theorem for the function $\mathbf{v}\mathbf{=}\mathbf{\left(}\mathbf{xy}\mathbf{\right)}\mathbf{i}\mathbf{+}\mathbf{\left(}\mathbf{2}\mathbf{yz}\mathbf{\right)}\mathbf{j}\mathbf{+}\mathbf{\left(}\mathbf{3}\mathbf{zx}\mathbf{\right)}\mathbf{k}\mathbf{.}$ .Take as your volume the cube shown in Fig. 1.30, with sides of length 2.

Test Stokes' theorem for the function , using the triangular shaded area of Fig. 1.34.

Test Stokes' theorem for the function $\mathit{v}\mathbf{=}\left(xy\right)\mathit{i}\mathbf{ }\mathbf{+}\left(2yz\right)\mathbf{ }\mathit{j}\mathbf{+}\left(3zx\right)\mathbf{ }\mathit{k}$ , using the triangular shaded area of Fig. 1.34.

Question:  Check Corollary 1 by using the same function and boundary line as in Ex. 1.11, but integrating over the five faces of the cube in Fig. 1.35. The back of the cube is open.

(a) Show that

$\underset{\mathbf{s}}{\mathbf{\int }}\mathit{f}\mathbf{(}\mathbf{\nabla }\mathbf{\times }\mathit{A}\mathbf{)}\mathbf{-}\mathit{d}\mathit{a}\mathbf{=}\underset{\mathbf{s}}{\mathbf{\int }}\mathbf{[}\mathbf{A}\mathbf{\times }\left(\nabla f\right)\mathbf{]}\mathbf{-}\mathit{d}\mathit{a}\mathbf{+}\underset{\mathbf{P}}{\mathbf{\int }}\mathit{f}\mathit{A}\mathbf{-}\mathit{d}\mathit{l}$

(b) Show that

$\underset{\mathbf{v}}{\mathbf{\int }}\mathit{B}\mathbf{(}\mathbf{\nabla }\mathbf{\times }\mathbf{A}\mathbf{)}\mathbf{-}\mathit{d}\mathit{\tau }\mathbf{=}\underset{\mathbf{v}}{\mathbf{\int }}\mathit{A}\mathbf{-}\mathbf{(}\mathbf{\nabla }\mathbf{\times }\mathbf{B}\mathbf{)}\mathbf{-}\mathit{d}\mathit{\tau }\mathbf{+}\underset{\mathbf{s}}{\mathbf{\int }}\mathit{A}\mathbf{\times }\mathit{B}\mathbf{-}\mathit{d}\mathit{a}$

Express the unit vectors  in terms of x, y, z (that is, derive Eq. 1.64). Check your answers several ways         ( $r.r=$^? 1, $\theta .\varphi =$^?  $r x \theta =$^?$\varphi$),  .Also work out the inverse formulas, giving x, y, z in terms of  $r,\theta ,\varphi$ (and $\theta ,\varphi$).        

Express the unit vectors  $\mathit{r}\mathbf{,}\mathbf{ }\mathit{\theta }\mathbf{,}\mathbf{ }\mathit{\varphi }$in terms of x, y, z (that is, deriveEq. 1.64). Check your answers several ways ( $\mathit{r}\mathbf{·}\mathit{r}\stackrel{\mathbf{?}}{\mathbf{=}}\mathbf{1}$, $\mathit{\theta }\mathbf{·}\mathit{\varphi }\stackrel{\mathbf{?}}{\mathbf{=}\mathbf{0}}$, $\mathit{r}\mathbf{\times }\mathit{\theta }\stackrel{\mathbf{?}}{\mathbf{=}\mathbf{\varphi }}$).Also work out the inverse formulas, giving x, y, z in terms of  $\mathit{r}\mathbf{,}\mathbf{ }\mathit{\theta }\mathbf{,}\mathbf{ }\mathit{\varphi }$ (and $\mathbf{ }\mathit{\theta }\mathbf{,}\mathbf{ }\mathit{\varphi }$ ).

Question: Find formulas for  $\mathit{r}\mathbf{,}\mathit{\theta }\mathbf{,}\mathit{\varphi }$ in terms of x, y, z (the inverse, in other words, of Eq. 1.62)

(a) Check the divergence theorem for the function ${\mathit{v}}_{\mathbf{1}}\mathbf{=}{\mathit{r}}^{\mathbf{2}}\mathbf{ }\hat{\mathbf{r}}$ , using as your volume the sphere of radius R, centred at the origin.

(b) Do the same for  ${\mathit{v}}_{\mathbf{2}}\mathbf{=}\mathbf{\left(}\frac{\mathbf{1}}{{\mathbf{r}}^{\mathbf{2}}}\mathbf{\right)}\hat{\mathbf{r}}$. (If the answer surprises you, look back at Prob. 1.16)

Compute the divergence of the function $\mathit{v}\mathbf{=}\mathbf{(}\mathbf{r}\mathbf{ }\mathbf{c}\mathbf{o}\mathbf{s}\mathbf{ }\mathbf{\theta }\mathbf{)}\hat{\mathbf{r}}\mathbf{+}\mathbf{(}\mathit{r}\mathbf{ }\mathit{s}\mathit{i}\mathit{n}\mathbf{ }\mathit{\theta }\mathbf{)}\hat{\mathbf{\theta }}\mathbf{+}\mathit{r}\mathbf{ }\mathit{s}\mathit{i}\mathit{n}\mathit{\theta }\mathit{c}\mathit{o}\mathit{s}\mathit{\varphi }\hat{\mathbf{\varphi }}$

Check the divergence theorem for this function, using as your volume the inverted hemispherical bowl of radius R, resting on the xy plane and centered at the origin (Fig. 1.40).

Question:  (a) Find the divergence of the function  

$\mathit{v}\mathbf{=}\mathit{s}\mathbf{ }\mathbf{(}\mathbf{2}\mathbf{+}\mathit{s}\mathit{i}{\mathit{n}}^{\mathbf{2}}\mathit{\varphi }\mathbf{)}\hat{\mathbf{s}}\mathbf{+}\mathbf{\left(}\mathit{s}\mathit{s}\mathit{i}\mathit{n}\mathit{\varphi }\mathit{c}\mathit{o}\mathit{s}\mathit{\varphi }\mathbf{\right)}\hat{\mathbf{\varphi }}\mathbf{+}\mathbf{3}\mathit{z}\hat{\mathbf{z}}$

(b) Test the divergence theorem for this function, using the quarter-cylinder (radius 2, height 5) shown in Fig. 1.43.

(c) Find the curl of v.

Compute the gradient and Laplacian of the function$\mathit{T}\mathbf{=}\mathit{r}\mathbf{ }\mathbf{(}\mathit{c}\mathit{o}\mathit{s}\mathit{\theta }\mathbf{+}\mathit{s}\mathit{i}\mathit{n}\mathbf{ }\mathit{\theta }\mathit{c}\mathit{o}\mathit{s}\mathbf{ }\mathit{\varphi }\mathbf{)}$. Check the Laplacian by converting T to Cartesian coordinates and using Eq. 1.42. Test the gradient theorem for this function, using the path shown in Fig. 1.41, from (0, 0, 0) to (0, 0, 2).