Exercises

1. Find the Cartesian components of the basis vectors ${\bf e}_r$ , ${\bf e}_\theta$ , and ${\bf e}_z$ of the cylindrical coordinate system. Verify that the vectors are mutually orthogonal. Do the same for the basis vectors ${\bf e}_r$ , ${\bf e}_\theta$ , and ${\bf e}_\phi$ of the spherical coordinate system.

2. Use cylindrical coordinates to prove that the volume of a right cylinder of radius $a$ and length $l$ is $\pi\,a^{\,2}\,l$ . Demonstrate that the moment of inertia of a uniform cylinder of mass $M$ and radius $a$ about its symmetry axis is $(1/2)\,M\,a^{\,2}$ .

3. Use spherical coordinates to prove that the volume of a sphere of radius $a$ is $(4/3)\,\pi\,a^{\,3}$ . Demonstrate that the moment of inertia of a uniform sphere of mass $M$ and radius $a$ about an axis passing through its center is $(2/5)\,M\,a^{\,2}$ .

4. For what value(s) of $n$ is $\nabla\cdot(r^{\,n}\,{\bf e}_r)=0$ , where $r$ is a spherical coordinate?

5. For what value(s) of $n$ is $\nabla\times(r^{\,n}\,{\bf e}_r)={\bf0}$ , where $r$ is a spherical coordinate?

6. 1. Find a vector field ${\bf F} = F_r(r)\,{\bf e}_r$ satisfying $\nabla\cdot {\bf F} = r^{\,m}$ for $m\geq 0$ . Here, $r$ is a spherical coordinate.
   2. Use the divergence theorem to show that
      $$\int_V r^{\,m}\,dV = \frac{1}{m+3}\int_S r^{\,m+1}\,{\bf e}_r\cdot d{\bf S},$$
      where $V$ is a volume enclosed by a surface $S$ .
   3. Use the previous result (for $m=0$ ) to demonstrate that the volume of a right cone is one third the volume of the right cylinder having the same base and height.

7. The electric field generated by a $z$ -directed electric dipole of moment $p$ , located at the origin, is
   $${\bf E}({\bf r})= \frac{1}{4\pi\,\epsilon_0} \left[\frac{3\,({\bf e}_r\cdot {\bf p})\,{\bf e}_r-{\bf p}}{r^3}\right],$$
   where ${\bf p} = p\,{\bf e}_z$ , and $r$ is a spherical coordinate. Find the components of ${\bf E}({\bf r})$ in the spherical coordinate system. Calculate $\nabla\cdot{\bf E}$ and $\nabla\times {\bf E}$ .

8. Show that the parabolic cylindrical coordinates $u$ , $v$ , $z$ , defined by the equations $x=(u^{\,2}-v^{\,2})/2$ , $y=u\,v$ , $z=z$ , where $x$ , $y$ , $z$ are Cartesian coordinates, are orthogonal. Find the scale factors $h_u$ , $h_v$ , $h_z$ . What shapes are the $u={\rm constant}$ and $v={\rm constant}$ surfaces? Write an expression for $\nabla^{\,2} f$ in parabolic cylindrical coordinates.

9. Show that the elliptic cylindrical coordinates $\xi$ , $\eta$ , $z$ , defined by the equations $x=\cosh\xi\,\cos\eta$ , $y=\sinh\xi\,\sin\eta$ , $z=z$ , where $x$ , $y$ , $z$ are Cartesian coordinates, and $0\leq \xi\leq \infty$ , $-\pi<\eta \leq \pi$ , are orthogonal. Find the scale factors $h_\xi$ , $h_\eta$ , $h_z$ . What shapes are the $\xi={\rm constant}$ and $\eta={\rm constant}$ surfaces? Write an expression for $\nabla f$ in elliptical cylindrical coordinates.