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Next: Ellipsoidal Potential Theory Up: Non-Cartesian Coordinates Previous: Spherical Coordinates

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?

    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

      $\displaystyle \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.

  6. The electric field generated by a $ z$ -directed electric dipole of moment $ p$ , located at the origin, is

    $\displaystyle {\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}$ .

  7. 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.

  8. 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.

next up previous
Next: Ellipsoidal Potential Theory Up: Non-Cartesian Coordinates Previous: Spherical Coordinates
Richard Fitzpatrick 2016-01-22