4 02/07 F

1. Derive $\vec n {\rm d}S$ in terms of ${\rm d}\theta$ and ${\rm d}\phi$, where $(r,\theta,\phi)$ are spherical coordinates. Assume that the surface is given by some relationship $F(r,\theta,\phi)$ = constant. Use the formulae given earlier in class for $\vec n {\rm d}S$ in terms of two parameters $u$ and $v$. The formula requires you to differentiate $\vec r$ with respect to the parameters. Now in spherical,
   
   $$\vec r = r \hat\imath_r$$
   
   
   From class, the derivatives of $\hat\imath_r$ are
   
   $$\frac{\partial \hat\imath_r}{\partial r} = 0 \qquad \fra... ...al \hat\imath_r}{\partial\phi} = \sin\theta \hat\imath_\phi$$
   
   
   To get the dervatives of $r$, note that certainly, on the surface, $r$ will be some function $r(\theta,\phi)$. To get formulae for its derivatives, differentiate the constant function $F$:
   
   $$\frac{\partial F}{\partial r} \frac{\partial r}{\partial \... ...l r}{\partial \phi} + \frac{\partial F}{\partial \theta} = 0$$
   
   
   Write the obtained expression for $\vec n {\rm d}S$ in terms of the gradient of F. (The expression for the gradient of F in spherical coordinates can be found in mathematical handbooks.) Compare with the Eulerian expression,
   
   $$\vec n {\rm d}S = \frac{\nabla F}{F_z} {\rm d}x {\rm d}y$$
   
   
   as derived in class. Here ${\rm d}x{\rm d}y$ can be denoted symbolically as ${\rm d}S_z$: it is the area of a surface of constant $z$ of dimensions ${\rm d}x\times{\rm d}y$. (In other words, it is the projection of surface element ${\rm d}S$ on a surface of constant $z$.) What is the equivalent to ${\rm d}S_z$ in your spherical coordinates expression?

2. 1st Ed: p160, q38, 2nd Ed: p183, q38. Simplify as much as possible. Sketch each surface, taking the $z$-axis upwards.

3. Finish finding the derivatives of the unit vectors of the spherical coordinate system using the class formulae. Then finish 1st Ed p160 q47, 2nd Ed p183 q47, as started in class, by finding the acceleration. Note that the metric indices $h_i$ for spherical coordinates are in mathematical handbooks. Also,
   
   $$\frac{\partial {\hat\imath}_i}{\partial u_i} = \frac{1}{h_... ...frac{1}{h_i} \frac{\partial h_j}{\partial u_i} {\hat\imath}_j$$
   
   

4. Express the acceleration in terms of the spherical velocity components $v_r,v_\theta,v_\phi$ and their first time derivatives, instead of time derivatives of position coordinates. Like $a_r = \dot v_r + \ldots$, etc. This is how you do it in fluid mechanics, where time-derivatives of particle position coordinates are normally not used. (So, get rid of the position coordinates with dots on them in favor of the velocity components.)