5 10/02 F

1. 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. As noted in class,
   
   $$\frac{\partial {\hat\imath}_i}{\partial u_i} = \frac{1}{h_... ...frac{1}{h_i} \frac{\partial h_j}{\partial u_i} {\hat\imath}_j$$
   
   

2. 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 all position coordinates with dots on them in favor of the velocity components and position coordinates without dots.) Hint: you may want to differentiate the expressions for the velocity components with respect to time to get expressions for the second order derivatives of the position coordinates. Then get rid of the second order derivatives first.

3. Notes 18.2.1.1 Show all details.

4. Notes 18.2.1.2 Show all details.

5. Notes 18.2.2.1 Show all details.

6. Notes 18.2.3.1 Show all details.