4 02/06 W

1. p160, q38
2. Finish finding the derivatives of the unit vectors of the spherical coordinate system using the class formulae.
3. p160, q47 (finish)
4. Express the acceleration in terms of the spherical velocity components and their first time derivatives, instead of derivatives of position. Like $a_r = \dot v_r + \ldots$, etc. This is how you do it in fluid mechanics, where particle position coordinates are normally not used.
5. Derive the scale factors $h_r$, $h_\theta$ and $h_z$ of cylindrical coordinates.
6. Use them to find the Laplacian in cylindrical coordinates.
7. Two-dimensional steady temperature distributions in a simple plate must satisfy a PDE called the Laplace equation $k\nabla^2T=0$. Which of the following possibilities are potential steady temperature distributions in a plate?
   1. $T=\sqrt{2}y$
   2. $T=2\theta/\pi$
   3. $T=\sin\theta$
8. Suppose additionally that the plate is triangular in shape and that the following boundary conditions (BC) must be satisfied.
   1. the temperature is zero on the side $y=0$ of the plate;
   2. the heat flux coming out of the side $x=1$ of the plate is zero;
   3. the heat flux entering the third side $x=y$ is constant and equal to one, (per unit length in the $z$-direction).
   Write these BC as mathematical equations for the temperature distribution $T(x,y)$. According to Fourier’s law, the heat flux per unit cross-sectional area is $-k\nabla T$, where the heat conduction constant $k$ is here assumed to be one.
9. Which ones of the three solutions satisfy both the PDE and BC? Show why.