User:EGM6321.f09.Team3.cgs/HW5

= Problem 1 =

{| class="toccolours collapsible collapsed" width="80%" style="text-align:left" !Please click on "show" to see the problem solution

Given
$$ \Delta\psi=\frac{1}{h_1h_2h_3} \sum_{i=1}^3 \frac{\partial}{\partial\xi_i}\lbrack\frac{h_1h_2h_3}{(h_i)^2}\frac{\partial\psi}{\partial\xi_i}\rbrack$$

Find
$$i_2 $$ and $$i_3$$

Solution
We know that $$ h_1=1$$, $$ h_2=r cos\theta$$, and $$h_3=r$$

We also know that $$\xi_1=r$$, $$\xi_2=\varphi$$, and $$\xi_3=\theta$$

Equation 1 in full form (without the summation)

$$ \Delta\psi=\frac{1}{h_1h_2h_3} \lbrack\frac{\partial}{\partial\xi_1} \frac{h_1h_2h_3}{(h_1)^2}\frac{\partial\psi}{\partial\xi_1}\rbrack+ \frac{1}{h_1h_2h_3} \lbrack\frac{\partial}{\partial\xi_2} \frac{h_1h_2h_3}{(h_2)^2}\frac{\partial\psi}{\partial\xi_2}\rbrack+\frac{1}{h_1h_2h_3} \lbrack\frac{\partial}{\partial\xi_3} \frac{h_1h_2h_3}{(h_3)^2}\frac{\partial\psi}{\partial\xi_3}\rbrack$$ (Equation 2)

If we solve equation 2 using these relations we arrive at the following equation:

$$ \Delta\psi=\frac{1}{r^2cos\theta} \lbrack\frac{\partial}{\partial r} (\frac{r^2cos\theta}{(1)^2}\frac{\partial \psi}{\partial r})\rbrack+ \frac{1}{r^2cos\theta} \lbrack\frac{\partial}{\partial\varphi} (\frac{r^2cos\theta}{(r cos\theta)^2}\frac{\partial\psi}{\partial\varphi})\rbrack+\frac{1}{r^2cos\theta} \lbrack\frac{\partial}{\partial\theta}( \frac{r^2cos\theta}{(r)^2}\frac{\partial\psi}{\partial\theta})\rbrack$$ (Equation 2)

$$ \Delta\psi=\underbrace{\frac{1}{r^2cos\theta} \lbrack\frac{\partial}{\partial r} (r^2cos\theta\frac{\partial \psi}{\partial r})\rbrack}_{i_1}+ \underbrace{\frac{1}{r^2cos\theta} \lbrack\frac{\partial^2\psi}{\partial\varphi^2}\rbrack}_{i_2}+\underbrace{\frac{1}{r^2cos\theta} \lbrack\frac{\partial}{\partial\theta}( \cos\theta\frac{\partial\psi}{\partial\theta})\rbrack}_{i_3}$$ (Equation 3)

Therefore,

$$i_1=\frac{1}{r^2cos\theta} \lbrack\frac{\partial}{\partial r} (r^2cos\theta\frac{\partial \psi}{\partial r})\rbrack$$

$$i_2=\frac{1}{r^2cos\theta} \lbrack\frac{\partial^2\psi}{\partial\varphi^2}\rbrack$$

$$i_3=\frac{1}{r^2cos\theta} \lbrack\frac{\partial}{\partial\theta}( \cos\theta\frac{\partial\psi}{\partial\theta})\rbrack$$

= Problem 2 =
 * }

{| class="toccolours collapsible collapsed" width="80%" style="text-align:left" !Please click on "show" to see the problem solution

Given
$$ Homework goes here $$

Find
m & n such that the ODE is exact.