Question

1. in polar coordinates the gradient of a function can be computed with the formula: $|

abla u|^{2}=u_{r}^{2}+\frac{1}{r^{2}}u_{\theta}^{2}$. use this formula to find $|
abla u|^{2}$. then find it directly by first converting the function into cartesian coordinates. (a) $u(r,\theta)=r^{2}cos\thetasin\theta$. (b) $u(r,\theta)=e^{r^{2}}sin^{2}\theta$. (c) $u(r,\theta)=r\theta$.

Explanation:

Step1: Recall the relationships between polar and Cartesian coordinates

$x = r\cos\theta$, $y = r\sin\theta$, $r^{2}=x^{2}+y^{2}$, $\theta=\arctan(\frac{y}{x})$ and the partial - derivative formulas for polar - to - Cartesian conversion. Also, recall that $\|
abla u\|^{2}=u_{r}^{2}+\frac{1}{r^{2}}u_{\theta}^{2}$.

Step2: For $u(r,\theta)=r^{2}\cos\theta\sin\theta$

First, find the partial derivatives with respect to $r$ and $\theta$.
$u_{r} = 2r\cos\theta\sin\theta$, $u_{\theta}=r^{2}(\cos^{2}\theta - \sin^{2}\theta)$.
Then, $\|
abla u\|^{2}=(2r\cos\theta\sin\theta)^{2}+\frac{1}{r^{2}}(r^{2}(\cos^{2}\theta - \sin^{2}\theta))^{2}$.
$=4r^{2}\cos^{2}\theta\sin^{2}\theta+r^{2}(\cos^{2}\theta - \sin^{2}\theta)^{2}$.
$=4r^{2}\cos^{2}\theta\sin^{2}\theta+r^{2}(\cos^{4}\theta - 2\cos^{2}\theta\sin^{2}\theta+\sin^{4}\theta)$.
$=r^{2}(\cos^{4}\theta + 2\cos^{2}\theta\sin^{2}\theta+\sin^{4}\theta)$.
$=r^{2}(\cos^{2}\theta+\sin^{2}\theta)^{2}=r^{2}$.

Step3: For $u(r,\theta)=e^{r^{2}}\sin^{2}\theta$

$u_{r}=2re^{r^{2}}\sin^{2}\theta$, $u_{\theta}=2e^{r^{2}}\sin\theta\cos\theta$.
$\|
abla u\|^{2}=(2re^{r^{2}}\sin^{2}\theta)^{2}+\frac{1}{r^{2}}(2e^{r^{2}}\sin\theta\cos\theta)^{2}$.
$=4r^{2}e^{2r^{2}}\sin^{4}\theta+\frac{4}{r^{2}}e^{2r^{2}}\sin^{2}\theta\cos^{2}\theta$.
$=4e^{2r^{2}}\sin^{2}\theta(r^{2}\sin^{2}\theta+\frac{\cos^{2}\theta}{r^{2}})$.

Step4: For $u(r,\theta)=r\theta$

$u_{r}=\theta$, $u_{\theta}=r$.
$\|
abla u\|^{2}=\theta^{2}+\frac{1}{r^{2}}\times r^{2}=\theta^{2} + 1$.

Answer:

For $u(r,\theta)=r^{2}\cos\theta\sin\theta$, $\|
abla u\|^{2}=r^{2}$; for $u(r,\theta)=e^{r^{2}}\sin^{2}\theta$, $\|
abla u\|^{2}=4e^{2r^{2}}\sin^{2}\theta(r^{2}\sin^{2}\theta+\frac{\cos^{2}\theta}{r^{2}})$; for $u(r,\theta)=r\theta$, $\|
abla u\|^{2}=\theta^{2}+1$