My first question is:
1) Is $f(x,y)=\frac {xy(x^{2}-y^{2})}{{(x^{2}+y^{2})}^{3/2}}$ differntiable at (0,0)?
Considering polar co-ordinates: $x=r\cos \theta$ and $y=r\sin \theta$.
$\Rightarrow$ $f(x,y)= \frac {r\cos \theta r\sin \theta (r^{2}\cos ^{2}\theta-r^{2}\sin ^{2}\theta)}{{(r^{2}\cos ^{2}\theta+r^{2}\sin ^{2}\theta)}^{3/2}}$ = $\frac {r^{4}\cos\theta\sin\theta(\cos ^{2}\theta-\sin ^{2}\theta)}{r^{3}}$ = $r\cos\theta\sin\theta(\cos ^{2}\theta-\sin ^{2}\theta)$
Hence linear in r, therfore there doesn't exist and unique tangent plane at (0,0), therefore not differnetiable there.
2) Considering $f(x,y)= \frac {x^{3}y}{x^{6}+y^{2}}$ (x,y) $\neq$ 0 and 0 if (x,y)=(0,0). If you were to plot the function $\theta \rightarrow f(r\cos\theta, r\sin\theta)$ for $\theta \in [0,2\pi]$ what might the plot look like. Justify your answer.
Bit ensure on what this plot would be and the reason. Im guessing at the crinkle function?
Many thanks in advance.