I am stuck with an exercise that requires me to find the Laplacian $\Delta u=(D_x^2u+D_y^2u)$ of a 2d-function $u$ in polar coordinates (in the standard Euclidean plane).
I found the following article on the net, and tried to follow its logic, but I could not understand two steps: http://www.sci.brooklyn.cuny.edu/~mate/misc/laplacian_polarcoord_higherdim.pdf
at first, the representation of $D_x$ and $D_y$ in terms of $r, \theta$, at the bottom of page 2:
$D_y=\sin\theta D_r+\frac{\cos\theta}{r}D_\theta$ and
$D_x=\cos\theta D_r-\frac{\sin\theta}{r}D_\theta$ . When I draw a sketch of the plane with a circle and all the coordinates, I get that it should be $D_y=\sin\theta D_r+r\ \cos\theta D_\theta$, because the larger the radius is, the greater will be the impact of a change in $\theta$ on a change in $y$. What am I making wrong here?
And then, secondly, what looks like an easy multiplication, namely taking the square of the above terms (on the top of page 3 in the link):
$D_y^2=(\sin\theta D_r+\frac{\cos\theta}{r}D_\theta)(\sin\theta D_r+\frac{\cos\theta}{r}D_\theta)$
$=\sin^2\theta D_r^2+\frac{2\sin\theta \cos\theta}{r}D_\theta D_r+\frac{\sin^2\theta}{r^2}D_\theta^2 \\ -\frac{\cos\theta}{r^2}D_\theta + \frac{\cos^2\theta}{r}D_r - \frac{\cos\theta \sin\theta}{r^2}D_\theta$
and similarly for $D_x$.
I don't see from where the last three summands come, what I see is: $(a+b)(a+b)=a^2+2ab+b^2+c+d+e$ and I cannot see the context of $c,d,e.$
It would be great if you could explain those issues to me!