Graphing polar coordinates

Question

i) f(r,θ) = (rcosθ,rsinθ)

Sketch the image of f of the set S=[1,2]×[0,π].

ii) sketch $f(x,y)=(x^2-y^2,2xy)$

for the set $ S = \{(x, y) : x^2 + y^2 < a^2;\; x,y\ge 0\}$

I have found Df matrix and its det of the jacobian matrix but not sure if it helps me to sketch the graph. Is there an app for two-dimension polar coordinates graph? And how would i graph this if i have to do it by hand?

Answer 1

It is quite elementary and you don't need calculus: for each fixed $r\in[1,2]$, you obtain the semi-circle centred at the origin with radius $r$, in the upper semi-plane above the $x$-axis. Hence, varying $r$, you obtain the semi-annulus between the circles with radii $1$ and $2$, located in the upper semi-plane.

Edit:

For question ii), use polar coordinates: the domain of $f$ is the set of points $(x,y)$ with polar coordinates $r,\theta$ such that $0\le r <a$ and $0\le \theta\le\frac\pi 2$. Now $$f(x,y)=\bigl(r^2(\cos^2\theta-\sin^2\theta), 2r^2\cos\theta\sin\theta\bigr)=r^2(\cos2\theta, \sin2\theta).$$ Further, $0\le 2\theta\le \pi$ and $0\le r^2<a^2$. So we obtain the open semi-circle in the upper semi-plane with radius $a^2$.