Let $f : \mathbb{R}^2\to \mathbb{R}^2$ be defined by the equation $f(r,\theta) = (r \cos \theta, r \sin \theta)$.

Question

Let $f : \mathbb{R}^2\to \mathbb{R}^2$ be defined by the equation $f(r,\theta) = (r \cos \theta, r \sin \theta)$. It is called the polar coordinate transformation.

(a) Calculate $Df$ and $\text{det} Df$.

(b) Sketch the image under $f$ of the set $S = [1 , 2] \times [0, \pi]$. [Hint: Find the images under $f$ of the line segments that bound $S$ .]

I have calculated that $Df(r,θ)$ is the matrix

$$\begin{bmatrix} \cos\theta &-r \sin\theta \\ \sin\theta & r\cos\theta \end{bmatrix}$$

so its determinant is $r$.

I do not know how I can solve (b), could someone help me please? Thank you very much.

Edit: I think that in (b), the image is this:

Answer 1

Hint: In polar coordinates, the image of $S$ is the region bounded by $r = 1$, $r = 2$, $\theta = 0$ and $\theta = \pi$. That is, we can describe the region as the set of points $(r,\theta)$ such that $1 \leq r \leq 2$ and $0 \leq \theta \leq \pi$.