Reparametrization of a disk.

Question

I know what is a reparametrization of a curve, $\gamma(t)$, is. But what is the definition of a reparametrization of a domain $X(u,v) = (x(u,v), y(u,v))?$

For example, how do we reparametrize the disk $D_r = \{(x,y): x^2+y^2<r.\}$

Note that this is a question my friend who is taking calculus-III asked and I never came across such a concept.

Answer 1

$\bullet$ Let $f: U \subset \mathbb{R}^m \to \mathbb{R}^n$ and $g: U' \subset \mathbb{R}^m \to \mathbb{R}^n$ be $\mathcal{C}^k$. We say $f$ is a reparametrization of $g$ if there exists a $\mathcal{C}^k$-diffeomorphism $\varphi: U' \to U$ such that,

$$ (g \circ \varphi)(x^1,...,x^n) =f(y^1,...,y^m) $$

where we let $f = f(y^1,...,y^m)$ and $g = g(x^1,...,x^n)$ i.e just establishing the coordinates on these functions respectively.

$\bullet$ To reparametrize the disk, you notice that $(x,y)$ are traditionally take to be the identity variables of $\mathbb{R}^2$ i.e both $x,y: \mathbb{R}^2 \to \mathbb{R}^2$ defined by $x(a,b) = a$ and $y(a,b) = b$. Hence, to reparametrize the disk, you can use any diffeomorphism of $\mathbb{R}^2$. The most common is polar coordinates i.e,

$$x = R \cos \theta, y = R \sin \theta \Rightarrow D = \{(R, \theta): R<r\}$$

$\bullet$ The last thing I'm left with is to show you that this polar coordinate map is the reparametrization of some other map. Well,

$$ (f \circ \varphi^{-1})(R, \theta) = f(x,y)$$

where $R = \sqrt{x^2+y^2}$ and $\theta = \tan^{-1}(y/x)$ i.e,

$$\varphi(x,y) = \left(\sqrt{x^2+y^2}, \tan^{-1}(y/x)\right) = (R, \theta)$$

Answer 2

As an example.. Any contour line can be expressed either in the first or second system of parametrizations.

Say a circle through origin in first system is

$$ r = a \, \cos\theta $$

With

$$ r \rightarrow \sqrt {x^2+y^2} $$ $$ \theta \rightarrow \tan^{-1} \frac {y}{x} $$

it transforms to

$$ a \,x= (x^2+y^2) $$

in the second system.

Now, if you want to go back to first system, you re-parametrize by plugging in

$$ x= r \cos \theta, \, y= r \, \sin \theta $$

to get back first original form

$$ r = a \, \cos\theta $$

Note that $ y= \cos x $ and $ r= \cos \theta $ are two entirely different curves.