202303141200 Exercise 7.7.1

Consider the coordinate transformation $(x,y)=G(u,v)$ given by

$G(u,v)=(au,bv)$ for $(u,v)\in\mathbb{R}^2$

where $a$ and $b$ are positive constants.

(a) Prove that $G$ is $\textrm{1--1}$ on $\mathbb{R}^2$ .
(b) Show that $G$ maps the circular disc

$K=\{(u,v)\in\mathbb{R}^2 : u^2+v^2\leqslant 1\}$

in the $u$ , $v$ plane onto the elliptical disc

$G(K)=\{(x,y)\in\mathbb{R}^2 : x^2/a^2+y^2/b^2\leqslant 1\}$

in the $x$ , $y$ plane.
(c) Calculate $\mathrm{det}\,J_{G,(u,v)}$ .
(d) Calculate, using Theorem 7.7.5, the area of the elliptical disc $G(K)$ . (Assume known that the area of $K$ is $\pi$ .)

_{Extracted from P. R. Baxandall. (1986). Vector Calculus.}

Roughwork.

Symbolically,

$\forall\, a,b\in X,\quad f(a)=f(b)\Rightarrow a=b$ ,

or the contrapositive

$\forall\, a,b\in X,\quad a\neq b\Rightarrow f(a)\neq f(b)$ .

_{Wikipedia on Injective function}

(a)

From $G:\mathbb{R}^2\to\mathbb{R}^2:(u,v)\mapsto (x=au,y=bv)$ ,

$\begin{aligned} & \enspace & (x_1,y_1)=G(u_1,v_1) & = G(u_2,v_2)=(x_2,y_2) \\ \Rightarrow & \enspace & (au_1,bv_1) & = (au_2,bv_2) \\ \Rightarrow & \enspace & (u_1,v_1) & = (u_2,v_2) \\ \end{aligned}$

(b) Obvious.

(c)

$\begin{aligned} J_{G}(u,v) & = \begin{bmatrix} \partial_u(au)&\partial_v(au) \\ \partial_u(bv) & \partial_v(bv) \\ \end{bmatrix} \\ & = \begin{bmatrix} a & 0 \\ 0 & b \\ \end{bmatrix} \\ \mathrm{det}\,J_{G,(u,v)} & = ab \\ \end{aligned}$

(d)

Theorem (Change of variables). Let $G:K\subseteq \mathbb{R}^2\to\mathbb{R}^2$ be a $C^1$ function defined on a compact set $K$ in $\mathbb{R}^2$ , and let $D\subseteq K$ be an open subset of $\mathbb{R}^2$ such that