Maps

Definition: $T:D^{\ast }\to {\mathbb{R}}^2$ is one-to-one if $T(u,v)=T(u^{\prime },v^{\prime })⟺(u,v)=(u^{\prime },v^{\prime })$
Definition: $T:D^{\ast }\to D\subset {\mathbb{R}}^2$ is onto if for any $(u^{\prime },v^{\prime })\in D$ there exists $T(u,v)=(u^{\prime },v^{\prime })$

Say we have a region $x^2+y^2=r^2$ for $0\le r\le 1$. It’s easy to derive a transformation $T(r,\theta )=(x,y)$ so we’d like to be able to use an inequality like ${\iint }_{D}dxdy={\int }_0^{2\pi }{\int }_0^{1}T(r,\theta )drd\theta$ which would be quite easier to solve.

Look at how a small parallelogram maps through the transformation. If it begins with area $\Delta r\Delta \theta$ then it will map to a parallelogram with area $\left|\det \left(\begin{array}{cc}\frac{\partial x}{\partial r}& \frac{\partial y}{\partial r}\\ \frac{\partial x}{\partial \theta }& \frac{\partial y}{\partial \theta }\end{array}\right)\right|\Delta r\Delta \theta$. This is called the Jacobian matrix

Change of variables is pretty simple for single-variable functions
$f:[x(a),x(b)]\to \mathbb{R}$, $u\in [a,b]$, $u\to x(u)$ is differentiable
${\int }_{x(a)}^{x(b)}f(x)dx={\int }_a^{b}f(x(u))\frac{dx}{du}du$

Theorem: Let $D$ and $D^{\ast }$ be two elementary region and $T:D^{\ast }\to D$ such that $D=T(D^{\ast })$. Then for any integrable function $f:D\to \mathbb{R}$, ${\iint }_{D}f(x,y)dxdy={\iint }_{D^{\ast }}f(x(u,v),y(u,v))\left|\frac{\partial (x,y)}{\partial (u,v)}\right|dudv$ where $\left|\frac{\partial (x,y)}{\partial (u,v)}\right|$ represents the absolute value of the determinant of the Jacobian matrix of this transformation.

Example: Let $P$ be a parallelogram bound by $y=2x,y=2x-2,y=x,y=x+1$
Evaluate ${\iint }_{P}xydxdy$ by making a change of variable $x=u-v,y=2u-v$

Since this is a linear transformation, the resulting region will also be a parallelogram. Calculating the vertices reveals that it is a rectangle.
${\iint }_{P}xydxdy={\iint }_{P^{\ast }}(u-v)(2u-v)\mathbf{J}dudv={\int }_0^{-2}{\int }_0^1(u-v)(2u-v)dudv=-7$

Definition: The cylindrical coordinates of $(x,y,z)\in {\mathbb{R}}^3$ are given by $x=r\cos \theta ,y=r\sin \theta ,z=z$. In the other direction, $(r,\theta ,z)=(\sqrt{x^2+y^2},{\tan }^{-1}\frac{y}{x},z)$.

This is a three variable transformation, so the Jacobian matrix $\left|\frac{\partial (x,y,z)}{\partial (r,\theta ,z)}\right|$ will be a bit larger $\left|\det \left(\begin{array}{ccc}\frac{\partial x}{\partial r}& \frac{\partial y}{\partial r}& \frac{\partial z}{\partial r}\\ \frac{\partial x}{\partial \theta }& \frac{\partial y}{\partial \theta }& \frac{\partial z}{\partial \theta }\\ \frac{\partial x}{\partial z}& \frac{\partial y}{\partial z}& \frac{\partial z}{\partial z}\end{array}\right)\right|=r$

Definition: The spherical coordinates of $(x,y,z)$ are given by $(\rho ,\theta ,\varphi )$ where $x=\rho \cos \theta \sin \varphi ,y=\rho \sin \theta \sin \varphi ,z=\rho \cos \varphi$

The Jacobian $\left|\frac{\partial (x,y,z)}{\partial (\rho ,\theta ,\varphi )}\right|={\rho }^2\sin \varphi$

$\theta$ gives the angle with the $x$-axis, $\varphi$ gives the angle with the $z$-axis (physics convention does the opposite)

Example: Calculate ${\iiint }_{W}dxdydz$ where $W=\left\{(x,y,z):x^2+y^2+z^2\le r^2\right\}$
${\int }_0^{\pi }{\int }_0^{2\pi }{\int }_0^r{\rho }^2\sin \varphi d\rho d\theta d\varphi =\frac{r^3}{3}{\int }_0^{\pi }{\int }_0^{2\pi }d\theta \sin \varphi d\varphi =\frac{4\pi r^3}{3}$