Summary of Change of Variables in Multiple Integrals

A transformation $T$ is a function that transforms a region $G$ in one plane (space) into a region $R$ in another plane (space) by a change of variables.
A transformation $T:G\rightarrow{R}$ defined as $T(u,v)=(x,y)$ $(\text{or }T(u,v,w)=(x,y,z))($ is said to be a one-to-one transformation if no two points map to the same image point.
If $f$ is continuous on $R$ , then $\underset{R}{\displaystyle\iint} f(x,y) dA =\underset{S}{\displaystyle\iint} f\left(g(u,v),h(u,v)\right)\left\arrowvert\frac{\partial(x,y)}{\partial(u,v)}\right\arrowvert du dv$
If $F$ is continuous on $R$ , then $R$ , then $\underset{R}{\displaystyle\iiint} F(x,y,z) dV =\underset{G}{\displaystyle\iiint} F\left(g(u,v,w),h(u,v,w),k(u,v,w)\right)\left\arrowvert\frac{\partial(x,y,z)}{\partial(u,v,w)}\right\arrowvert du dv dw = \underset{G}{\displaystyle\iiint}H(u,v,w)|J(u,v,w)| du dv dw$

Glossary

Jacobian: the Jacobian $J(u ,v)$ in two variables is a $2{\times}2$ determinant:; $J(u,v) = \begin{vmatrix}\frac{dx}{du} & \frac{dy}{du}\\\frac{dx}{dv} & \frac{dy}{dv}\end{vmatrix}$; the Jacobian $J(u ,v, w)$ in three variables is a $3{\times}3$ determinant:; $J(u,v,w)=\begin{vmatrix}\frac{dx}{du} & \frac{dy}{du} & \frac{dz}{du}\\\frac{dx}{dv} & \frac{dy}{dv} & \frac{dz}{dv}\\\frac{dx}{dw} & \frac{dy}{dw} & \frac{dz}{dw}\end{vmatrix}$

one-to-one transformation: a transformation $T : G {\rightarrow} R$ defined as $T(u, v) = (x, y)$ is said to be one-to-one if no two points map to the same image point

planar transformation: a function $T$ that transforms a region $G$ in one plane into a region $R$ in another plane by a change of variables

transformation: a function that transforms a region $G$ in one plane into a region $R$ in another plane by a change of variables