Summary of Change of Variables in Multiple Integrals

A transformation [latex]T[/latex] is a function that transforms a region [latex]G[/latex] in one plane (space) into a region [latex]R[/latex] in another plane (space) by a change of variables.
A transformation [latex]T:G\rightarrow{R}[/latex] defined as [latex]T(u,v)=(x,y)[/latex] [latex](\text{or }T(u,v,w)=(x,y,z))([/latex] is said to be a one-to-one transformation if no two points map to the same image point.
If [latex]f[/latex] is continuous on [latex]R[/latex], then [latex]\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[/latex]
If [latex]F[/latex] is continuous on [latex]R[/latex], then [latex]R[/latex], then [latex]\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[/latex]

Glossary

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

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

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

transformation: a function that transforms a region [latex]G[/latex] in one plane into a region [latex]R[/latex] in another plane by a change of variables