Jacobian

Jacobian as Substitution in multi dimensional Integrals

Picture $f(x,y)$ as function of substitution variables such that $f(x(u,v),y(u,v))$. In an integral $\int f(x(u,v),y(u,v))dxdy$, before substituting for the $u,v,w,...$ of our choice, we need to change the differentials accordingly. The factor $J(u,v,...)$, that is appended to the new differentials making the substitution possible, is the Jacobian of that Substitution. The entire process follows the formula

$$\int f(x(u,v),y(u,v))dxdy\to \int u\cdot J(u,v)\cdot dudv$$

where the Jacobian is every initial variable’s definition in terms of the substitution variables, say $x\to x(u,v)$ partially differentiated with respect to every substitution variable.

$$J(u,v)=\left[\begin{array}{c}\frac{\partial x}{\partial u}\frac{\partial x}{\partial v}\\ \frac{\partial y}{\partial u}\frac{\partial y}{\partial v}\end{array}\right]$$

Requirements of this method

• You need to define a substitution variable with a given value, say $u=2x^2,v=3y+x,w=...$
  • Solve the system of equations for the initial variables say $x=...,y=...$

Substituting bounds

Sketching or picturing the new region and comparing to the old region in geometric space. → Generation of new bounds