Change-of-Variable Formula (Jacobian Matrix)

The Change-of-Variable Formula

Suppose that the variables $x$ and $y$ are related to the variables $u$ and $v$ by the equations $x=x(u,v),y=y(u,v)$. Then

${\iint }_{R_{xy}}f(x,y)dxdy={\iint }_{R_{uv}}f[x(u,v),y(u,v)]\left|\frac{\partial (x,y)}{\partial (u,v)}\right|dudv$

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

This function is called the Jacobian of the transformation, denoted by $J$.

The Jacobian is a factor that is introduced to compensate for the distortion of the domain that occurs when we move it from one coordinate system to another.

Restriction to formula

There is one important restriction on the transformation $(x,y)\to (u,v)$; we must not have $J=0$ at any point on the interior of $R_{uv}$​. This condition ensures that the transformation is invertible on the domain of integration.

This is where the jacobian comes from when moving from one coordinate system to another