Change of Variables

Substitution Theorem

Suppose $\vec{T}(u,v)$ is a 1-1, Differentiable Transformation that maps the Region $G$ in the $uv$ plane to the region $R$ in the $xy$ plane. Then:

$${\iint }_{R}f(x,y)dxdy={\iint }_{G}f(\vec{T}(u,v))\det \left(J_{\vec{T}}\right)dudv$$

where $\det J_{\vec{T}}$ is the Jacobian Determinant of $\vec{T}(u,v)$

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Derivative of Inverse Coordinate Transformation

If $\vec{T}(u,v)$ is a 1-1 Differentiable Transformation that maps a region $G$ in the uv-plane to a region $R$ in the xy-plane and $\vec{T}(u_0,v_0)=(x_0,y_0)$, then we have

$$\left|\det (D\vec{T}(u_0,v_0))\right|=\frac{1}{\left|\det (D{\vec{T}}^{-1}(x_0,y_0))\right|}$$Link to original