Mr. Integral
Math notes/Calculus III/Double Integrals

General Changes of Variables and Jacobians — Cheat sheet

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Core rule

dA=∣det⁡∂(x,y)∂(u,v)∣dudv.dA=\left|\det\frac{\partial(x,y)}{\partial(u,v)}\right|du dv.

Watch for

Use the absolute forward Jacobian for area. Check one-to-one coverage and distinguish forward from inverse determinants.