Change of variables

A suitable coordinate change transforms volume elements through the absolute Jacobian determinant.

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Change of variables, Jacobian determinant, Double and triple integrals｜701号时钟草
701号时钟草 · English · Explanation
Connection evidence
Reviewed English material v1: card "The goal of change of variables in double integrals" at 0:00 — At the start, the video uses $\iint_{T(D)}$ dxdy = $\iint_D$ ?\,dudv to highlight the central task: not only must an appropriate transformation T be found, but the scaling factor relating the area elements before and after the transformation must also be determined.

Double and triple integrals
Integration over planar or spatial regions accumulates density into total quantities.
Jacobian matrix
The derivative matrix of a multivariable map captures its local linear action.
Antiderivatives
An antiderivative has the given function as its derivative; on an interval, any two antiderivatives differ by a constant.
Chain rule
The derivative of a differentiable composition combines the derivative of the outer function with that of the inner function.
Continuity
Continuity at a point means that the function value equals its limit there; continuity on an interval requires this at each relevant point.