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$$\iint_Df(x,y)dxdy = \iint_\omega f(g(s,t))\cdot|\det(J)|dtds$$

Where $J$ is Jacobi matrix: $\begin{bmatrix}x_s \quad x_t \\y_s \quad y_t \end{bmatrix}$

My question is why do we need Jacobi matrix in the first place? What exactly is going on when we are doing this. I did a couple of examples and know how to use it, but I still don't quite understand how it works. What would we use if we would want to calculate a triple integral or "$n$" integral. What about only a single integral, would we still need the Jacobi matrix?

Lorago
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  • The key insight comes from linear transformations and that their absolute value determinants give the "scale factor" for areas/volumes. The Jacobian determinant is there in all dimensions. See this answer of mine for some heuristic remarks which you may find helpful. Also, littleO's answer gives more details, regarding Riemann sums, than my linked answer, so you may find that useful. But in any case, you MUST understand things at the special case when $g$ is linear first. – peek-a-boo Feb 07 '22 at 22:44
  • It is the multivariable version of the derivative that occurs in $du = g'(x)dx$ when you carried out the 1-var substitution $u=g(x)$ back in the first semester of calculus. – Randall Feb 07 '22 at 22:45

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