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A multivariable substitution rule involving the Jacobian determinant.
Let be open sets, let be a diffeomorphism, and let be continuous with compact support. The change of variables formula is
where is the Jacobian determinant of at .
This is the multivariable generalization of the one-dimensional substitution rule, and it is fundamental for computing multiple Riemann integrals under smooth coordinate changes.
An open set in a topological space is a subset such that .
Let and be smooth manifolds. A map is a diffeomorphism if
If is a diffeomorphism, then for each its differential is a linear isomorphism , where denotes the tangent space.
For a differentiable map , where , the Jacobian determinant of at is the determinant of its Jacobian matrix:
The Jacobian determinant controls local invertibility and local volume scaling: nonvanishing of is the hypothesis of the inverse function theorem, and it appears in the change of variables formula for integrals.
Substitution rule: Let , let be continuously differentiable and strictly increasing, and set and . If is continuous, then
This is the one-dimensional instance of the general change of variables formula, and it is closely tied to the chain rule and fundamental theorem of calculus II.
A multiple Riemann integral of a bounded function over a rectangle is a number such that for every there exists with the property that for every rectangular partition of with mesh and every choice of tags (sample points) in each subrectangle , the corresponding Riemann sum
satisfies . In this case one writes .