Find a concept
Start typing to search the mathematical index.
Determinant of the Jacobian matrix for a map from Rn to Rn
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.
Let be open, let , and let . The map is differentiable at if there is a linear map such that
where is small enough that , and is the Euclidean norm.
The map , which is necessarily unique, is the Fréchet derivative . In standard coordinates it is represented by the Jacobian matrix.
A determinant is a function that assigns to each matrix over a field the scalar
where is the set of permutations of and is the sign of .
A Jacobian matrix of a map (with ) at a point is the matrix
provided these partial derivatives exist.
When is differentiable at , the Fréchet derivative is a linear map, and is the matrix that represents in the standard bases of and . For scalar-valued , is closely related (up to transpose conventions) to the gradient.
Inverse function theorem in : Let be an open set and let be continuously differentiable. If the Jacobian determinant is nonzero at some , then there exist neighborhoods of and of such that restricts to a bijection whose inverse is continuously differentiable. Moreover,
Thus, near , the map is a diffeomorphism onto its image; in particular it is a local homeomorphism.
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 .
A linear map is a function between vector spaces over the same field such that for all and ,