Jacobian matrix
In vector calculus, the Jacobian matrix (/dʒᵻˈkoʊbiən/, /jᵻˈkoʊbiən/) is the matrix of all first-order partial derivatives of a vector-valued function. When the matrix is a square matrix, both the matrix and its determinant are referred to as the Jacobian in literature.
Suppose f : ℝn → ℝm is a function which takes as input the vector x ∈ ℝn and produces as output the vector f(x) ∈ ℝm. Then the Jacobian matrix J of f is an m×n matrix, usually defined and arranged as follows:
or, component-wise:
This matrix, whose entries are functions of x, is also denoted by Df, Jf, and ∂(f1,...,fm)/∂(x1,...,xn). (Note that some literature defines the Jacobian as the transpose of the matrix given above.)
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