Jacobian matrix and determinant

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 → ℝ^m is a function which takes as input the vector x ∈ ℝⁿ and produces as output the vector f(x) ∈ ℝ^m. 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, J_f, and ∂(f₁,...,f_m)/∂(x₁,...,x_n). (Note that some literature defines the Jacobian as the transpose of the matrix given above.)

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