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    Jacobian matrix and determinant - Wikipedia

    Motivation

    The Jacobian can be understood by considering a unit area in the new coordinate space; and examining … See more

    Definition

    Suppose f : R → R is a function such that each of its first-order partial derivatives exists on R . This function takes a point x ∈ R as input and produces the vector f(x) ∈ R as output. Then the Jacobian matrix of f, denoted Jf … See more

    Jacobian matrix

    The Jacobian of a vector-valued function in several variables generalizes the gradient of a scalar-valued function in several variables, which in turn generalizes the derivative of a scalar-valued function of a single varia… See more

    Jacobian determinant

    If m = n, then f is a function from R to itself and the Jacobian matrix is a square matrix. We can then form its determinant, known as the Jacobian determinant. The Jacobian determinant is sometimes simply referred to as "the Jac… See more

    Inverse

    According to the inverse function theorem, the matrix inverse of the Jacobian matrix of an invertible function f : R → R is the Jacobian matrix of the inverse function. That is, the Jacobian matrix of the inverse function at … See more

    Critical points

    If f : R → R is a differentiable function, a critical point of f is a point where the rank of the Jacobian matrix is not maximal. This means that the rank at the critical point is lower than the rank at some neighbour point. In ot… See more

    Examples

    Consider a function f : R → R , with (x, y) ↦ (f1(x, y), f2(x, y)), given by
    Then we have
    and
    The Jacobian matrix of f is
    and the Jac… See more

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