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  2. Jacobi's formula - Wikipedia

    en.wikipedia.org/wiki/Jacobi's_formula

    In matrix calculus, Jacobi's formula expresses the derivative of the determinant of a matrix A in terms of the adjugate of A and the derivative of A. [1]If A is a differentiable map from the real numbers to n × n matrices, then

  3. Matrix calculus - Wikipedia

    en.wikipedia.org/wiki/Matrix_calculus

    In mathematics, matrix calculus is a specialized notation for doing multivariable calculus, especially over spaces of matrices.It collects the various partial derivatives of a single function with respect to many variables, and/or of a multivariate function with respect to a single variable, into vectors and matrices that can be treated as single entities.

  4. Jacobian matrix and determinant - Wikipedia

    en.wikipedia.org/wiki/Jacobian_matrix_and...

    In vector calculus, the Jacobian matrix (/ dʒ ə ˈ k oʊ b i ə n /, [1] [2] [3] / dʒ ɪ-, j ɪ-/) of a vector-valued function of several variables is the matrix of all its first-order partial derivatives. When this matrix is square, that is, when the function takes the same number of variables as input as the number of vector components of ...

  5. Hessian matrix - Wikipedia

    en.wikipedia.org/wiki/Hessian_matrix

    The Hessian matrix plays an important role in Morse theory and catastrophe theory, because its kernel and eigenvalues allow classification of the critical points. [2] [3] [4] The determinant of the Hessian matrix, when evaluated at a critical point of a function, is equal to the Gaussian curvature of the function considered as a manifold. The ...

  6. Numerical differentiation - Wikipedia

    en.wikipedia.org/wiki/Numerical_differentiation

    Finite difference estimation of derivative. In numerical analysis, numerical differentiation algorithms estimate the derivative of a mathematical function or subroutine using values of the function and perhaps other knowledge about the function.

  7. Notation for differentiation - Wikipedia

    en.wikipedia.org/wiki/Notation_for_differentiation

    for the nth derivative. When f is a function of several variables, it is common to use "∂", a stylized cursive lower-case d, rather than "D". As above, the subscripts denote the derivatives that are being taken. For example, the second partial derivatives of a function f(x, y) are: [6]

  8. Matrix differential equation - Wikipedia

    en.wikipedia.org/wiki/Matrix_differential_equation

    A differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders. A matrix differential equation contains more than one function stacked into vector form with a matrix relating the functions to their derivatives. For ...

  9. Wronskian - Wikipedia

    en.wikipedia.org/wiki/Wronskian

    This is the determinant of the matrix constructed by placing the functions in the first row, the first derivatives of the functions in the second row, and so on through the () derivative, thus forming a square matrix. When the functions f i are solutions of a linear differential equation, the Wrońskian can be found explicitly using Abel's ...