Jacobian matrix

The Jacobian matrix collects the first partial derivatives of a multivariable function. It describes the function’s best linear approximation near a point and how that approximation changes directions and scale.

Total derivative: The linear map that best approximates a differentiable function near a point. The Jacobian matrix represents this derivative in coordinates.
Partial derivative: The derivative of a function with respect to one variable while other variables are held fixed. Each Jacobian entry is a partial derivative of one output with respect to one input.
Newton's method: An iterative method that uses derivatives to approximate roots of equations. For systems of equations, each update solves a linear system built from the Jacobian.
Hessian matrix: The matrix of second partial derivatives of a scalar-valued function. It captures curvature, whereas a Jacobian records first-order change.
Chain rule: A rule for differentiating a composition of functions. Jacobians of composed functions multiply in the same order as their derivative maps.
Multivariable function: A function whose inputs or outputs include more than one real variable. The Jacobian organizes first derivatives for functions with multiple inputs and outputs.
Change of variables: A method for rewriting integrals using a transformation between coordinate systems. The absolute Jacobian determinant corrects for local changes in volume.
Jacobian matrix of a vector field: The matrix of spatial first derivatives of a vector-valued field. This common case specializes the general Jacobian to a field with matching input and output dimensions.
Jacobian determinant: The determinant of a square Jacobian matrix, measuring signed local volume scaling. It distills the matrix’s local volume change into one number.
Linear map: A function between vector spaces that preserves vector addition and scalar multiplication. A Jacobian acts as the linear map approximating a nonlinear function locally.

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