Jacobian matrix and determinant

The Jacobian matrix collects a function’s first-order partial derivatives, while its determinant gives the transformation’s local signed volume-scaling factor.

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Total derivative: The linear map that best approximates a differentiable function near a point. The Jacobian matrix represents this local linear approximation in coordinates.
Change of variables: A method for rewriting integrals using a transformation of variables and its local volume factor. The absolute Jacobian determinant corrects for volume changes in transformed integrals.
Partial derivative: A derivative measuring change with respect to one variable while other variables are held fixed. Each Jacobian entry differentiates one output component with respect to one input coordinate.
Hessian matrix: A square matrix of second-order partial derivatives of a scalar-valued function. The Hessian describes second-order variation, unlike the Jacobian’s first-order behavior.
Rotation matrix: A matrix representing a rotation of Euclidean space while preserving lengths and orientation. Its determinant is one, showing that rotation preserves oriented volume.
Chain rule: A rule for differentiating a composition of functions. Jacobian matrices of successive transformations multiply according to this rule.
Multivariable integration: Integration of functions over regions in spaces with multiple dimensions. Jacobian determinants enable integrals over regions described by transformed coordinates.
Differentiable function: A function that admits a linear approximation whose error becomes negligible relative to distance from the point. The Jacobian describes the linear approximation when the function is differentiable.
Gradient: A vector of first-order partial derivatives of a scalar-valued function. The gradient handles one scalar output; the Jacobian also covers vector-valued outputs.
Reflection (mathematics): A transformation that fixes a subspace and reverses direction across it. Its determinant is negative, illustrating local orientation reversal.