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The 38 pages that link to Jacobian matrix, each with the reason it gives.

• Coordinate systemRelated: Its entries govern local effects of changing coordinates, including area and volume scaling.

• DeterminantRelated: Its determinant gives the function's local signed volume scaling.

• Partial derivativeRelated: It extends the collection of first derivatives to functions with multiple outputs.

• Chain ruleRelated: For vector-valued compositions, the chain rule becomes multiplication of Jacobian matrices.

• Tangent spaceRelated: Its coordinate transformation law ensures tangent vectors do not depend on a chosen chart.

• DivergenceRelated: The divergence is the trace of the vector field’s Jacobian matrix.

• Solid angleRelated: Coordinate transformations explain why spherical area includes a sine factor.

• Condition numberRelated: Its norm measures local sensitivity for differentiable problems.

• Directional derivativeRelated: For differentiable maps, it sends a direction to the resulting first-order change.

• Hessian matrixRelated: The Hessian is the Jacobian of the gradient vector.

• Identity matrixRelated: The identity Jacobian describes a local map whose first-order effect leaves coordinates unchanged.

• Poincaré mapRelated: The map’s Jacobian near a periodic point determines local stability.

• Implicit function theoremRelated: The theorem tests the Jacobian with respect to the variables to be determined.

• Conformal mapRelated: Its form for a holomorphic map explains why local stretching is uniform in every direction.

• Coordinate chartRelated: Its entries describe how coordinates change under a differentiable transition map.

• BackpropagationRelated: Each layer’s local Jacobian maps downstream error signals to derivatives with respect to its inputs.

• Cauchy–Riemann equationsRelated: The equations constrain the Jacobian of a complex function to have a special matrix form.

• Lyapunov exponentRelated: Its action propagates small perturbations in discrete-time systems.

• Inverse function theoremRelated: Its determinant tests invertibility for maps between equal-dimensional spaces.

• Vector calculusRelated: Gradient, divergence, and curl are formed from first derivatives represented by Jacobians.

• Total derivativeRelated: Multiplying Jacobians composes the component changes used to calculate a total derivative.

• Implicit differentiationRelated: For systems of implicit equations, Jacobians extend the derivative relations to several variables.

• Coordinate singularityRelated: A degenerate coordinate transformation can signal that a chart has failed locally.

• Singularity theoryRelated: Its rank detects many singular points of maps between smooth spaces.

• Multiple integralRelated: Its determinant supplies the local area or volume scaling in coordinate transformations.

• Dynamical systems theoryRelated: Near an equilibrium, its eigenvalues help determine local stability.

• Implicit functionRelated: Its relevant determinant tests whether the equation can be solved locally for selected variables.

• Jacobian conjectureRelated: Its determinant supplies the conjecture’s local invertibility condition.

• Nonlinear least squaresRelated: Its columns describe how residuals change with each parameter.

• Vector-valued functionRelated: For several inputs, it represents the function’s local linear change.

• Birch's theoremRelated: Its rank detects singular points of the system of forms.

• Hartman–Grobman theoremRelated: At the fixed point, this matrix supplies the linear system used in the theorem.