= Jacobian Matrices and Determinants =

A '''Jacobian matrix''' is a square matrix of [[Calculus/PartialDerivative|partial derivatives]].

A Jacobian is most commonly used to describe a [[Calculus/CoordinateSystem#Changing_System|change of coordinate systems]] or a [[LinearAlgebra/Basis#Change_of_Basis|change of basis]]. As the result, the [[LinearAlgebra/Determinant|determinant]] of such a matrix is called '''Jacobian determinant'''.

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== Description ==

Consider a vector-valued function, such as a [[Calculus/ParametricEquation|parameterized]] function, but for simplicity let ''F(X) = <f,,1,,(X), f,,2,,(X)>'' where ''X'' is a vector as ''X = <x,,1,,``, x,,2,,``, ... x,,n,,>''. A [[Calculus/Gradient|gradient]] can be calculated for each of the component function.

{{attachment:jacobian.svg}}



=== Relation to Hessian Matrices ===

A [[Calculus/HessianMatrix|Hessian matrix]] is also a square matrix of partial derivatives. Importantly however, it is formed by all possible second-order derivatives of a scalar-valued function. As such, it is more closely related to a gradient than a Jacobian matrix.

To express all possible second-order derivatives of a vector-valued function, a tensor would in fact be required.

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== Usage ==



=== Change of Coordinate Systems ===

Some differentiation problems are more easily solved in a different [[Calculus/CoordinateSystem|coordinate system]]. The transformation of points is straightforward and known. The trick is that space (i.e., area in 2 dimensions, volume in 3, and so on) was also transformed. For a given transformation, the '''Jacobian matrix''' contains all [[Calculus/PartialDerivative|partial derivatives]] involved.

The [[LinearAlgebra/Determinant|determinant]] of any matrix describes its scaling factor in space. For a Jacobian matrix, this is the '''Jacobian determinant'''. Note that determinants can be positive or negative. A negative Jacobian determinant simply means that space was flipped; the true scaling factor is given by the absolute value of the Jacobian determinant.

Consider the transformation from polar to Cartesian coordinates. The Jacobian matrix and determinant are given by:

{{attachment:polar.svg}}

Therefore ''dxdy = rdrdθ''.

The Jacobian determinant also explains the [[Calculus/ChainRule|chain rule]].



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