= Jacobian Matrices and Determinants =

A '''Jacobian matrix''' is a square matrix of [[Calculus/PartialDerivative|partial derivatives]] describing a [[Calculus/CoordinateSystem#Changing_System|change of coordinate systems]] or, more generally, a [[LinearAlgebra/Basis#Change_of_Basis|change of basis]].

The [[LinearAlgebra/Determinant|determinant]] of such a matrix is the '''Jacobian determinant'''.

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== Description ==

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.

Generically, consider the transformation from ''(x,y)'' coordinates to ''(u,v)'' coordinates. The Jacobian matrix is given by:

{{{
┌        ┐
| ∂x  ∂x |
| ――  ―― |
| ∂u  ∂v |
|        |
| ∂y  ∂y |
| ――  ―― |
| ∂u  ∂v |
└        ┘
}}}

The [[LinearAlgebra/Determinant|determinant]] of any matrix describes its scaling factor in space. For a Jacobian matrix, this is the '''Jacobian determinant'''.

{{{
    | ∂x ∂x |
    | ―― ―― |
    | ∂u ∂v |   ∂x ∂y   ∂x ∂y
det |       | = ―― ―― - ―― ――
    | ∂y ∂y |   ∂u ∂v   ∂v ∂u
    | ―― ―― |
    | ∂u ∂v |
}}}

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:

{{{
    | ∂x ∂x |       
    | ―― ―― |
    | ∂θ ∂r |       | cosθ -r*sinθ |
det |       | = det | sinθ  r*cosθ | = (cosθ)(r*cosθ) - (-r*sinθ)(sinθ) = r
    | ∂y ∂y |
    | ―― ―― |
    | ∂θ ∂r |
}}}

Therefore ''dxdy = rdrdθ''.

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



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CategoryRicottone