Jacobian Matrices and Determinants

A Jacobian matrix is a square matrix of partial derivatives describing a change of coordinate systems or, more generally, a change of basis.

The determinant of such a matrix is the Jacobian determinant.

Description

Some differentiation problems are more easily solved in a different 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 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 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 chain rule.

CategoryRicottone