General

Many of the properties described here depend on the nature of a Jacobian matrix associated with a transformation; there are potentially two of these matrices, corresponding with the forward and inverse mappings. Using the notation of Equation , the Jacobian matrix ${\bf J}_F$ associated with the forward mapping is the $n \times m$ matrix of partial derivatives:

$$\textbf{J}_F = \left[ \begin{array}{cccc} \frac{\partial y_{... ...ts & \frac{\partial y_{n}}{\partial x_{m}} \end{array} \right]$$ (5)

while that associated with the inverse mapping ${\bf J}_I$ is the equivalent $m \times n$ matrix obtained by inter-changing input and output variables ($x$ and $y$) throughout.

The significance of these matrices can be seen by considering a simple linear mapping in two dimensions. Such a mapping is capable of representing a combination of a shift of origin, magnification, rotation, reflection and shearing deformation:

$$\begin{array}{lll} y_1 & = & a x_1 + b x_2 + c \\ y_2 & = & d x_1 + e x_2 + f \end{array}$$ (6)

It may be re-written as the matrix equation:

$$\begin{array}{cc} \left[ \begin{array}{c} y_1 \\ y_2 \end{ar... ...begin{array}{cc} a & b \\ d & e \end{array} \right] \end{array}$$ (7)

The Jacobian matrix $\mathbf{J}$ therefore contains the coefficients which define this mapping and determine its character, apart from a shift of origin. The determinant of $\mathbf{J}$ ( $\det\mathbf{J}=ae-bd$) is the signed ``area scale factor'' which the mapping introduces (i.e. the area of the parallelogram produced when the mapping acts on a unit square). In more than two dimensions, $\det\mathbf{J}$ would be the equivalent ``volume scale factor''.

If the mapping is not linear, then the Jacobian matrix will vary from point to point. Nevertheless, it may still be regarded as a local linear approximation to the true mapping (apart from re-location of the origin) and $\det \textbf{J}$ can still be interpreted as the local area (or volume) scale factor, which may now change from point to point.