# Dictionary:Jacobian

For a set of transform equations ${\displaystyle y_{i}=y_{i}(x_{1},x_{2},\dots ,x_{N})}$, the Jacobian matrix J is:
${\displaystyle {\textbf {J}}=\left\|J_{ik}\right\|=\left\|{\frac {\partial y_{i}}{\partial x_{k}}}\right\|}$.
The Jacobian is a measure of the change in the ith data point as the kth parameter is changed; it is a measure of how strongly data depend on a parameter. In iterative solutions, it can be used to indicate the degree of convergence. In inverting electromagnetic data, columns are generally ordered (first) resistivities, (second) thicknesses, (third) calibration factors (in the case of joint inversions); see Raiche et al. (1985). The Jacobian matrix transforms one vector into another, as in a coordinate transform. The inverse of the Jacobian is sometimes called the data influence matrix as it shows how a small change in the data would influence the inversion result. Vanishing of the determinant of the Jacobian shows that relations are not independent. For a 2x2 matrix, independence is shown by ${\displaystyle ad-bc\neq 0}$. Named for Karl Gustav Jacob Jacobi (1804-1851), German mathematician.