The matrix formed by all first-order partial derivatives of a vector-valued function.
The Jacobian can be interpreted as a “map” of how a system will respond to small changes in the value of its independent variables: a matrix of “influence coefficients” or “sensitivity coefficients”.
If time is an independent variable, the Jacobean provides limited insight into how the system is postured to respond to changes in circumstance in the very near future. Jacobians that are more numerically stable can be used to predict changes farther into the future. Jacobians that are less numerically stable represent systems that are more responsive to changes as they occur.