The Jacobian matrix contains information about the local behavior of a function. The Jacobian matrix can be seen as a representation of some local factor of change. It consists of first order partial derivatives. If we take the partial derivatives from the first order partial derivatives, we get the second order partial derivatives, which are used in the Hessian matrix. The Hessian matrix is used for the Second Partial Derivative Test with which we can test, whether a point x is a local maximum, minimum or a so called saddle point .
With the Jacobian matrix we can convert from one coordinate system into another
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