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The Jacobian

Given a vector of functions, each a function of a vector of n variables, i.e.,

where

the Jacobian J(f)(x) is defined as:

You may think of this as repeating the column vector of the functions in f(x) n times, and the first column takes the partial derivative of each of these functions w.r.t. to the first variable, x1, the second column takes the partial derivative with respect to x2, and so on.

For example, if x = (u, v, w)T and

then the Jacobian is:

Note that the first column is the partial derivative of each of the three functions in f with respect to the variable u, and so on.