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, *x*_{1}, the second column takes
the partial derivative with respect to *x*_{2}, 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.