[ODE] ODE's Jacobian matrices

[Aaron Dwyer]

A question for the math & ODE-internals inclined:

I am confused about the calculation of ODE's
constraint jacobians.  I've read "How to make new
joints in ODE".  The suggested procedure for
calculating jacobians there is to first create a
position constraint equation C(q) = 0, then take the
time derivative of that, then rearrange until you have
a matrix times linear velocity and a matrix times
angular velocity for the bodies involved.  These
matrices are then your jacobians.  This doesn't agree
with any of my studies with regards to jacobian
matrices...

When I have a set of functions C1 through Cm of
variables q1 through qn, I calculate the jacobian by
taking the partial derivative of each C1 through Cm
with respect to q1 through qn, and I'm left with an m
row, n column matrix which is my jacobian.  

What is the connection between these two methods?  How
does the ODE procedure avoid the need to take partial
derivatives?

-Aaron Dwyer