HERE IS A LIST OF ALL THE KINDS OF "CHAIN RULE" IN THE VECTOR CALC BOOK.
----WD Smith, math 127, Spr 2003----------------------------------------


Capital letters are vectors, lowercase are scalars
. = dot product
X = cross product.

(d/dt) [f(g(t)] = g'(t) f'(g(t))    usual scalar chain rule.

Thm 15.4.1:
(d/dt) [f(G(t))] = G'(t) . GRADf( G(t) )

Here GRADf should be interpreted as a single function name
(just like G' is a single function name).

(d/dt) [U(F(t)] = (F'(t) . GRAD) U.

JacobianMatrix [U(F(x,y,z))] = JacobianMatrix[U at F] JacobianMatrix[F at x,y,z]

(d/dt) [U(F(t))] = JacobianMatrix[U at F]  (d/dt)F    <-- this is a matrix-vector product

The rows of the Jacobian matrix of F are the gradients of each
of the components of (the vector-valued function) F.
It is often written "dF/dX" where both F and X are vectors,
which is a useful abuse of notation, and you can avoid getting in
trouble by always writing arrows on your vectors (here, I 
have used capital letters, since no arrows on this typewriter).