[BUG] multivariable chain rule

[Tevian Dray]

Here are some thoughts about Phil's thought-provoking examples.

I would argue that the first example, AS WORDED, isn't about the chain rule at
all, but rather about the properties of the gradient.  This is really about
directional derivatives, but where the "direction" is the velocity.  We like
to formulate all questions about the gradient in terms of our "Master
Formula", which says
	dp = (grad p) dot (d rvec)
where rvec is the position vector.  (We prefer to make "d rvec" fundamental
rather than rvec, but that's another issue.)  Most texts will introduce
directional derivatives (the above formula "divided" by ds) as a property of
the gradient, whereas Phil's problem (the above formula "divided" by dt)
becomes instead an example of the chain rule.

There is no essential difference between these two classes of problems!

For me, Phil's example provides an excellent reason for NOT restricting the
notion of directional derivative to unit vectors, but in this case his example
becomes "just" another directional derivative problem.  Alternatively, if this
is really about the chain rule, then directional derivatives are also, and
should be taught that way.

I must confess that I was not familiar with this usage of "Jacobian" to refer
to the matrix of partial derivatives of an arbitrary vector field, but it
seems like reasonable terminology.  This really is the chain rule, in the form

                              dH^i dx^j   dH^i
                              ---- ---- = ----
                              dx^j  dt     dt

And it falls into the category in my original post, since one does not
explicitly know the functional dependence.  But note also that this problem
does not in any way depend on knowing whether the coordinates are rectangular!

Returning to the first example, it is apparently assumed without comment that
"galactic coordinates" are rectangular.  OK, that's a common assumption in
calculus courses, but it is important to realize that this turns out NOT to be
relevant!  The only place where the choice of coordinates matter in this
problem is in the assertion that the derivatives dp/dx^i are "del p".  Remove
this assertion (by explicitly writing dp/dx^i), and these problems really are
the same, with a vector field in one case and a scalar field in the other.

So yes, I think Phil has provided two good examples of the use of the chain
rule in situations where one does not know the functional dependence.  But I
personally like the first problem better as currently worded, in which case I
view it as "really" about directional derivatives, not chain rule.  So the
question I still have is whether one needs two separate concepts here, or
whether one is enough.

Tevian