[BUG] multivariable chain rule

[Matthias Kawski]

On Sun, 3 Apr 2005, Tevian Dray wrote:

> Can anyone provide a good example of the use of the
> multivariable chain rule? Seems to me that most purported
> examples don't actually involve the chain rule at all.  For
> instance, if V=IR and 1/R=1/R1+1/R2, the computation of dV/dt
> doesn't really use the chain rule (with V=V(I,R) and everything
> else a function of t), but rather the product and power rules.
> In fact, I claim the only time one actually needs the
> multivariable chain rule is when one does NOT explicitly know
> the functional dependence.  In the above example, that means not
> knowing how V depends on I and R.  This is very relevant for
> theoretical discussions, such as changing variables in a PDE.
> But what purpose does it serve in a multivariable calculus
> course?
>
> Tevian

I agree -- if explicit formulae are provided for all functions
then one might generally as well substitute these into each other
and use simply the single-variable chain-rule for any derivatives.

Thus the emphasis when selling the multi-variable chain-rule
should be on the theoretical side. I like to spend much time
on the example z=f(g(t),h(t)), from many points of view, even-
tually leading to the product of a 1x2 and a 2x1 matrix for
the derivatives in coordinates (main use: grad perp to level
curve).
Similarly, I like to change to polar/spherical coord's (incl.
partials, gradient, div, curl) first with geometric approximate
arguments (pictures)... but in the end I like the algebraic
confirmation using the chain rule (closely intertwined with
differentials).

I don't recall ever having had real success with more abstract
arguments with general expressions (no "explicit" formulas) or
implicitly defined functions. While I personally greatly enjoy
the identity (dV/dp)*(dp/dT)*(dT/dV) = -1 (immediate from the
chain-rule and implict differentiation), I don't see how this
can be more than a cute side-remark in a mainstream multi-variable
calculus course.
In summary, I recommend to honestly "sell" the chain rule as
primarily used for more theoretical arguments but don't go too
far with such theory.

Matthias
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Matthias Kawski                http://math.asu.edu/~kawski
Dept. of Mathematics and Statistics         kawski at asu.edu
Arizona State University            office: (480) 965 3376
Tempe, Arizona 85287-1804           home:   (480) 893 0107
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