Differentiation in several variables

Taking derivatives in several variables is:
- computational extremely easy
- extremely useful in applications
- requires doing some geometry
- deeply clarifies the geometry of what's happening
It should be done in all calculus classes.

Here's how I teach it:

- study functions $\bR^2 \to \bR$
  (show 'em visually as elevation maps;
   draw contour lines, talk about mountain charts,
     (draw a hill; draw a cliff)
   joke about my mishap in Yerushalem from not noticing that;
   joke that now I understand why getting a footbath was such a big
   deal back then)
NB: isopleths are contour lines;
isallopleths are perpendicular to these; they're integral curves for
the gradient vector field (that gives the speed parametrization).

- define partial derivatives
  some useful notation: $f_x := \frac{\partial f}{\partial x}$

- look at the -total- derivative, $df$, and see how it is a functional
  on tangent vectors: it shows how an infintesimal move in a given
  direction affects the value of $f$
  $df=\frac{\partial f}{\partial x} dx + \frac{\partial f}{\partial y} dy$
  (implicitly there's linear algebra going on... (pun intended))
  [! yes, this may be people's first functional !]
  [Note that -this easy computation- really clarifies what it means by
   ``the derivative is a local linear approximation'',
   at least in the sense of -several dimensions-
     (but not in the sense of asymptotics: you need Taylor for that)
   (and makes the definition of derivative in higher dimensions
    -make sense-)]

- note that moving in the direction $\grad f$
  (not so coincidentally, but confusingly, the same coordinates as $df$)
  maximizes growth (for given length; lil' tricky to prove)

- and that moving perpendicular to it results in zero growth,
  infintesimally
  (this proves that grad maximizes growth)

- conclude that $df = 0$ infintesimally keeps you on a level set,
  and thus defines the tangent line (more generally, hyperplane)
  to a level set at point.

- note that the same philosophy allows you to do Lagrange multipliers:
  so long as the steepest ascent for $f$ (the objective function) is
  not the same as for $g$ (the constraint), there's some direction you
  can go to increase $f$: when they're equal, you're stuck.
