[BUG] A geometric approach to gradients

[Tevian Dray]

>>>>> Martin Jackson writes:

    MJ> Tevian, can you give us more detail on how you want students to
    MJ> think about your RESOLUTION?  In particular, how do you want them to
    MJ> think about counting "the number of contour lines" in computing the
    MJ> number per unit distance?  Does Delta f enter into this explicitly?

Yes, I think it must.  I like talking about the "number" of contour lines
because I think it's easier for students to comprehend -- you can easily see
how many lines are crossed, and it's then quite intuitive to see directional
derivatives as projections, that is, dot products.  I called it "resolution"
because the natural units are something like "lines per inch".  But of course
what's really being described is the change in the value of the function.

Be aware that I made this argument initially for a linear function, for which
I didn't have to distinguish between the graph and the tangent plane.  Any
effort to make this argument for a nonlinear function should really address
the fact that the "contours" on the tangent plane only approximate the change
in the function.  I didn't address this, and felt that most students weren't
bothered by it.  Not sure if this is your point, nor how careful one must be
here.

Tevian