[BUG] A geometric approach to gradients

[Matthias Kawski]

On Tue, 22 Feb 2005, Martin Jackson wrote:

I very enjoyed reading your development and may borrow some
details from you next I time I teach this...

But one comment re your question in item 3:

> 3.  I've defined gradient in the linear case to have direction
> perpendicular to the level curves.  Tevian defines the direction to
> be "that in which f increases the fastest."  I'm not sure if one
> approach is significantly better than the other.  For me, the
> direction that is geometrically obvious in looking at the contour
> diagram is the one perpendicular to the level curves so this seems
> like the more natural definition.  The fact that this direction gives
> the greatest rate of change requires a bit of reasoning and seems
> more natural as a conclusion.

As long as the units in x and y direction are the same
(meters or dollars), there is no real issue. But as soon
as the variables are e.g. (pressure,volume) or (voltage,
current) then perpendicular (and dot product) make(s) no
sense anymore. But reading contour diagrams in such
settings (thermo, or electrical devices) is still important,
and there is the still need for derivatives and tangent
plane / linearization. Here working with differentials
is still fine, but gradients ask for much caution (e.g.
fastest increase looses meaning if we have no "unit"
length in the domain in directiosn other than parallel
to the axes.)

More generally, the perpendicularity issue appears as soon
as anything is graphed using different scales along the
axes. E.g. plot the lines y=10x and y=-10x on the window
[-1,1] x [-10,10].

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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