[BUG] grad(f) and df

[Smith, Alexander J.]

It is amusing that df and linearization always makes sense, but grad(f) only makes sense if the "units" are the same, or equivalently, if there is a notion of angle or perpendicularity. I enjoy bringing contour diagrams that can be found in the Harvard calc books into class that have axes like "growing-degree days" and "rainfall", with the countours correspond to crop production in something like bushels or tons. A favorite extra credit assignment of mine is to write a paragraph explaining why gradients do not make sense here, even though the differential does make sense. Warning: be ready for some really bad answers because the point is subtle for undergrads. 

You can't add apples and oranges, and you can't add feet and pounds, but with without any extra work you can multiply feet and pounds.

Alex

-----Original Message-----
From: bug-bounces at science.oregonstate.edu on behalf of Martin Jackson
Sent: Tue 2/22/2005 4:17 PM
To: bug at science.oregonstate.edu
Subject: Re: [BUG] A geometric approach to gradients
 
>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
>**********************************************************
>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
>**********************************************************

Good point.  It also serves to remind me that I need to consider 
students other than those majoring in math and physics.  Economics 
and chemistry majors probably don't use much geometric reasoning when 
applying multivariate calculus ideas in their own disciplines.  For 
these students, seeing df as a sum of (rate with respect to 
variable)*(change in variable) terms might be more useful than df as 
gradient dotted with vector dr.  Does anyone on this list know if 
economists think geometrically about Lagrange multipiers?  Do chemist 
think geometrically about derivatives in, say, thermodynamics?
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