[BUG] lagrange multipliers and econ

[Smith, Alexander J.]

It would seem that Lagrange multipliers do make sense, even if the gradient does not.

To find a critical point to  z=f(x,y) subject to the constraint g(x,y)=c more or less  boils down to the observation that if the level curve of f cuts the curve g(x,y)=c transversally, then you can't possibly be at a critical point because an infinitesimal step along the constraint curve will take you to a higher or lower level in z. Thus a min or max will be at a place where the level curve of f is tangential to g(x,y)=c. This much makes sense without a gradient. 

It is be convenient to quantify the condition that the two curves meet tangentially by saying that their gradients are tangential, but this is just gravy. 

.................


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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