5

I find the field of partial differentiation of complicated multivariable expressions to have many subtleties which lead to unexpected results, or to my applying operations incorrectly.

I'm wondering if there is some established set of exercises which address the common "gatcha" situation. I'm particularly interested in inverse and implicit mappings.

Any suggestions? I have Edwards's Advanced Calculus of Several Variables, and will admit to not having worked many of the exercises (yet).

Steven Thomas Hatton
  • 3,353
  • 4
    I don't have the same impression, except I do believe that bad notation as well as poor explanations sometimes cause unnecessary confusion. What's an example of a "gotcha" situation you have encountered? – littleO Jun 14 '17 at 20:12
  • Agree with OP. I actually never realized what vast swaths of complex analysis were essentially specialized results for multivariate functions--largely because of incompetent education in both--until I worked through Ahlfors' book on the subject. In retrospect the connection seems obvious. – JMJ Jun 14 '17 at 20:16
  • @littleO There isn't space in the comment field to provide and example but these are two recent examples Inverting a derivative matrix or Un-proving 1=-1 – Steven Thomas Hatton Jun 14 '17 at 20:57
  • 2
    You might be interested in this partial derivative paradox that a former student of mine stumbled on in April 1998. – Dave L. Renfro Jun 14 '17 at 21:01
  • 1
    In the "1=-1" post it seems that the confusion was caused by giving two different functions the same name. For some reason people do this a lot when computing partial derivatives (especially when using the chain rule). It can cause confusion, but it's easily avoided by using more careful notation. – littleO Jun 15 '17 at 03:28
  • I agree with littleO's assessment that confusions with partial derivatives are largely due to ambiguous notation. I suggest reading the preface to Sussman and Wisdom's Structure and Interpretation of Classical Mechanics, and following their recommendation to read Spivak's Calculus on Manifolds, which uses an unambiguous functional notation instead. (links are to full text) –  Dec 21 '17 at 23:36

0 Answers0