[U_Calculus_20150520DCPL] Different Coordinate Systems, Same Lengths

Question

Introduction

     I got this question from a student who as studying AP Calculus under a school teacher who went beyond the syllabus.  If we try to do this question directly, it will be a tedious mess without any insight.  Is calculus just a mindless torture?  Is there a better way to look at the problem?

Solution

Remarks

     Observe that both the LHS and the RHS are squares of lengths of the vector gradient  Ñw  of  w  in their respective coordinate systems (polar for LHS and rectangular for RHS).  The key insight is that the Jacobian-like matrix  J  represents a rotation, which common sense tells us preserves lengths.  So it is not surprising that the LHS and RHS are equal.  This is one of the “evidences” that the vector gradient is a concept that transcends coordinate systems, and represents something “real and physical”.  Indeed the gradient  Ñw  is the vector that represents the change of  w  per unit distance in its direction of maxium increase.

HeuristicsUsed

H04. Look for pattern(s)

H09. Restate the problem in another way

H10. Simplify the problem

H12* Think of a related problem

H13* Use Equation / write a Mathematical Sentence

Suitable Levels

* University Level Mathematics  (Calculus, Vector Calculus)

* AP Calculus students who wish to stretch themselves / are being stretched

* other syllabuses that involve calculus and coordinate systems