Wednesday, May 20, 2015
[U_Calculus_20150520DCPL] Different Coordinate Systems, Same Lengths
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?
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.
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
* 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