[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

[JCH2CNAGDH_20150402] Vector Rotation via Complex Numbers

Question

The points  D  and  G  in an Argand diagram represent the complex numbers  3i  and  6 + 7i  respectively.  DG  is in fact a diagonal of the square  DFGH.  Find the complex numbers represented by  F  and  H.

Introduction

     This is a question that tests students’ concept of the geometry complex numbers.  The key to solving this question is to understand that multiplying a complex number  re^iq  gives a stretching effect, as well as a rotating effect.  The stretch is by  r  times  and the rotation is by  q  radians anti-clockwise.  If  r = 1,  then there is just rotation and effectively no stretching.

Important Principles

Solution