[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

[H2_20150512GSS] A Square Spiral in Geometric Progression

Question

Introduction

     This is a type of question that has become quite common in H2 Mathematics at Junior College level.  I was not told where this question came from, but I seem to get a bout of déjà vu.  It is quite a common practice for the Singapore junior colleges to mimic questions from the GCE ‘A’ Levels and/or one another.

Solution

Summary

     The keys to solving this question are:-

          (1)  if a coordinate frame is needed but not given it the question, we invent our own.

          (2)  see whether we need to care about the direction or not, and then apply the
                 geometric series formula.

          (3)  When applying the geometric series formula, always identify correctly the first term,
                 the common ratio, and the number of terms to be summed.

Heuristics Used

H02. Use a diagram / model

H04. Look for pattern(s)

H09. Restate the problem in another way

H10. Simplify the problem

H11. Solve part of the problem

H13* Use Equation / write a Mathematical Sentence

Suitable Levels

* GCE ‘A’ Level H2 Mathematics

* other syllabuses that involve geometric series