Commentary on Appendix A: Mapping S2 and R3

Sometimes I almost despair reading this book. I cannot understand part of it or find a solution. When I do, the feeling of joy is superb. There is much air punching. This is a case in point.

Why am I here?

I'm still thinking about commutators and wanted to understand Appendix B on Diffeomorphisms and Lie Derivatives. Lie (pronounced lee) brackets are the same as commutators. Appendix B starts by saying "we continue the explorations of the previous Appendix", so I was thrown back. Appendix A is fairly comprehensible and there is a good example at the end. I'll restate the first part here:

The problem part 1

Consider the two-sphere embedded in ℝ³, thought of as the locus of points a unit distance from the origin. If we put coordinates θ, ф on S² (the sphere) and x,y,z on ℝ³, the map  is given by

ψ (θ, ф) = (sinθ cosф, sinθ sinф, cosθ)          (1)

In the book this equation is written as

    ф(θ, ф) = (sinθ cosф, sinθ sinф, cosθ)            (A.11)

ф is used in the equation to mean a map and a coordinate. I will avoid this.

Sticking the sphere into  in this way induces a metric on , which is just the pull-back of the flat-space metric. The simple-minded way to find this is to start with the metric

             ds² = dx² + dy² + dz²                               (1a)

and substitute (1) into this expression yielding a metric dθ² + sin²θ dф² on S².

This was not so simple and I struggled with the substitution for days. (1) is easy to show algebraically or geometrically. Getting to the metric in polar coordinates involves various differentiation rules: (trig functions and the product rule), but then they need to be applied to infinitesimals (dx, dθ etc) rather than proper derivatives (dx/dy etc). That aspect was very novel to me.

Along the way I made an interesting observation about writing the matrix for ∂_xy_α. It is the transpose of the matrix for ∂_xy_α. I will add this to my up coming opus on Tensors, Matrices and Indexes.

Five pages. You can read it here: Commentary App A Mapping S2 and R3.pdf

It contains a very useful corollary

It contains a very useful corollary on how to get the metric of a non-Cartesian system.

If we can transform non-Cartesian components p ^j into Cartesian x ⁱ with equations

x ⁱ = f ⁱ (p ^j)                                                (20)

i = 1..m,   j = 1..n 

Then we can construct the (n x m) pullback operator

Multiplying this by its transpose gives the metric of the non-Cartesian system. 

Of necessity this is a (n x m) x (m x n) = (n x n) matrix, as it must be.