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.

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 (

## 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 ℝ
is given by

^{3}, thought of as the locus of points a unit distance from the origin. If we put coordinates*θ, ф*on*S*^{2}(the sphere) and*x,y,z*on ℝ^{3}, the map*ψ*(

*θ, ф*) = (sin

*θ*cos

*ф*, sin

*θ*sin

*ф*, cos

*θ*) (1)

In the book this equation is written as

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

*ф*(

*θ, ф*) = (sin

*θ*cos

*ф*, sin

*θ*sin

*ф*, cos

*θ*) (A.11)

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

Sticking the sphere into

*ds² = dx² + dy² + dz²*(1a)

and substitute (1) into this expression yielding a metric .

*d**θ**² +*sin²*θ**d**ф**²*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, d**θ*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

*∂*. It is the transpose of the matrix for_{x}y_{α}*∂*. I will add this to my up coming opus on Tensors, Matrices and Indexes._{x}y_{α}
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*into Cartesian^{j}*x*with equations^{i}*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.