# The Lie bracket of vector fields (A) and (B) is defined as their commutator, ([A, B]=) (A

## Question:

The Lie bracket of vector fields \(A\) and \(B\) is defined as their commutator, \([A, B]=\) \(A B-B A\). The Lie bracket of two basis vectors vanishes for a coordinate basis but not for a non-coordinate basis. For the 2D plane parameterized with cartesian \((x, y)\) or polar \((r, \theta)\) coordinates and the bases

show that \(\left(e_{1}, e_{2}\right)_{1}\) and \(\left(e_{1}, e_{2}\right)_{2}\) are coordinate bases, but that \(\left(e_{1}, e_{2}\right)_{3}\) is not.

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**Related Book For**

## Symmetry Broken Symmetry And Topology In Modern Physics A First Course

**ISBN:** 9781316518618

1st Edition

**Authors:** Mike Guidry, Yang Sun