A coordinate system has basis vectors and metric

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Tensors and differential geometry · Tensors and differential geometry

Invariance, Ch General bases, the metric and tensors in spacetime

Problem

A coordinate system

u^i

has basis vectors

\mathbf e_i = \partial\mathbf r/\partial u^i

and metric

g_{ij} = \mathbf e_i\cdot\mathbf e_j

. A second, primed coordinate system

u'^i

is related to

u^i

by smooth, invertible functions

u^i(u'^1,\dots,u'^n)

. ((a)) Starting from the chain rule applied to

\mathbf e_i

, derive how

g'_{ij}\equiv\mathbf e'_i\cdot\mathbf e'_j

is related to

g_{ij}

, and identify the result as the transformation law for a tensor with two lower indices. ((b)) Cartesian coordinates

(x,y)

, in metres, have

g_{ij}=\delta_{ij}

. Apply your law to plane polar coordinates

(r,\theta)

, defined by

x=r\cos\theta

y=r\sin\theta

with

r

in metres and

\theta

in radians, and check that it reproduces

g_{rr}

g_{r\theta}

and

g_{\theta\theta}

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