The two stereographic charts of the chapter are related on their overlap by

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

The Geometry of Gravity, Ch Manifolds, vectors, tensor fields and the metric

Problem

R_S=a^2/R_N

, with the same

\phi

in both. Consider the point where

R_N=2a

(so

R_S=a/2

), and at that point the tangent vector

\mathbf V=\partial_{R_N}

and the covector

\boldsymbol\omega=dR_N

(the gradient of the coordinate function

R_N

itself). ((a)) Starting from the basis chain rules

\partial_{\mu'}=(\partial x^\mu/\partial x^{\mu'})\partial_\mu

and

dx^{\mu'}=(\partial x^{\mu'}/\partial x^\mu)dx^\mu

, derive general formulas for

V^{R_S}

in terms of

V^{R_N}

and for

\omega_{R_S}

in terms of

\omega_{R_N}

, for a vector or covector with no

\phi

component. ((b)) Evaluate the Jacobian

\partial R_S/\partial R_N

and the reciprocal Jacobian

\partial R_N/\partial R_S

R_N=2a

, and use them to find

V^{R_S}

and

\omega_{R_S}

there. ((c)) Compute

\boldsymbol\omega(\mathbf V)

directly in the

(R_N,\phi)

chart, and again from the components found in part (b) in the

(R_S,\phi)

chart. Explain why the two must agree, whatever the two charts are.

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