1. Let S be a smooth regular surface. Suppose there is a coordinate charton S defined as i : U → S, where xu · Xv:0, xu · Xuf (v), x, · Xy = h(v),f and h are smooth functions. Write down the Christoffel symbols of first andsecond kind under this coordinate using f, h and their derivatives.

We have; xu.xu=f(v)xv.xv=h(v)and,xu.xv=0⇒xv.xu=0considering S to be a torsion free surfaceSo, the…

1. Let S be a smooth regular surface. Suppose there is a coordinate chart on S defined as i : U –→ S, where xu · Xv f and h are smooth functions. Write down the Christoffel symbols of first and second kind under this coordinate using f, h and their derivatives. 0, xu · Xu f(v), xv · Xv h(v),

1. Let S be a smooth regular surface. Suppose there is a coordinate chart on S defined as i : U –→ S, where xu · Xv f and h are smooth functions. Write down the Christoffel symbols of first and second kind under this coordinate using f, h and their derivatives. 0, xu · Xu f(v), xv · Xv h(v),

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

1. Let S be a smooth regular surface. Suppose there is a coordinate chart
on S defined as i : U → S, where xu · Xv
:0, xu · Xu
f (v), x, · Xy = h(v),
f and h are smooth functions. Write down the Christoffel symbols of first and
second kind under this coordinate using f, h and their derivatives.

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