A Student's Guide to General Relativity by Norman Gray

Author:Norman Gray
Language: eng
Format: epub, pdf
Tags: science, Cosmology, Physics, Mathematical & Computational, General, Relativity
Publisher: Cambridge University Press
Published: 2019-01-03T03:30:24.610000+00:00

to transform the components of the vectors and which you obtained in (c), into the polar basis {er, eθ}, and show that

[u++]

Exercise 3.5 (§ 3.1.4)Define a scalar field, ϕ, by

for cartesian coordinates {x, y}.

(a) From Eq. (3.5), the ith component of the gradient one-form is obtained by taking the contraction of the gradient with the basis vector ei = ∂/∂xi. Thus write down the components of the gradient one-form with respect to the cartesian basis.

(b) The result of Exercise 2.10 says that the transformation law for the components of a one-form is

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