Introduction to Electrodynamics
Introduction to Electrodynamics
4th Edition
ISBN: 9781108420419
Author: David J. Griffiths
Publisher: Cambridge University Press
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Chapter 1.4, Problem 1.42P
To determine

To Express: The cylindrical unit vectors s^ , ϕ , z^ in terms of x^ , y^ , z^ . Invert to get x^ , y^ , z^ in terms of s^ , ϕ , z^ .

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(a) Express the spherical unit vectors ê, ê, in terms of the Cartesian unit vectors ✰, ŷ, 2 (that is, derive Eq. 1.64 of Griffiths). Also work out the inverse formulas, giving ✰, ŷ, 2 in terms of f, 0, $ (and 0, $). Calculate af/00 and af/ap, and express them in terms of spherical unit vectors. (b) Express the cylindrical unit vectors ŝ, , 2 in terms of the Cartesian unit vec- tors î, ŷ, 2 (that is, derive Eq. 1.75 of Griffiths). Also work out the inverse formulas, giving x, ŷ, 2 in terms of ŝ, $, 2 (and ). Show that af/0 = $.
Problem 1.45. As an illustration of why it matters which variables you hold fixec when taking partial derivatives, consider the following mathematical example. Le w = ry and z = yz. (a) Write w purely in terms of x and z, and then purely in terms of y and z. (b) Compute the partial derivatives and and show that they are not equal. (Hint: To compute (ðu/dz)y, use i formula for w in terms of r and y, not z. Similarly, compute (ðu/ar}: from a formula for w in terms of only z and z.)
1.1 Give a physical interpretation of what is meant by the curl of a vector. 1.2 Suppose a vector function A is given by A = -y i + x j and another vector function B is given by B = x j. Calculate (i) the curl of A: and (ii) the curl of B: 1.3 In which direction are the curls pointing? Hence what can you say about their divergence and why?

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Introduction to Electrodynamics

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