Introduction to Electrodynamics
4th Edition
ISBN: 9781108420419
Author: David J. Griffiths
Publisher: Cambridge University Press
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Chapter 1.4, Problem 1.41P
To determine
Gradient and Laplace of the function in spherical and Cartesian coordinates.
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Students have asked these similar questions
(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 = $.
1.32 In the Eigen vector equation AX = X, the operator A is given
32
A =
41
Find:
(a) The Eigen values
(b) The Eigen vector X
(c) The modal matrix C and it's inverse C-1
(d) The product C-1 AC
Problem 2.16 The divergence of a vector F is given by
1 a(pF,) , 13F, , ƏF,
V.F =
+
az
Evaluate V · F for
F(p.0.2) = pp+z sin ø4+Jpzk.
Answer: 2+zcoso/p+VP/z.
Chapter 1 Solutions
Introduction to Electrodynamics
Ch. 1.1 - Using the definitions in Eqs. 1.1 and 1.4, and...Ch. 1.1 - Prob. 1.2PCh. 1.1 - Prob. 1.3PCh. 1.1 - Prob. 1.4PCh. 1.1 - Prob. 1.5PCh. 1.1 - Prob. 1.6PCh. 1.1 - Prob. 1.7PCh. 1.1 - Prob. 1.8PCh. 1.1 - Prob. 1.9PCh. 1.1 - Prob. 1.10P
Ch. 1.2 - Prob. 1.11PCh. 1.2 - The height of a certain hill (in feet) is given by...Ch. 1.2 - Prob. 1.13PCh. 1.2 - Prob. 1.14PCh. 1.2 - Prob. 1.15PCh. 1.2 - Prob. 1.16PCh. 1.2 - Prob. 1.17PCh. 1.2 - Prob. 1.18PCh. 1.2 - Prob. 1.19PCh. 1.2 - Prob. 1.20PCh. 1.2 - Prob. 1.21PCh. 1.2 - Prob. 1.22PCh. 1.2 - Prob. 1.23PCh. 1.2 - Prob. 1.24PCh. 1.2 - Prob. 1.25PCh. 1.2 - Prob. 1.26PCh. 1.2 - Prob. 1.27PCh. 1.2 - Prob. 1.28PCh. 1.3 - Prob. 1.29PCh. 1.3 - Prob. 1.30PCh. 1.3 - Prob. 1.31PCh. 1.3 - Prob. 1.32PCh. 1.3 - Prob. 1.33PCh. 1.3 - Prob. 1.34PCh. 1.3 - Prob. 1.35PCh. 1.3 - Prob. 1.36PCh. 1.4 - Prob. 1.37PCh. 1.4 - Express the unit vectors in terms of (that is,...Ch. 1.4 - Prob. 1.39PCh. 1.4 - Prob. 1.40PCh. 1.4 - Prob. 1.41PCh. 1.4 - Prob. 1.42PCh. 1.4 - Prob. 1.43PCh. 1.5 - Evaluate the following integrals:
(a)
(b)
(c)...Ch. 1.5 - Prob. 1.45PCh. 1.5 - (a) Show that .
[Hint: Use integration by...Ch. 1.5 - Prob. 1.47PCh. 1.5 - Prob. 1.48PCh. 1.5 - Prob. 1.49PCh. 1.6 - (a) Let and . Calculate the divergence and curl...Ch. 1.6 - Prob. 1.51PCh. 1.6 - Prob. 1.52PCh. 1.6 - Prob. 1.53PCh. 1.6 - Prob. 1.54PCh. 1.6 - Prob. 1.55PCh. 1.6 - Prob. 1.56PCh. 1.6 - Prob. 1.57PCh. 1.6 - Prob. 1.58PCh. 1.6 - Prob. 1.59PCh. 1.6 - Prob. 1.60PCh. 1.6 - Prob. 1.61PCh. 1.6 - Prob. 1.62PCh. 1.6 - Prob. 1.63PCh. 1.6 - Prob. 1.64P
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- 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?arrow_forwardProblem 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.)arrow_forwardthe surface S is defined by the sphere R? = x? + y + z?. Do the integral directly and also by using Gauss's theorem. 1-38. Find the value of the integral fs (V x A) da if the vector A = yi + zj + xk and Sis the surface defined by the paraboloid z = 1- x-y', where z2 0.arrow_forward
- 1.38 You lay a rectangular board on the horizontal floor and then tilt the board about one edge until it slopes at angle 0 with the horizontal. Choose your origin at one of the two corners that touch the floor, the x axis pointing along the bottom edge of the board, the y axis pointing up the slope, and the z axis normal to the board. You now kick a frictionless puck that is resting at O so that it slides across the board with initial velocity (vox, voy, 0). Write down Newton's second law using the given coordinates and then find how long the puck takes to return to the floor level and how far it is from O when it does so.arrow_forwardProblem 1.31 Check the fundamental theorem for gradients, usingT =x²+4xy+2yz³, the points a = (0,0, 0), b=(1, 1,1), and the three paths in Fig. 1.28: (a) (0,0, 0) -→ (1.0,0) → (1. 1,0) → (1, 1. 1); (b) (0, 0, 0) → (0,0, 1) → (0, 1, 1) → (1, 1, 1); (c) the parabolic path z = x2; y =x. (a) (b) Figure 1.28arrow_forwardProblem 1.32 Check the fundamental theorem for gradients, using T =x²+ ANy +2yz' the points a = (0, 0. 0), b = (1,D, and the three paths in Fig. 1.28: Ka) (0.0.0) → K1,0.0)→ (1,0)– (1,1,1). (b) (0,0, 0) → KOJ 0, 1) → (0. 1! 1)÷(1: (c) the parabolic path z = x: N= (a) (b) (c) FIGURE 1.28 Eunda mor orom for vorgenoarrow_forward
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