Introduction to Electrodynamics
4th Edition
ISBN: 9781108420419
Author: David J. Griffiths
Publisher: Cambridge University Press
expand_more
expand_more
format_list_bulleted
Question
Chapter 1.4, Problem 1.42P
To determine
To Express: The cylindrical unit vectors
Expert Solution & Answer
Want to see the full answer?
Check out a sample textbook solutionStudents 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 = $.
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?
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
Knowledge Booster
Similar questions
- Prove that for any vector u, u = (u e,)e, +(u ·e, )e, + (u e,)e,. [Hint: write u in component form.] Find the projection of the vector u e, -2e, +e, on the vector 2. 3. v = 4e, - 4e, + 7e,.arrow_forwardProblem 1.15 Calculate the divergence of the following vector functions: (a) va = x² + 3xz²ŷ - 2xz2.arrow_forwardProblem 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.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_forward1.4 Draw the cubic (111), (011) planes and the [011] direction on the diagram of a cube in figur (attached sheet). Clearly show the primitive vectors on the diagram.arrow_forward1.32 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 ACarrow_forward
- Find the angle between u = 3e, +2e, - 6e, and v = 4e, – 3e, +e, . Write down an expression for a unit vector parallel to the resultant of two vectors u and v (in symbolic notation). Find this vector when u = 2e, +4e, - 5e,, 4. 5. V = e, +2e, +3e, (in component form). Check that your final vector is indeed a unit vector.arrow_forward1.4 Please answer these (a,b,). Explain well your calculations and steps! Thank you!!! A) Uses the scalar vector product to check if the three points J(-3, -4), K(5, 0) and L (2,6) are the vertices of a right angle. B) The speed of an aircraft is 415 km/h according to a 120° cap. A wind of 30 km/h blows at 45°. C) Determines the velocity vector resulting from the aircraft in relation to the North. (Do this problem by algebraic representation.) 3. Determines a unit vector going in the opposite direction to the vector at = 1-2,4].arrow_forwardGiven the points T (5, 20°, 120°) to P (2, 80°, 30°): 1.) find the unit vector in cartesian coordinates at T that is directedtoward P. 2.) find the unit vector in spherical coordinates at T that is directedtoward P.arrow_forward
- Express the ff. as a single 2 × 2 matrix. exp (it)arrow_forwardLet vectors A =(2,1,−4), B =(−3,0,1), and C =(−1,−1,2).Calculate the following: What is the angle θAB between vector A and vector B?arrow_forwardFind the vectors that satisfy the stated conditions. NOTE: Enter the exact answers using the provided round brackets.. (a) Oppositely directed to v = (7, -20) and half the length of v. W = (?, ?) (b) Length √13 and same direction as v = (1, 0, -5). (?, ?, ?) W =arrow_forward
arrow_back_ios
SEE MORE QUESTIONS
arrow_forward_ios
Recommended textbooks for you
- College PhysicsPhysicsISBN:9781305952300Author:Raymond A. Serway, Chris VuillePublisher:Cengage LearningUniversity Physics (14th Edition)PhysicsISBN:9780133969290Author:Hugh D. Young, Roger A. FreedmanPublisher:PEARSONIntroduction To Quantum MechanicsPhysicsISBN:9781107189638Author:Griffiths, David J., Schroeter, Darrell F.Publisher:Cambridge University Press
- Physics for Scientists and EngineersPhysicsISBN:9781337553278Author:Raymond A. Serway, John W. JewettPublisher:Cengage LearningLecture- Tutorials for Introductory AstronomyPhysicsISBN:9780321820464Author:Edward E. Prather, Tim P. Slater, Jeff P. Adams, Gina BrissendenPublisher:Addison-WesleyCollege Physics: A Strategic Approach (4th Editio...PhysicsISBN:9780134609034Author:Randall D. Knight (Professor Emeritus), Brian Jones, Stuart FieldPublisher:PEARSON
College Physics
Physics
ISBN:9781305952300
Author:Raymond A. Serway, Chris Vuille
Publisher:Cengage Learning
University Physics (14th Edition)
Physics
ISBN:9780133969290
Author:Hugh D. Young, Roger A. Freedman
Publisher:PEARSON
Introduction To Quantum Mechanics
Physics
ISBN:9781107189638
Author:Griffiths, David J., Schroeter, Darrell F.
Publisher:Cambridge University Press
Physics for Scientists and Engineers
Physics
ISBN:9781337553278
Author:Raymond A. Serway, John W. Jewett
Publisher:Cengage Learning
Lecture- Tutorials for Introductory Astronomy
Physics
ISBN:9780321820464
Author:Edward E. Prather, Tim P. Slater, Jeff P. Adams, Gina Brissenden
Publisher:Addison-Wesley
College Physics: A Strategic Approach (4th Editio...
Physics
ISBN:9780134609034
Author:Randall D. Knight (Professor Emeritus), Brian Jones, Stuart Field
Publisher:PEARSON