Mathematical Methods in the Physical Sciences
3rd Edition
ISBN: 9780471198260
Author: Mary L. Boas
Publisher: Wiley, John & Sons, Incorporated
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Textbook Question
Chapter 13.6, Problem 8P
Do Problem 6 in polar coordinates to find the eigenfunctions and energy eigenvalues of a particle in a circular box
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Suppose that i = Ax. Draw the phase portrait where the 2 x 2 matrix A has
complex eigenvalue A = +5i with i 0.
Find all values of the angle 0 (in radians, with 0 < 0 < 2n) for which the matrix
Cos e -sin 0
A =
sin 0
Cos O
has real eigenvalues. (Enter your answers as a comma-separated list.)
rad
Interpret your answer geometrically.
O The only rotation that sends vectors to multiples of themselves is the 180° rotation.
O The only rotations that send vectors to multiples of themselves are the identity and the 180° rotation.
O There are no rotations that send vectors to multiples of themselves.
The only rotation that sends vectors to multiples of themselves is the identity rotation.
Need Help?
Read It
Find all values of the angle 0 (in radians, with 0 < 0 < 2n) for which the matrix
cos e -sin 0
A =
sin 0
cos e
has real eigenvalues. (Enter your answers as a comma-separated list.)
rad
Interpret your answer geometrically.
O The only rotation that sends vectors to multiples of themselves is the 180° rotation.
O The only rotation that sends vectors to multiples of themselves is the identity rotation.
O There are no rotations that send vectors to multiples of themselves.
O The only rotations that send vectors to multiples of themselves are the identity and the 180° rotation.
Chapter 13 Solutions
Mathematical Methods in the Physical Sciences
Ch. 13.1 - Assume from electrostatics the equations E=/0 and...Ch. 13.1 - Show that the expression u=sin(xvt) describing a...Ch. 13.1 - Assume from electrodynamics the following...Ch. 13.1 - Obtain the heat flow equation (1.3) as follows:...Ch. 13.2 - After you find the series solution of a problem,...Ch. 13.2 - T=0,0x10,100,10x20. Solve the semi-infinite plate...Ch. 13.2 - Solve the semi-infinite plate problem if the...Ch. 13.2 - Solve the semi-infinite plate problem if the...Ch. 13.2 - Show that the solutions of (2.5) can also be...Ch. 13.2 - Show that the series in (2.12) can be summed to...
Ch. 13.2 - Solve Problem 3 if the plate is cut off at height...Ch. 13.2 - Find the steady-state temperature distribution in...Ch. 13.2 - Solve Problem 2 if the plate is cut off at height...Ch. 13.2 - Find the steady-state temperature distribution in...Ch. 13.2 - Find the steady-state temperature distribution in...Ch. 13.2 - Find the temperature distribution in a rectangular...Ch. 13.2 - Find the steady-state temperature distribution in...Ch. 13.2 - In the rectangular plate problem, we have so far...Ch. 13.2 - Consider a finite plate, 10cm by 30cm, with two...Ch. 13.2 - Show that there is only one function u which...Ch. 13.3 - Verify the coefficients in equation (3.14).Ch. 13.3 - A bar 10 cm long with insulated sides is initially...Ch. 13.3 - In the initial steady state of an infinite slab of...Ch. 13.3 - At t=0, two flat slabs each 5cm thick, one at 0...Ch. 13.3 - Prob. 5PCh. 13.3 - Show that the following problem is easily solved...Ch. 13.3 - A bar of length l with insulated sides has its...Ch. 13.3 - A bar of length 2 is initially at 0. From t=0 on,...Ch. 13.3 - Solve Problem 8 if, for t0, the x=0 end of the bar...Ch. 13.3 - Separate the wave equation (1.4) into a space...Ch. 13.3 - Solve the particle in a box problem to find (x,t)...Ch. 13.3 - Do Problem 11 if (x,0)=sin2x on (0,1).Ch. 13.4 - Complete the plucked string problem to get...Ch. 13.4 - A string of length l has a zero initial velocity...Ch. 13.4 - Solve Problem 2 if the initial displacement is:Ch. 13.4 - Solve Problem 2 if the initial displacement is :Ch. 13.4 - A string of length l is initially stretched...Ch. 13.4 - Do Problem 5 if the initial velocity V(x)=(y/t)t=0...Ch. 13.4 - Solve Problem 5 if the initial velocity is:Ch. 13.4 - Solve Problem 5 if the initial velocity is...Ch. 13.4 - In each of the Problems 1 to 8,find the frequency...Ch. 13.4 - Verify that, if k=nT, then the sum of the two...Ch. 13.4 - Verify (4.16) and find a similar formula for a...Ch. 13.4 - In Sections 2, 3, 4, we have solved a number of...Ch. 13.4 - Do Problem 12 for f(x)=1cos2x on (0,).Ch. 13.4 - Do Problem 12 for f(x)=xx3 on (0, 1).Ch. 13.5 - Compute numerically the coefficients (5.16) of the...Ch. 13.5 - Find the steady-state temperature distribution in...Ch. 13.5 - Find the steady-state temperature distribution in...Ch. 13.5 - A flat circular plate of radius a is initially at...Ch. 13.5 - Do Problem 4 if the initial temperature...Ch. 13.5 - Consider Problem 4 if the initial temperature...Ch. 13.5 - Find the steady-state temperature distribution in...Ch. 13.5 - Water at 100 is flowing through a long pipe of...Ch. 13.5 - Find the steady-state distribution of temperature...Ch. 13.5 - A cube is originally at 100. From t=0 on, the...Ch. 13.5 - The following two R(r) equations arise in various...Ch. 13.5 - Separate Laplaces equation in two dimensions in...Ch. 13.5 - Find the steady-state distribution of temperature...Ch. 13.5 - Find the steady state temperature distribution in...Ch. 13.5 - Solve Problem 14 if the temperatures of the two...Ch. 13.6 - Continue Figure 6.1 to show the fundamental modes...Ch. 13.6 - Prob. 2PCh. 13.6 - Separate the wave equation in two-dimensional...Ch. 13.6 - Find the characteristic frequencies for sound...Ch. 13.6 - A square membrane of side l is distorted into the...Ch. 13.6 - Let V=0 in the Schrödinger equation (3.22) and...Ch. 13.6 - In your Problem 6 solutions, find some examples of...Ch. 13.6 - Do Problem 6 in polar coordinates to find the...Ch. 13.7 - Find the steady-state temperature distribution...Ch. 13.7 - Find the steady-state temperature distribution...Ch. 13.7 - Find the steady-state temperature distribution...Ch. 13.7 - Find the steady-state temperature distribution...Ch. 13.7 - Find the steady-state temperature distribution...Ch. 13.7 - Find the steady-state temperature distribution...Ch. 13.7 - Find the steady-state temperature distribution...Ch. 13.7 - Find the steady-state temperature distribution...Ch. 13.7 - Find the steady-state temperature distribution...Ch. 13.7 - Find the steady-state temperature distribution...Ch. 13.7 - Find the steady-state temperature distribution...Ch. 13.7 - Do Problem 11 if the curved surface is held at...Ch. 13.7 - Find the electrostatic potential outside a...Ch. 13.7 - Find the steady-state temperature distribution in...Ch. 13.7 - A sphere initially at 0 has its surface kept at...Ch. 13.7 - Separate the wave equation in spherical...Ch. 13.7 - Do Problem 6.6 in 3 dimensional rectangular...Ch. 13.7 - Prob. 18PCh. 13.7 - Find the eigenfunctions and energy eigenvalues for...Ch. 13.7 - Write the Schrödinger equation (3.22) if is a...Ch. 13.7 - Prob. 21PCh. 13.7 - Find the energy eigenvalues and eigen functions...Ch. 13.8 - Show that the gravitational potential V=Gm/r...Ch. 13.8 - Using the formulas of Chapter 12, Section 5, sum...Ch. 13.8 - Do the problem in Example 1 for the case of a...Ch. 13.8 - Prob. 4PCh. 13.8 - Find the method of images for problem 4.Ch. 13.8 - Substitute (8.25) into (8.22) and use (8.23) and...Ch. 13.8 - Verify that the Green function in (8.29) is zero...Ch. 13.8 - Show that the Green function (8.28) which is zero...Ch. 13.8 - Show that our results can be extended to find the...Ch. 13.9 - Verify that (9.15) follows from (9.14). Hint: Use...Ch. 13.9 - A metal plate covering the first quadrant has the...Ch. 13.9 - Consider the heat flow problem of Section 3. Solve...Ch. 13.9 - A semi-infinite bar is initially at temperature...Ch. 13.9 - Prob. 5PCh. 13.9 - Continue the problem of Example 2 in the following...Ch. 13.9 - Continue with Problem 4 as in Problem 6.Ch. 13.10 - Find the steady-state temperature distribution in...Ch. 13.10 - Solve Problem 1 if T=0 for x=0,x=1,y=0, and T=1x...Ch. 13.10 - Solve Problem 1 if the sides x=0 and x=1 are...Ch. 13.10 - Find the steady-state temperature distribution in...Ch. 13.10 - A bar of length l is initially at 0. From t=0 on,...Ch. 13.10 - Do Problem 5 if the x=0 end is insulated and the...Ch. 13.10 - Solve Problem 2 if the sides x=0 and x=1 are...Ch. 13.10 - A slab of thickness 10cm has its two faces at 10...Ch. 13.10 - A string of length l has initial displacement...Ch. 13.10 - Solve Problem 5.7 if half the curved surface of...Ch. 13.10 - The series in Problem 5.12 can be summed (see...Ch. 13.10 - A plate in the shape of a quarter circle has...Ch. 13.10 - Sum the series in Problem 12 to get...Ch. 13.10 - A long cylinder has been cut into quarter...Ch. 13.10 - Repeat Problems 12 and 13 for a plate in the shape...Ch. 13.10 - Consider the normal modes of vibration for a...Ch. 13.10 - Sketch some of the normal modes of vibration for a...Ch. 13.10 - Repeat Problem 17 for a membrane in the shape of a...Ch. 13.10 - Prob. 19MPCh. 13.10 - Use Problem 7.16 to find the characteristic...Ch. 13.10 - The surface temperature of a sphere of radius 1 is...Ch. 13.10 - Find the interior temperature in a hemisphere if...Ch. 13.10 - Find the steady-state temperature in the region...Ch. 13.10 - Find the general solution for the steady-state...Ch. 13.10 - The Klein-Gordon equation is 2u=1/v22u/t2+2u. This...Ch. 13.10 - Prob. 26MPCh. 13.10 - Do Problem 26 for a rectangular membrane.Ch. 13.10 - Find the steady-state temperature in a...
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- Find all values of the angle 0 (in radians, with 0 < 0 < 27) for which the matrix cos e -sin 0 A = sin 0 cos 8 has real eigenvalues. (Enter your answers as a comma-separated list.) rad Interpret your answer geometrically. O The only rotation that sends vectors to multiples of themselves is the 180° rotation. O The only rotations that send vectors to multiples of themselves are the identity and the 180° rotation. O The only rotation that sends vectors to multiples of themselves is the identity rotation. O There are no rotations that send vectors to multiples of themselves. Need Help? Read Itarrow_forwardFind all values of the angle 0 (in radians, with 0arrow_forwardFind the eigenvalues and the eigenvectors of A and A²: A 1 3 = 2 0 λι 2 = A² = 2 -31 6 λι 2 with X1 and with x2 with X1 ။ = with x2 andarrow_forwardH.W.: prove that the eigenfunctions , (x) =cos x) and ,(x) =cos (x) cOs are orthogonal.arrow_forwardProve that the Y,.0 spherical harmonic is an eigenfunction of the A2 (Legendrian) operator by computing its eigenvalue. Circle the eigenvalue carefully. 1 1 sin0 sin0 d0 sin20 ao² 1/2 3 Y1,0 coso %3Darrow_forwardFind all values of the angle 6 (in radians, with 0 0 < 27) for which the matrix Cos a -sin 0 sin e COs e has real eigenvalues. (Enter your answers as a comma-separated list.) rad Interpret your answer geometrically. O The only rotation that sends vectors to multiples of themselves is the 180° rotation. O The only rotation that sends vectors to multiples of themselves is the identity rotation. O There are no rotations that send vectors to multiples of themselves. O The only rotations that send vectors to multiples of themselves are the identity and the 180° rotation.arrow_forward= (x - y, 3x). Then find all eigenvalues and eigen- Problem 21. Let T(x, y) vectors of T.arrow_forwardFind all values of the angle 8 (in radians, with 0 ≤ 0 < 2π) for which the matrix cos Ꮎ -sin Ꮎ [ sin 0 cos 0 has real eigenvalues. (Enter your answers as a comma-separated list.) 0 = A = rad Interpret your answer geometrically. O There are no rotations that send vectors to multiples of themselves. O The only rotation that sends vectors to multiples of themselves is the identity rotation. O The only rotations that send vectors to multiples of themselves are the identity and the 180° rotation. O The only rotation that sends vectors to multiples of themselves is the 180° rotation.arrow_forwardShot.- that A and AT have the same eigenvalues.arrow_forwardShow all work and every step. That is the only way I will know how you solved the problem. Directions: Find the bases for the eigenspaces of A.arrow_forward. Let A = 2 150 00 00 0 52000 CATL co 9 11 -1 0 -13 7 2 0 3 . Find all the eigenvalues of A5. Justify your answer.arrow_forwardFind all values of the angle e (in radians, with o se < 2n) for which the matrix cos e -sin e sin e cos e A = has real eigenvalues. (Enter your answers as a comma-separated list.) rad Interpret your answer geometrically. O The only rotation that sends vectors to multiples of themselves is the identity rotation. There are no rotations that send vectors to multiples of themselves. The only rotation that sends vectors to multiples of themselves is the 180° rotation. O The only rotations that send vectors to multiples of themselves are the identity and the 180° rotation.arrow_forwardarrow_back_iosSEE MORE QUESTIONSarrow_forward_ios
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