Mathematical Methods in the Physical Sciences
3rd Edition
ISBN: 9780471198260
Author: Mary L. Boas
Publisher: Wiley, John & Sons, Incorporated
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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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Prove:
{a,{a,b}}={c,{c,d}} ⟺ a=c and b=d.
3. Define a function T : R³ → R² by
T(x, y, z) = (x + y + z, x + 2y − 3z).
(1) Show that T is a linear transformation.
—
(2) Find all vectors in the kernel of T.
(3) Show that T is onto.
(4) Find the matrix representation of T relative to the standard ba-
sis of R³ and R².
5. Use the Gram-Schmidt process to find an orthonormal basis for the
subspace of R4 spanned by the vectors
(1,0,1,1),
(1,0,1,0),
(0, 0, 1, 1).
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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