Mathematical Methods in the Physical Sciences
3rd Edition
ISBN: 9780471198260
Author: Mary L. Boas
Publisher: Wiley, John & Sons, Incorporated
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Chapter 13.10, Problem 20MP
Use Problem 7.16 to find the characteristic vibration frequencies of sound in a spherical cavity.
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7. [10 marks]
Let G
=
(V,E) be a 3-connected graph. We prove that for every x, y, z Є V, there is a
cycle in G on which x, y, and z all lie.
(a) First prove that there are two internally disjoint xy-paths Po and P₁.
(b) If z is on either Po or P₁, then combining Po and P₁ produces a cycle on which
x, y, and z all lie. So assume that z is not on Po and not on P₁. Now prove that
there are three paths Qo, Q1, and Q2 such that:
⚫each Qi starts at z;
• each Qi ends at a vertex w; that is on Po or on P₁, where wo, w₁, and w₂ are
distinct;
the paths Qo, Q1, Q2 are disjoint from each other (except at the start vertex
2) and are disjoint from the paths Po and P₁ (except at the end vertices wo,
W1, and w₂).
(c) Use paths Po, P₁, Qo, Q1, and Q2 to prove that there is a cycle on which x, y, and
z all lie. (To do this, notice that two of the w; must be on the same Pj.)
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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