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 11.11, Problem 10P
Use Stirling’s formula to find
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1) a) Find a matrix P such that PT AP orthogonally diagonalizes the following matrix
A.
= [{² 1]
A =
b) Verify that PT AP gives the correct diagonal form.
2
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2) Given the following matrices A =
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1] an
and B =
0
1
-3
2
find the following matrices:
a) (AB) b) (BA)T
3) Find the inverse of the following matrix A using Gauss-Jordan elimination or
adjoint of the matrix and check the correctness of your answer (Hint: AA¯¹ = I).
[1 1 1
A = 3 5 4
L3 6 5
4) Solve the following system of linear equations using any one of Cramer's Rule,
Gaussian Elimination, Gauss-Jordan Elimination or Inverse Matrix methods and
check the correctness of your answer.
4x-y-z=1
2x + 2y + 3z = 10
5x-2y-2z = -1
5) a) Describe the zero vector and the additive inverse of a vector in the vector
space, M3,3.
b) Determine if the following set S is a subspace of M3,3 with the standard
operations. Show all appropriate supporting work.
13) Let U = {j, k, l, m, n, o, p} be the universal set. Let V = {m, o,p), W = {l,o, k}, and X = {j,k). List the elements of
the following sets and the cardinal number of each set.
a) W° and n(W)
b) (VUW) and n((V U W)')
c) VUWUX and n(V U W UX)
d) vnWnX and n(V WnX)
9) Use the Venn Diagram given below to determine the number elements in each of the following sets.
a) n(A).
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Chapter 11 Solutions
Mathematical Methods in the Physical Sciences
Ch. 11.3 - The integral in ( 3.1) is improper because of the...Ch. 11.3 - Use the recursion relation (3.4), and if needed,...Ch. 11.3 - Use the recursion relation (3.4), and if needed,...Ch. 11.3 - Use the recursion relation (3.4), and if needed,...Ch. 11.3 - Use the recursion relation (3.4), and if needed,...Ch. 11.3 - Use the recursion relation (3.4), and if needed,...Ch. 11.3 - Use the recursion relation (3.4), and if needed,...Ch. 11.3 - Express each of the following integrals as a ...Ch. 11.3 - Express each of the following integrals as a T...Ch. 11.3 - Express each of the following integrals as a ...
Ch. 11.3 - Express each of the following integrals as a ...Ch. 11.3 - Express each of the following integrals as a ...Ch. 11.3 - Express each of the following integrals as a ...Ch. 11.3 - Express each of the following integrals as a ...Ch. 11.3 - Express each of the following integrals as a ...Ch. 11.3 - A particle starting from rest at x=1 moves along...Ch. 11.3 - Express as a function 01ln1xp1dx, Hint: See...Ch. 11.5 - Using (5.3) with (3.4) and (4.1), find...Ch. 11.5 - Without computer or tables, but just using facts...Ch. 11.5 - In Chapter 1, equations(13.5)and (13.6), we...Ch. 11.5 - Prove that, for positive integral n:...Ch. 11.5 - Use (5.4) to show that (a) 12n12+n=(1)n if n= a...Ch. 11.5 - Prove...Ch. 11.5 - In the Table of Laplace Transforms (end of Chapter...Ch. 11.6 - Prove that B(p,q)=B(q,p). Hint: Put x=1y in...Ch. 11.6 - Prove equation (6.5) (6.5)B(p,q)=0yp1dy(1+y)p+q.Ch. 11.6 - Show that for integral n, m...Ch. 11.7 - Express the following integrals as B functions,...Ch. 11.7 - Express the following integrals as B functions,...Ch. 11.7 - Express the following integrals as B functions,...Ch. 11.7 - Express the following integrals as B functions,...Ch. 11.7 - Express the following integrals as B functions,...Ch. 11.7 - Express the following integrals as B functions,...Ch. 11.7 - Express the following integrals as B functions,...Ch. 11.7 - Express the following integrals as B functions,...Ch. 11.7 - Prove B(n,n)=Bn,12/22n1. Hint: In (6.4), use the...Ch. 11.7 - Computer plot the graph of x3+y3=8. Write the...Ch. 11.7 - Computer plot the graph of x3+y3=8. Write the...Ch. 11.7 - Computer plot the graph of x3+y3=8. Write the...Ch. 11.7 - Computer plot the graph of x3+y3=8. Write the...Ch. 11.8 - Complete the pendulum problem to find the period...Ch. 11.8 - Suppose that a car with a door open at right...Ch. 11.8 - The figure is part of a cycloid with parametric...Ch. 11.9 - Sketch or computer plot a graph of the function...Ch. 11.9 - Verify equations (9.2),(9.3), and (9.4). Hint:...Ch. 11.9 - Prove that erf(x) is an odd function of x. Hint:...Ch. 11.9 - Show that ey2/2dy=2 (a) by using (9.5) and (9.2a);...Ch. 11.9 - Replace x by $i x in(9.1)andlet t = i u$ to show...Ch. 11.9 - Assuming that x is real, show the following...Ch. 11.10 - Carry through the algebra to get equation (10.4).Ch. 11.10 - The integral xtp1etdt=(p,x) is called an...Ch. 11.10 - Express the complementary error function erfc (x)...Ch. 11.10 - En(x)=1exttndt,n=0,1,2,, and Ei(x)=xettdt, and...Ch. 11.10 - 2(a) Express E1(x) as an incomplete function. (b)...Ch. 11.10 - The logarithmic integral is li(x)=0xdtlnt. Express...Ch. 11.10 - Computer plot graphs of (a) En(x) for n=0 to 10...Ch. 11.11 - Use the term 1/(12p) in (11.5) to show that the...Ch. 11.11 - (a) To see the results in Problem 1 graphically,...Ch. 11.11 - In statistical mechanics, we frequently use the...Ch. 11.11 - Use Stirlings formula to evaluate...Ch. 11.11 - Use Stirlings formula to evaluate limnn+32n(n+1).Ch. 11.11 - Use equations (3.4) and (11.5) to show that...Ch. 11.11 - The function (p)=ddpln(p) is called the digamma...Ch. 11.11 - Sketch or computer plot a graph of y=lnx for x0....Ch. 11.11 - The following expression occurs in statistical...Ch. 11.11 - Use Stirlings formula to find limn(n!)1/n/n.Ch. 11.12 - Expand the integrands of K and E [see ( 12.3 )] in...Ch. 11.12 - Use a graph of sin2 and the text discussion just...Ch. 11.12 - Computer plot graphs of K(k) and E(k) in (12.3)...Ch. 11.12 - In Problems 4 to 13, identify each of the...Ch. 11.12 - In Problems 4 to 13, identify each of the...Ch. 11.12 - In Problems 4 to 13, identify each of the...Ch. 11.12 - In Problems 4 to 13, identify each of the...Ch. 11.12 - In Problems 4 to 13, identify each of the...Ch. 11.12 - In Problems 4 to 13, identify each of the...Ch. 11.12 - In Problems 4 to 13, identify each of the...Ch. 11.12 - In Problems 4 to 13, identify each of the...Ch. 11.12 - In Problems 4 to 13, identify each of the...Ch. 11.12 - In Problems 4 to 13, identify each of the...Ch. 11.12 - Find the circumference of the ellipse 4x2+9y2=36.Ch. 11.12 - Find the length of arc of the ellipse x2+y2/4=1...Ch. 11.12 - Find the are length of one arch of y=sinx.Ch. 11.12 - Write the integral in equation (12.7) as an...Ch. 11.12 - Computer plot graphs of sn u, cn u, and dn u, for...Ch. 11.12 - If u=ln(sec+tan), then is a function of u called...Ch. 11.12 - Show that for k=0:u=F(,0)=,snu=sinu,cnu=cosu,dnu=1...Ch. 11.12 - Show that the four answers given in Section 1 for...Ch. 11.12 - In the pendulum problem, =sing/lt is an...Ch. 11.12 - A uniform solid sphere of density 12 is floating...Ch. 11.12 - Sometimes you may find the notation F(,k) in...Ch. 11.12 - As in Problem $24,$ show that...Ch. 11.13 - Show that $ 0ymdy(1+y)n+1=1(nm)C(n,m) $ for...Ch. 11.13 - Show that B(m,n)B(m+n,k)=B(n,k)B(n+k,m).Ch. 11.13 - Use Stirlings formula to show that...Ch. 11.13 - Verify the asymptotic series 0etdt(1+xt)~ (1)nn!xn...Ch. 11.13 - Use gamma and beta function formulas to show that...Ch. 11.13 - Generalize Problem 5 to show that...Ch. 11.13 - Identify each of the following integrals or...Ch. 11.13 - Identify each of the following integrals or...Ch. 11.13 - Identify each of the following integrals or...Ch. 11.13 - Identify each of the following integrals or...Ch. 11.13 - Identify each of the following integrals or...Ch. 11.13 - Identify each of the following integrals or...Ch. 11.13 - Identify each of the following integrals or...Ch. 11.13 - Identify each of the following integrals or...Ch. 11.13 - Identify each of the following integrals or...Ch. 11.13 - Identify each of the following integrals or...Ch. 11.13 - Identify each of the following integrals or...Ch. 11.13 - Identify each of the following integrals or...Ch. 11.13 - Identify each of the following integrals or...Ch. 11.13 - Identify each of the following integrals or...Ch. 11.13 - Identify each of the following integrals or...Ch. 11.13 - Identify each of the following integrals or...Ch. 11.13 - Find an expression for the exact value of (55.5)...Ch. 11.13 - Using your result in Problem 23 and equation...Ch. 11.13 - As in problems 23 and 24, find expressions for the...
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