Introductory Mathematics for Engineering Applications
1st Edition
ISBN: 9781118141809
Author: Nathan Klingbeil
Publisher: WILEY
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Chapter 7, Problem 31P
Repeat problem P7-30 if
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5. (a) State the Residue Theorem. Your answer should include all the conditions required
for the theorem to hold.
(4 marks)
(b) Let y be the square contour with vertices at -3, -3i, 3 and 3i, described in the
anti-clockwise direction. Evaluate
に
dz.
You must check all of the conditions of any results that you use.
(5 marks)
(c) Evaluate
L
You must check all of the conditions of any results that you use.
ཙ
x sin(Tx)
x²+2x+5
da.
(11 marks)
3. (a) Lety: [a, b] C be a contour. Let L(y) denote the length of y. Give a formula
for L(y).
(1 mark)
(b) Let UCC be open. Let f: U→C be continuous. Let y: [a,b] → U be a
contour. Suppose there exists a finite real number M such that |f(z)| < M for
all z in the image of y. Prove that
<
||, f(z)dz| ≤ ML(y).
(3 marks)
(c) State and prove Liouville's theorem. You may use Cauchy's integral formula without
proof.
(d) Let R0. Let w € C. Let
(10 marks)
U = { z Є C : | z − w| < R} .
Let f UC be a holomorphic function such that
0 < |ƒ(w)| < |f(z)|
for all z Є U. Show, using the local maximum modulus principle, that f is constant.
(6 marks)
3. (a) Let A be an algebra. Define the notion of an A-module M. When is a module M
a simple module?
(b) State and prove Schur's Lemma for simple modules.
(c) Let AM(K) and M = K" the natural A-module.
(i) Show that M is a simple K-module.
(ii) Prove that if ƒ € Endд(M) then ƒ can be written as f(m) = am, where a
is a matrix in the centre of M, (K).
[Recall that the centre, Z(M,(K)) == {a Mn(K) | ab
M,,(K)}.]
= ba for all bЄ
(iii) Explain briefly why this means End₁(M) K, assuming that Z(M,,(K))~
K as K-algebras.
Is this consistent with Schur's lemma?
Chapter 7 Solutions
Introductory Mathematics for Engineering Applications
Ch. 7 - Consider the two-loop circuit shown in Fig P7.1....Ch. 7 - Consider the two-loop circuit shown in Fig P7.2....Ch. 7 - Consider the two-loop circuit shown in Fig P7.3....Ch. 7 - Consider the two-node circuit shown in Fig P7.4....Ch. 7 - Consider the two-node circuit shown in Fig P7.5....Ch. 7 - Consider the two-node circuit shown in Fig P7.6....Ch. 7 - An Analysis of the circuit shown in Fig P7.7....Ch. 7 - A summing Op-Amp circuit is shown in Fig. 7.6. An...Ch. 7 - A summing Op-Amp circuit is shown in Fig. 7.6. An...Ch. 7 - A 20 kg object is suspended by two cables as shown...
Ch. 7 - A 100 lb weight is suspended by two cables as...Ch. 7 - A two-bar truss Supports a weight of W = 750 lb as...Ch. 7 - A 10 kg object is suspended from a two-bar truss...Ch. 7 - A F = 100 N force is applied to a two-bar truss...Ch. 7 - A force F = 200 lb is applied to a two-bar truss...Ch. 7 - The weight of a vehicle is supported by reaction...Ch. 7 - Solve problem P7.16 if the system of equations are...Ch. 7 - Solve problem P7.16 if the system of equations are...Ch. 7 - A vehicle weighing W = 10 kN is parked on an...Ch. 7 - A vehicle weighing W = 2000 lb is parked on an...Ch. 7 - A 100 lb crate weighing W = 100 lb is pushed...Ch. 7 - A crate weighing W = 500 N is pushed against an...Ch. 7 - A 100 kg desk rests on a horizontal plane with a...Ch. 7 - A 200 kg desk rests on a horizontal plane with a...Ch. 7 - The weight of a vehicle is supported by reaction...Ch. 7 - Repeat problem P7-25, if l1=2m, l2=1.5m, k=1.5m,...Ch. 7 - Repeat problem P7-25, if l1=2m, l2=2m, k=1.5m,...Ch. 7 - Repeat problem P7-25, if l1=2m, l2=2m, k=1.5m,...Ch. 7 - Prob. 29PCh. 7 - An environmental engineer wishes to blend a single...Ch. 7 - Repeat problem P7-30 if V=400L, C=0.2, c1=0.2, and...Ch. 7 - Consider the two-loop circuit shown in Fig. P7.32....Ch. 7 - Consider the two-loop circuit shown in Fig. P7.33....Ch. 7 - Consider the two-node circuit shown in Fig. P7.34....Ch. 7 - A mechanical system along with its free body...Ch. 7 - Fig. P7.36 shows a system with two-mass elements...Ch. 7 - A co-op Student at a composite materials...Ch. 7 - Repeat parts (a)—(d) of problem P7-37 if...Ch. 7 - A structural engineer is performing a finite...Ch. 7 - A structural engineer is designing a crane that is...
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