Mathematics for Machine Technology
7th Edition
ISBN: 9781133281450
Author: John C. Peterson, Robert D. Smith
Publisher: Cengage Learning
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Textbook Question
Chapter 5, Problem 5A
Determine the length A in Figure 5-3. All dimensions are in inches.
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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?
(a) State, without proof, Cauchy's theorem, Cauchy's integral formula and Cauchy's
integral formula for derivatives. Your answer should include all the conditions
required for the results to hold.
(8 marks)
(b) Let U{z EC: |z| -1}. Let 12 be the triangular contour with vertices at
0, 2-2 and 2+2i, parametrized in the anticlockwise direction. Calculate
dz.
You must check the conditions of any results you use.
(d) Let U C. Calculate
Liz-1ym dz,
(z - 1) 10
(5 marks)
where 2 is the same as the previous part. You must check the conditions of any
results you use.
(4 marks)
(a) Suppose a function f: C→C has an isolated singularity at wЄ C. State what it
means for this singularity to be a pole of order k.
(2 marks)
(b) Let f have a pole of order k at wЄ C. Prove that the residue of f at w is given
by
1
res (f, w):
=
Z
dk
(k-1)! >wdzk−1
lim
-
[(z — w)* f(z)] .
(5 marks)
(c) Using the previous part, find the singularity of the function
9(z) =
COS(πZ)
e² (z - 1)²'
classify it and calculate its residue.
(5 marks)
(d) Let g(x)=sin(211). Find the residue of g at z = 1.
(3 marks)
(e) Classify the singularity of
cot(z)
h(z) =
Z
at the origin.
(5 marks)
Chapter 5 Solutions
Mathematics for Machine Technology
Ch. 5 - Express the mixed number 7316 as an improper...Ch. 5 - Add 15+213+41115 .Ch. 5 - Subtract 716332 .Ch. 5 - Multiply 56423 .Ch. 5 - Determine the length A in Figure 5-3. All...Ch. 5 - Determine the length B in Figure 5-3. All...Ch. 5 - Find the reciprocal of each of the fractions in...Ch. 5 - Find the reciprocal of each of the fractions in...Ch. 5 - Find the reciprocal of each of the fractions in...Ch. 5 - Find the reciprocal of each of the fractions in...
Ch. 5 - This casting in Figure 5-4 shows seven tapped...Ch. 5 - Bar stock is being cut on a lathe. The tool feeds...Ch. 5 - A groove 1516 inch deep is to be milled in a steel...Ch. 5 - This sheet metal section shown in Figure 5-5 has...Ch. 5 - The feed on a lathe is set for 164 inch. How many...Ch. 5 - How many complete pieces can be blanked from a...Ch. 5 - A slot is milled the full length of a steel plate...Ch. 5 - How many binding posts can be cut from a brass rod...Ch. 5 - A bar of steel 2314 feet long weighs 11012 pounds....Ch. 5 - A single-threaded (or single-start) square-thread...Ch. 5 - A double-threaded square-thread screw is shown in...
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