Mathematics for Machine Technology
7th Edition
ISBN: 9781133281450
Author: John C. Peterson, Robert D. Smith
Publisher: Cengage Learning
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Chapter 62, Problem 4A
Convert 12.02 sq in. to square centimeters.
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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 62 Solutions
Mathematics for Machine Technology
Ch. 62 - In order to make a conical duct a circular sheet...Ch. 62 - Prob. 2ACh. 62 - Prob. 3ACh. 62 - Convert 12.02 sq in. to square centimeters.Ch. 62 - Prob. 5ACh. 62 - Prob. 6ACh. 62 - Prob. 7ACh. 62 - Compute the volume of each sphere 7 through 14....Ch. 62 - Prob. 9ACh. 62 - Compute the volume of each sphere 7 through 14....
Ch. 62 - Prob. 11ACh. 62 - Compute the volume of each sphere 7 through 14....Ch. 62 - Prob. 13ACh. 62 - Compute the volume of each sphere 7 through 14....Ch. 62 - Prob. 15ACh. 62 - Solve these exercise. A vat in the shape of a...Ch. 62 - Prob. 17ACh. 62 - Solve these exercises. A company produces...Ch. 62 - Solve these exercises. Pieces in the shape of...Ch. 62 - Solve these exercises. A plastic products...Ch. 62 - Solve these exercises. Find the weight of the...Ch. 62 - Compute the capacity, in liters, of the container...Ch. 62 - A seamless brass tube and brass flange assembly is...Ch. 62 - Find the number of cubic centimeters of material...Ch. 62 - Find the weight of the cast-iron angle iron shown....Ch. 62 - Prob. 26ACh. 62 - Prob. 27ACh. 62 - Compute the number of cubic centimeters of...
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