Mathematical Methods in the Physical Sciences
3rd Edition
ISBN: 9780471198260
Author: Mary L. Boas
Publisher: Wiley, John & Sons, Incorporated
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Chapter 9.3, Problem 17P
Find the geodesics on the cone
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(b) In various places in this module, data on the silver content of coins
minted in the reign of the twelfth-century Byzantine king Manuel I
Comnenus have been considered. The full dataset is in the Minitab file
coins.mwx. The dataset includes, among others, the values of the
silver content of nine coins from the first coinage (variable Coin1) and
seven from the fourth coinage (variable Coin4) which was produced a
number of years later. (For the purposes of this question, you can
ignore the variables Coin2 and Coin3.) In particular, in Activity 8 and
Exercise 2 of Computer Book B, it was argued that the silver contents
in both the first and the fourth coinages can be assumed to be normally
distributed. The question of interest is whether there were differences in
the silver content of coins minted early and late in Manuel’s reign. You
are about to investigate this question using a two-sample t-interval.
(i) Using Minitab, find either the sample standard deviations of the
two variables…
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)
Chapter 9 Solutions
Mathematical Methods in the Physical Sciences
Ch. 9.1 - The speed of light in a medium of index of...Ch. 9.1 - The speed of light in a medium of index of...Ch. 9.1 - The speed of light in a medium of index of...Ch. 9.2 - Write and solve the Euler equations to make the...Ch. 9.2 - Write and solve the Euler equations to make the...Ch. 9.2 - Write and solve the Euler equations to make the...Ch. 9.2 - Write and solve the Euler equations to make the...Ch. 9.2 - Write and solve the Euler equations to make the...Ch. 9.2 - Write and solve the Euler equations to make the...Ch. 9.2 - Write and solve the Euler equations to make the...
Ch. 9.2 - Write and solve the Euler equations to make the...Ch. 9.2 - Write and solve the Euler equations to make the...Ch. 9.2 - Write and solve the Euler equations to make the...Ch. 9.3 - Change the independent variable to simplify the...Ch. 9.3 - Change the independent variable to simplify the...Ch. 9.3 - Change the independent variable to simplify the...Ch. 9.3 - Change the independent variable to simplify the...Ch. 9.3 - Write and solve the Euler equations to make the...Ch. 9.3 - Write and solve the Euler equations to make the...Ch. 9.3 - Write and solve the Euler equations to make the...Ch. 9.3 - Write and solve the Euler equations to make the...Ch. 9.3 - Write and solve the Euler equations to make the...Ch. 9.3 - Write and solve the Euler equations to make the...Ch. 9.3 - Use Fermats principle to find the path followed by...Ch. 9.3 - Use Fermats principle to find the path followed by...Ch. 9.3 - Use Fermats principle to find the path followed by...Ch. 9.3 - Use Fermats principle to find the path followed by...Ch. 9.3 - Find the geodesics on a plane using polar...Ch. 9.3 - Prob. 16PCh. 9.3 - Find the geodesics on the cone x2+y2=z2. Hint: Use...Ch. 9.3 - Find the geodesics on a sphere. Hints: Use...Ch. 9.4 - Verify equations (4.2).Ch. 9.4 - Show, in Figure 4.4, that for a point like...Ch. 9.4 - In the brachistochrone problem, show that if the...Ch. 9.4 - Consider a rapid transit system consisting of...Ch. 9.4 - In Problems 5 to 7, use Fermats principle to find...Ch. 9.4 - In Problems 5 to 7, use Fermats principle to find...Ch. 9.4 - In Problems 5 to 7, use Fermats principle to find...Ch. 9.5 - (a) Consider the case of two dependent variables....Ch. 9.5 - Set up Lagranges equations in cylindrical...Ch. 9.5 - Do Problem 2 in spherical coordinates.Ch. 9.5 - Use Lagranges equations to find the equation of...Ch. 9.5 - Find the equation of motion of a particle moving...Ch. 9.5 - A particle moves on the surface of a sphere of...Ch. 9.5 - Prove that a particle constrained to stay on a...Ch. 9.5 - Two particles each of mass m are connected by an...Ch. 9.5 - A mass m moves without friction on the surface of...Ch. 9.5 - Do Example 3 above, using cylindrical coordinates...Ch. 9.5 - A yo-yo (as shown) falls under gravity. Assume...Ch. 9.5 - Find the Lagrangian and Lagranges equations for a...Ch. 9.5 - A particle moves without friction under gravity on...Ch. 9.5 - 2A hoop of mass M and radius a rolls without...Ch. 9.5 - Generalize Problem 14 to any mass M of circular...Ch. 9.5 - Find the Lagrangian and the Lagrange equation for...Ch. 9.5 - A simple pendulum (Problem 4) is suspended from a...Ch. 9.5 - A hoop of mass m in a vertical plane rests on a...Ch. 9.5 - For the following problems, use the Lagrangian to...Ch. 9.5 - For the following problems, use the Lagrangian to...Ch. 9.5 - For the following problems, use the Lagrangian to...Ch. 9.5 - For the following problems, use the Lagrangian to...Ch. 9.5 - For the following problems, use the Lagrangian to...Ch. 9.5 - For the following problems, use the Lagrangian to...Ch. 9.5 - For the following problems, use the Lagrangian to...Ch. 9.6 - In Problems 1 and 2, given the length l of a curve...Ch. 9.6 - In Problems 1 and 2, given the length l of a curve...Ch. 9.6 - Given 10 cc of lead, find how to form it into a...Ch. 9.6 - Prob. 4PCh. 9.6 - A curve y=y(x), joining two points x1 and x2 on...Ch. 9.6 - In Problem 5, given the volume, find the shape of...Ch. 9.6 - Integrate (6.2), simplify the result and integrate...Ch. 9.8 - (a) In Section 3, we showed how to obtain a first...Ch. 9.8 - Find a first integral of the Euler equation to...Ch. 9.8 - Find a first integral of the Euler equation to...Ch. 9.8 - Find a first integral of the Euler equation to...Ch. 9.8 - Write and solve the Euler equations to make...Ch. 9.8 - Write and solve the Euler equations to make...Ch. 9.8 - Write and solve the Euler equations to make...Ch. 9.8 - Find the geodesics on the cylinder r=1+cos.Ch. 9.8 - Prob. 9MPCh. 9.8 - Find the geodesics on the parabolic cylinder y=x2.Ch. 9.8 - In Problems 11 to 18, use Fermats principle to...Ch. 9.8 - In Problems 11 to 18, use Fermats principle to...Ch. 9.8 - In Problems 11 to 18, use Fermats principle to...Ch. 9.8 - In Problems 11 to 18, use Fermats principle to...Ch. 9.8 - In Problems 11 to 18, use Fermats principle to...Ch. 9.8 - In Problems 11 to 18, use Fermats principle to...Ch. 9.8 - In Problems 11 to 18, use Fermats principle to...Ch. 9.8 - In Problems 11 to 18, use Fermats principle to...Ch. 9.8 - Find Lagranges equations in polar coordinates for...Ch. 9.8 - Repeat Problem 19 if V=K/r.Ch. 9.8 - Write Lagranges equations in cylindrical...Ch. 9.8 - In spherical coordinates, find the Lagrange...Ch. 9.8 - A particle slides without friction around a...Ch. 9.8 - Write and simplify the Euler equation to make...Ch. 9.8 - Prob. 25MPCh. 9.8 - A wire carrying a uniform distribution of positive...Ch. 9.8 - Find a first integral of the Euler equation for...Ch. 9.8 - Write the Lagrange equation for a particle moving...
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