Exam (elaborations)

Solutions for A Course in Mathematical Methods for Physicists, 1st Edition by Herman (All Chapters included)

0 purchase

Module
A Course in Mathematical Methods for Physicists 1e

Institution
A Course In Mathematical Methods For Physicists 1e

Complete Solutions Manual for A Course in Mathematical Methods for Physicists, 1st Edition by Russell L. Herman ; ISBN13: 9781466584679....(Full Chapters included from Chapter 1 to 11)...1 Introduction and Review 2 Free Fall and Harmonic Oscillators 3 Linear Algebra 4 Nonlinear Dynamics 5 The H...

[Show more]

Preview 4 out of 441 pages

View example

Uploaded on October 28, 2024
Number of pages 441
Written in 2013/2014
Type Exam (elaborations)
Contains Questions & answers

solutions
manual
answer guide
course
mathematical
methods
physicists
1st edition
herman
9781466584679
solutions manual
1e

Institution A Course in Mathematical Methods for Physicists 1e
Module A Course in Mathematical Methods for Physicists 1e

mizhouubcca

Member since 1 year 1069 documents sold

129

£24.13

Add to cart

Add to wishlist

100% satisfaction guarantee
Immediately available after payment
Both online and in PDF
No strings attached

SOLUTIONS MANUAL FOR

R . L . H E R M A N - V E R S I O N D AT E : N O V E M B E R 1 4 , 2 0 1 3

,ii

Contents

Preface iii

1 Introduction and Review 1

2 Free Fall and Harmonic Oscillators 25

3 Linear Algebra 69

4 Nonlinear Dynamics 111

5 The Harmonics of Vibrating Strings 141

6 Non-sinusoidal Harmonics 173

7 Complex Representations of Functions 219

8 Transform Techniques in Physics 245

9 Vector Analysis and EM Waves 291

10 Extrema and Variational Calculus 327

11 Problems in Higher Dimensions 357

A Review of Sequences and Infinite Series 379

B Quick Answers 395
Ch.1 Introduction . 395
Ch.2 Free Fall and Harmonic Oscillators 398
Ch.3 Linear Algebra 403
Ch.4 Nonlinear Dynamics 408
Ch.5 The Harmonics of Vibrating Strings 412
Ch.6 Non-sinusoidal Harmonics 415
Ch.7 Complex Representations of Functions 419
Ch.8 Transform Techniques in Physics 422
Ch.9 Vector Analysis and EM Waves 427
Ch.10 Extrema and Variational Calculus 431
Ch.11 Problems in Higher Dimensions 434
Ch. A Review of Sequences and Infinite Series 436

,1
Introduction and Review

1. Prove the following identities using only the definitions of the trigono-
metric functions, the Pythagorean identity, or the identities for sines and
cosines of sums of angles.

a. cos 2x = 2 cos2 x − 1.

cos 2x = cos2 x − sin2 x
= cos2 x − (1 − cos2 x )
= 2 cos2 x − 1.

b. sin 3x = A sin3 x + B sin x, for what values of A and B?

sin 3x = sin x cos 2x + sin 2x cos x
= sin x (cos2 x − sin2 x ) + 2 sin x cos2 x
= 3 sin x (1 − sin2 x ) − sin3 x
= 3 sin x − 4 sin3 x.

So, A = −4, B = 3.

θ π
c. sec θ + tan θ = tan + .
2 4

θ π

θ π
sin 2 + 4
tan + =
2 4 cos θ
+ π
2 4

sin 2θ cos π4 + sin π4 cos 2θ
=
cos 2θ cos π4 − sin π4 sin 2θ
sin 2θ + cos 2θ
=
cos 2θ − sin 2θ
!
sin 2θ + cos 2θ cos 2θ + sin 2θ
=
cos 2θ − sin 2θ cos 2θ + sin 2θ
2
cos 2θ + sin 2θ
=
cos2 θ
2 − sin2 θ
2
cos2 θ
2 + sin2 2θ + 2 sin 2θ cos 2θ
=
cos θ

, 2 mathematical methods for physicists

1 + sin θ
=
cos θ
= sec θ + tan θ.

2. Determine the exact values of
π
a. sin .
8

π 1 π
sin2 = 1 − cos
8 2 4
√ !
1 2
= 1−
2 2
1 √
= 2− 2 .
4
π 1
q √
Therefore, sin = 2− 2.
8 2
b. tan 15o .

tan 15o = tan(60o − 45o )
tan 60o − tan 45o
=
1 + tan 60o tan 45o
√
3−1
= √
1+ 3
√ √ !
3−1 1− 3
= √ √
1+ 3 1− 3
√
= 2 − 3.

One can get the same answer using
√
2 osin2 15o 1 − cos 30o 2− 3 √ 2
tan 15 = = = √ = ( 2 − 3) .
cos2 15o 1 + cos 30o 2+ 3

c. cos 105o .

1
cos2 105o (1 + cos 210o )
=
2
1
= (1 − cos 30o )
2
1 √
= (2 − 3).
4
p √
Therefore, cos 105o = − 21 2 − 3. One could also use

1
cos2 105o = sin2 15o = (1 − cos 30o ) .
2
Note that this answer can be denested:
1
q √ 1
q √
− 2− 3 = − 8−4 3
2 4

The benefits of buying summaries with Stuvia:

Guaranteed quality through customer reviews

Stuvia customers have reviewed more than 700,000 summaries. This how you know that you are buying the best documents.

Quick and easy check-out

You can quickly pay through credit card for the summaries. There is no membership needed.

Focus on what matters

Your fellow students write the study notes themselves, which is why the documents are always reliable and up-to-date. This ensures you quickly get to the core!

Frequently asked questions

What do I get when I buy this document?

You get a PDF, available immediately after your purchase. The purchased document is accessible anytime, anywhere and indefinitely through your profile.

Satisfaction guarantee: how does it work?

Our satisfaction guarantee ensures that you always find a study document that suits you well. You fill out a form, and our customer service team takes care of the rest.

Who am I buying these notes from?

Stuvia is a marketplace, so you are not buying this document from us, but from seller mizhouubcca. Stuvia facilitates payment to the seller.

Will I be stuck with a subscription?

No, you only buy these notes for £24.13. You're not tied to anything after your purchase.

Can Stuvia be trusted?

4.6 stars on Google & Trustpilot (+1000 reviews)

69052 documents were sold in the last 30 days

Founded in 2010, the go-to place to buy revision notes and other study material for 15 years now

Start selling

Exam (elaborations)

Solutions for A Course in Mathematical Methods for Physicists, 1st Edition by Herman (All Chapters included)

Document information

Subjects

Written for

Seller

Reviews received

Content preview