Mathematical tools for physics

486 Pages·2005·2.057 MB·English

by  Nearing J.

Mathematical Tools for Physics
by James Nearing
Physics Department
University of Miami
mailto:[email protected]
Copyright 2003, James Nearing
Permission to copy for
individual or classroom
use is granted.
QA 37.2
Contents
Introduction . . . . . . . . . . . . . . . . . iv 4 Differential Equations . . . . . . . . . . . . 83
Linear Constant-Coefficient
Bibliography . . . . . . . . . . . . . . . . vi
Forced Oscillations
Series Solutions
1 Basic Stuff . . . . . . . . . . . . . . . . . 1
Trigonometry via ODE’s
Trigonometry
Green’s Functions
Parametric Differentiation
Separation of Variables
Gaussian Integrals
Simultaneous Equations
erf and Gamma
Simultaneous ODE’s
Differentiating
Legendre’s Equation
Integrals
Polar Coordinates
5 Fourier Series . . . . . . . . . . . . . . . 118
Sketching Graphs
Examples
2 Infinite Series . . . . . . . . . . . . . . . . 30 Computing Fourier Series
The Basics Choice of Basis
Deriving Taylor Series Periodically Forced ODE’s
Convergence Return to Parseval
Series of Series Gibbs Phenomenon
Power series, two variables
6 Vector Spaces . . . . . . . . . . . . . . . 142
Stirling’s Approximation
The Underlying Idea
Useful Tricks
Axioms
Diffraction
Examples of Vector Spaces
Checking Results
Linear Independence
3 Complex Algebra . . . . . . . . . . . . . . 65 Norms
Complex Numbers Scalar Product
Some Functions Bases and Scalar Products
Applications of Euler’s Formula Gram-Schmidt Orthogonalization
Logarithms Cauchy-Schwartz inequality
Mapping Infinite Dimensions
i
7 Operators and Matrices . . . . . . . . . . 168 Integral Representation of Curl
The Idea of an Operator The Gradient
Definition of an Operator Shorter Cut for div and curl
Examples of Operators Identities for Vector Operators
Matrix Multiplication Applications to Gravity
Inverses Gravitational Potential
Areas, Volumes, Determinants Summation Convention
Matrices as Operators More Complicated Potentials
Eigenvalues and Eigenvectors
Change of Basis 10 Partial Differential Equations . . . . . . . 283
Summation Convention The Heat Equation
Can you Diagonalize a Matrix? Separation of Variables
Eigenvalues and Google Oscillating Temperatures
Spatial Temperature Distributions
8 Multivariable Calculus . . . . . . . . . . . 208 Specified Heat Flow
Partial Derivatives Electrostatics
Differentials
Chain Rule 11 Numerical Analysis . . . . . . . . . . . . 315
Geometric Interpretation Interpolation
Gradient Solving equations
Electrostatics Differentiation
Plane Polar Coordinates Integration
Cylindrical, Spherical Coordinates Differential Equations
Vectors: Cylindrical, Spherical Bases Fitting of Data
Gradient in other Coordinates Euclidean Fit
Maxima, Minima, Saddles Differentiating noisy data
Lagrange Multipliers Partial Differential Equations
Solid Angle
Rainbow 12 Tensors . . . . . . . . . . . . . . . . . . 354
3D Visualization Examples
Components
9 Vector Calculus 1 . . . . . . . . . . . . . 248 Relations between Tensors
Fluid Flow Non-Orthogonal Bases
Vector Derivatives Manifolds and Fields
Computing the divergence Coordinate Systems
ii
Basis Change Cauchy’s Residue Theorem
Branch Points
13 Vector Calculus 2 . . . . . . . . . . . . . 396
Other Integrals
Integrals
Other Results
Line Integrals
Gauss’s Theorem 15 Fourier Analysis . . . . . . . . . . . . . . 451
Stokes’ Theorem Fourier Transform
Reynolds’ Transport Theorem Convolution Theorem
Time-Series Analysis
14 Complex Variables . . . . . . . . . . . . . 418 Derivatives
Differentiation Green’s Functions
Integration Sine and Cosine Transforms
Power (Laurent) Series Weiner-Khinchine Theorem
Core Properties
Branch Points
iii
Introduction
I wrote this text for a one semester course at the sophomore-junior level. Our experience with students
taking our junior physics courses is that even if they’ve had the mathematical prerequisites, they usually need
more experience using the mathematics to handle it efficiently and to possess usable intuition about the processes
involved. If you’ve seen infinite series in a calculus course, you may have no idea that they’re good for anything.
If you’ve taken a differential equations course, which of the scores of techniques that you’ve seen are really used
a lot? The world is (at least) three dimensional so you clearly need to understand multiple integrals, but will
everything be rectangular?
How do you learn intuition?
When you’ve finished a problem and your answer agrees with the back of the book or with your friends
or even a teacher, you’re not done. The way do get an intuitive understanding of the mathematics and of the
physics is to analyze your solution thoroughly. Does it make sense? There are almost always several parameters
that enter the problem, so what happens to your solution when you push these parameters to their limits? In a
mechanics problem, what if one mass is much larger than another? Does your solution do the right thing? In
electromagnetism, if you make a couple of parameters equal to each other does it reduce everything to a simple,
special case? When you’re doing a surface integral should the answer be positive or negative and does your answer
agree?
When you address these questions to every problem you ever solve, you do several things. First, you’ll find
your own mistakes before someone else does. Second, you acquire an intuition about how the equations ought
to behave and how the world that they describe ought to behave. Third, It makes all your later efforts easier
because you will then have some clue about why the equations work the way they do. It reifies algebra.
Does it take extra time? Of course. It will however be some of the most valuable extra time you can spend.
Is it only the students in my classes, or is it a widespread phenomenon that no one is willing to sketch a
graph? (“Pulling teeth” is the clich´e that comes to mind.) Maybe you’ve never been taught that there are a few
basic methods that work, so look at section 1.8. And keep referring to it. This is one of those basic tools that is
far more important than you’ve ever been told. It is astounding how many problems become simpler after you’ve
sketched a graph. Also, until you’ve sketched some graphs of functions you really don’t know how they behave.
When I taught this course I didn’t do everything that I’m presenting here. The two chapters, Numerical
Analysis and Tensors, were not in my one semester course, and I didn’t cover all of the topics along the way. The
iv
last couple of chapters were added after the class was over. There is enough here to select from if this is a course
text. If you are reading this on your own then you can move through it as you please, though you will find that
the first five chapters are used more in the later parts than are chapters six and seven.
The pdf file that I’ve created is hyperlinked, so that you can click on an equation or section reference to
go to that point in the text. To return, there’s a Previous View button at the top or bottom of the reader or a
keyboard shortcut to do the same thing. [Command← on Mac, Alt← on Windows, Control← on Linux-GNU]
The contents and index pages are hyperlinked, and the contents also appear in the bookmark window.
If you’re using Acrobat Reader 5.0, you should enable the preference to smooth line art. Otherwise many
of the drawings will appear jagged. If you use 6.0 nothing seems to help.
I chose this font for the display version of the text because it appears better on the screen than does the
more common Times font. The choice of available mathematics fonts is more limited.
I have also provided a version of this text formatted for double-sided bound printing of the sort you can get
from commercial copiers.
I’d like to thank the students who found some, but probably not all, of the mistakes in the text. Also
Howard Gordon, who used it in his course and provided me with many suggestions for improvements.
v
Bibliography
Mathematical Methods for Physics and Engineering by Riley, Hobson, and Bence. Cambridge University
Press For the quantity of well-written material here, it is surprisingly inexpensive in paperback.
Mathematical Methods in the Physical Sciences by Boas. John Wiley Publ About the right level and
with a very useful selection of topics. If you know everything in here, you’ll find all your upper level courses much
easier.
Mathematical Methods for Physicists by Arfken and Weber. Academic Press At a slightly more advanced
level, but it is sufficiently thorough that will be a valuable reference work later.
Mathematical Methods in Physics by Mathews and Walker. More sophisticated in its approach to the
subject, but it has some beautiful insights. It’s considered a standard.
Schaum’s Outlines by various. There are many good and inexpensive books in this series, e.g. “Complex
Variables,” “Advanced Calculus,” ”German Grammar.” Amazon lists hundreds.
Visual Complex Analysis by Needham, Oxford University Press The title tells you the emphasis. Here the
geometry is paramount, but the traditional material is present too. It’s actually fun to read. (Well, I think so
anyway.) The Schaum text provides a complementary image of the subject.
Complex Analysis for Mathematics and Engineering by Mathews and Howell. Jones and Bartlett Press
Another very good choice for a text on complex variables.
Applied Analysis by Lanczos. Dover Publications This publisher has a large selection of moderately priced, high
quality books. More discursive than most books on numerical analysis, and shows great insight into the subject.
Linear Differential Operators by Lanczos. Dover publications As always with this author great insight and
unusual ways to look at the subject.
Numerical Methods that (usually) Work by Acton. Harper and Row Practical tools with more than the
usual discussion of what can (and will) go wrong.
vi
Numerical Recipes by Press et al. Cambridge Press The standard current compendium surveying techniques
and theory, with programs in one or another language.
A Brief on Tensor Analysis by James Simmonds. Springer This is the only text on tensors that I will
recommend. To anyone. Under any circumstances.
Linear Algebra Done Right by Axler. Springer Don’t let the title turn you away. It’s pretty good.
Advanced mathematical methods for scientists and engineers by Bender and Orszag. Springer Material
you won’t find anywhere else, and well-written. “...a sleazy approximation that provides good physical insight
into what’s going on in some system is far more useful than an unintelligible exact result.”
Probability Theory: A Concise Course by Rozanov. Dover Starts at the beginning and goes a long way in
148 pages. Clear and explicit and cheap.
vii
Basic Stuff
1.1 Trigonometry
The common trigonometric functions are familiar to you, but do you know some of the tricks to remember (or
to derive quickly) the common identities among them? Given the sine of an angle, what is its tangent? Given
its tangent, what is its cosine? All of these simple but occasionally useful relations can be derived in about two
seconds if you understand the idea behind one picture. Suppose for example that you know the tangent of θ, what
is sinθ? Draw a right triangle and designate the tangent of θ as x, so you can draw a triangle with tanθ = x/1.
√
The Pythagorean theorem says that the third side is 1+x2. You now read the
√
sine from the triangle as x/ 1+x2, so x
q
tanθ
sinθ = 1
p
1+tan2θ
Any other such relation is done the same way. You know the cosine, so what’s the cotangent? Draw a different
triangle where the cosine is x/1.
Radians
When you take the sine or cosine of an angle, what units do you use? Degrees? Radians? Other? And who
invented radians? Why is this the unit you see so often in calculus texts? That there are 360◦ in a circle is
something that you can blame on the Sumerians, but where did this other unit come from?
2θ
θ
s
R 2R
1
1—Basic Stuff 2
It results from one figure and the relation between the radius of the circle, the angle drawn, and the length
of the arc shown. If you remember the equation s = Rθ, does that mean that for a full circle θ = 360◦ so
s = 360R? No. For some reason this equation is valid only in radians. The reasoning comes down to a couple of
observations. You can see from the drawing that s is proportional to θ — double θ and you double s. The same
observation holds about the relation between s and R, a direct proportionality. Put these together in a single
equation and you can conclude that
s = CRθ
where C is some constant of proportionality. Now what is C?
You know that the whole circumference of the circle is 2πR, so if θ = 360◦, then
π
2πR = CR360◦, and C = degree−1
180
It has to have these units so that the left side, s, comes out as a length when the degree units cancel. This is an
awkward equation to work with, and it becomes very awkward when you try to do calculus.
d π
sinθ = cosθ
dθ 180
This is the reason that the radian was invented. The radian is the unit designed so that the proportionality
constant is one.
C = 1radian−1 then s = (cid:0)1radian−1(cid:1)Rθ
In practice, no one ever writes it this way. It’s the custom simply to omit the C and to say that s = Rθ with θ
restricted to radians — it saves a lot of writing. How big is a radian? A full circle has circumference 2πR, and
this is Rθ. It says that the angle for a full circle has 2πradians. One radian is then 360/2πdegrees, a bit under
60◦. Why do you always use radians in calculus? Only in this unit do you get simple relations for derivatives and
integrals of the trigonometric functions.
Hyperbolic Functions
The circular trigonometric functions, the sines, cosines, tangents, and their reciprocals are familiar, but their
hyperbolic counterparts are probably less so. They are related to the exponential function as
ex +e−x ex −e−x sinhx ex −e−x
coshx = , sinhx = , tanhx = = (1)
2 2 coshx ex +e−x