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The inverse hyperbolic functions are easier to evaluate than are the corresponding circularfunctions... The graph of aninverse function is the mirror image of the original function in th

Mathematical Tools for Physics

by James Nearing

Physics DepartmentUniversity of Miami

jnearing@miami.edu

www.physics.miami.edu/nearing/mathmethods/

Copyright 2003, James Nearing

Permission to copy forindividual or classroom

use is granted

QA 37.2

Rev Nov, 2006

Some General Methods

Trigonometry via ODE’s

Gibbs Phenomenon

6 Vector Spaces 120The Underlying Idea

AxiomsExamples of Vector SpacesLinear IndependenceNorms

Scalar ProductBases and Scalar ProductsGram-Schmidt OrthogonalizationCauchy-Schwartz inequalityInfinite Dimensions

7 Operators and Matrices 141The Idea of an Operator

Definition of an OperatorExamples of OperatorsMatrix MultiplicationInverses

Areas, Volumes, DeterminantsMatrices as Operators

Eigenvalues and EigenvectorsChange of Basis

Summation ConventionCan you Diagonalize a Matrix?

Eigenvalues and GoogleSpecial Operators

8 Multivariable Calculus 178Partial Derivatives

DifferentialsChain RuleGeometric InterpretationGradient

ElectrostaticsPlane Polar CoordinatesCylindrical, Spherical CoordinatesVectors: Cylindrical, Spherical BasesGradient in other CoordinatesMaxima, Minima, SaddlesLagrange MultipliersSolid Angle

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3D Visualization

9 Vector Calculus 1 212

Fluid Flow

Vector Derivatives

Computing the divergence

Integral Representation of Curl

The Gradient

Shorter Cut for div and curl

Identities for Vector Operators

Applications to Gravity

Gravitational Potential

Index Notation

More Complicated Potentials

10 Partial Differential Equations 243

The Heat Equation

Separation of Variables

Oscillating Temperatures

Spatial Temperature Distributions

Specified Heat Flow

Differentiating noisy data

Partial Differential Equations

14 Complex Variables 353Differentiation

IntegrationPower (Laurent) SeriesCore Properties

Branch PointsCauchy’s Residue TheoremBranch Points

Other IntegralsOther Results

15 Fourier Analysis 379Fourier Transform

Convolution TheoremTime-Series AnalysisDerivatives

Green’s FunctionsSine and Cosine TransformsWiener-Khinchine Theorem

16 Calculus of Variations 393Examples

Functional DerivativesBrachistochroneFermat’s PrincipleElectric FieldsDiscrete VersionClassical MechanicsEndpoint VariationKinks

Second Order

17 Densities and Distributions 420Density

FunctionalsGeneralizationDelta-function NotationAlternate ApproachDifferential EquationsUsing Fourier TransformsMore Dimensions

Index 441

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I wrote this text for a one semester course at the sophomore-junior level Our experiencewith students taking our junior physics courses is that even if they’ve had the mathematicalprerequisites, they usually need more experience using the mathematics to handle it efficientlyand to possess usable intuition about the processes involved If you’ve seen infinite series in acalculus course, you may have no idea that they’re good for anything If you’ve taken a differentialequations course, which of the scores of techniques that you’ve seen are really used a lot? Theworld is (at least) three dimensional so you clearly need to understand multiple integrals, but willeverything be rectangular?

How do you learn intuition?

When you’ve finished a problem and your answer agrees with the back of the book or withyour friends or even a teacher, you’re not done The way do get an intuitive understanding ofthe 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 yoursolution when you push these parameters to their limits? In a mechanics problem, what if onemass 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 anddoes 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 intuitionabout 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 theequations work the way they do It reifies algebra

Does it take extra time? Of course It will however be some of the most valuable extratime 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 beentaught 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 isastounding how many problems become simpler after you’ve sketched a graph Also, until you’vesketched 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 thetopics along the way Several more chapters were added after the class was over, so this is nowfar beyond a one semester text There is enough here to select from if this is a course text, but

if you are reading it on your own then you can move through it as you please, though you willfind that the first five chapters are used more in the later parts than are chapters six and seven.Chapters 8, 9, and 13 form a sort of package

The pdf file that I’ve placed online is hyperlinked, so that you can click on an equation orsection reference to go to that point in the text To return, there’s a Previous View button atthe top or bottom of the reader or a keyboard shortcut to do the same thing [Command← onMac, Alt← on Windows, Control on Linux-GNU] The contents and index pages are hyperlinked,

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If you’re using Acrobat Reader 7, the font smoothing should be adequate to read the textonline, but the navigation buttons may not work until a couple of point upgrades.

I chose this font for the display versions of the text because it appears better on the screenthan does the more common Times font The choice of available mathematics fonts is morelimited

I have also provided a version of this text formatted for double-sided bound printing of thesort you can get from commercial copiers

I’d like to thank the students who found some, but probably not all, of the mistakes in thetext Also Howard Gordon, who used it in his course and provided me with many suggestions forimprovements

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Mathematical Methods for Physics and Engineering by Riley, Hobson, and Bence bridge University Press For the quantity of well-written material here, it is surprisingly inexpen-sive in paperback.

Cam-Mathematical Methods in the Physical Sciences by Boas John Wiley Publ About theright level and with a very useful selection of topics If you know everything in here, you’ll findall your upper level courses much easier

Mathematical Methods for Physicists by Arfken and Weber Academic Press At a slightlymore 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 itsapproach 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,” and especially “AdvancedMathematics for Engineers and Scientists.” Amazon lists hundreds

Visual Complex Analysis by Needham, Oxford University Press The title tells you the phasis Here the geometry is paramount, but the traditional material is present too It’s actuallyfun to read (Well, I think so anyway.) The Schaum text provides a complementary image of thesubject

em-Complex Analysis for Mathematics and Engineering by Mathews and Howell Jones andBartlett Press Another very good choice for a text on complex variables Despite the title,mathematicians should find nothing wanting here

Applied Analysis by Lanczos Dover PublicationsThis publisher has a large selection of ately priced, high quality books More discursive than most books on numerical analysis, andshows great insight into the subject

moder-Linear Differential Operators by Lanczos Dover publications As always with this authorgreat insight and unusual ways to look at the subject

Numerical Methods that (usually) Work by Acton Harper and Row Practical tools withmore than the usual discussion of what can (and will) go wrong

Numerical Recipes by Press et al Cambridge Press The standard current compendiumsurveying 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 tensorsthat 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 prettygood

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Springer Material you won’t find anywhere else, with clear examples “ a sleazy tion that provides good physical insight into what’s going on in some system is far more usefulthan an unintelligible exact result.”

approxima-Probability Theory: A Concise Course by Rozanov Dover Starts at the beginning andgoes a long way in 148 pages Clear and explicit and cheap

Calculus of Variations by MacCluer Pearson Both clear and rigorous, showing how manydifferent types of problems come under this rubric, even “ operations research, a field begun bymathematicians, almost immediately abandoned to other disciplines once the field was determined

to be useful and profitable.”

Special Functions and Their Applications by Lebedev Dover The most important of thespecial functions developed in order to be useful, not just for sport

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1.1 Trigonometry

The common trigonometric functions are familiar to you, but do you know some of the tricks toremember (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 occasionallyuseful 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 sin θ? Draw a right triangle anddesignate the tangent of θ as x, so you can draw a triangle with tan θ = x/1

1θ

xThe Pythagorean theorem says that the third side is √

1 + x2.You now read the sine from the triangle as x/√

1 + x2, so

sin θ = p tan θ

1 + tan2θAny 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? Cycles?And who invented radians? Why is this the unit you see so often in calculus texts? That thereare 360◦ in a circle is something that you can blame on the Sumerians, but where did this otherunit come from?

s = CR θwhere 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

C = 1 radian−1 then s = 1 radian−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 circlehas circumference 2πR, and this is Rθ It says that the angle for a full circle has 2π radians.One radian is then 360/2π degrees, 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 trigonometricfunctions

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 functionas

ex− e−x

ex+ e−x (1)The other three functions are

sech x = 1

cosh x, csch x =

1sinh x, coth x =

1tanh xDrawing these is left to problem 4, with a stopover in section 1.8 of this chapter

Just as with the circular functions there are a bunch of identities relating these functions.For the analog of cos2θ + sin2θ = 1 you have

For a proof, simply substitute the definitions of cosh and sinh in terms of exponentials and watchthe terms cancel (See problem 4.23 for a different approach to these functions.) Similarly theother common trig identities have their counterpart here

1 + tan2θ = sec2θ has the analog 1 − tanh2θ = sech2θ (3)

The reason for this close parallel lies in the complex plane, because cos(ix) = cosh x and sin(ix) =

i sinh x See chapter three

The inverse hyperbolic functions are easier to evaluate than are the corresponding circularfunctions I’ll solve for the inverse hyperbolic sine as an example

y = sinh x means x = sinh−1y, y = e

x− e−x2Multiply by 2ex to get the quadratic equation

2exy = e2x− 1 or ex
2− 2y ex
− 1 = 0

The solutions to this are ex = y ±py2+ 1, and because py2+ 1 is always greater than |y|,you must take the positive sign to get a positive ex Take the logarithm of ex and

sinh

sinh−1

x = sinh−1y = ln y +py2+ 1

(−∞ < y < +∞)

As x goes through the values −∞ to +∞, the values that sinh x takes on go over the range

−∞ to +∞ This implies that the domain of sinh−1y is −∞ < y < +∞ The graph of aninverse function is the mirror image of the original function in the 45◦ line y = x, so if you havesketched the graphs of the original functions, the corresponding inverse functions are just thereflections in this diagonal line

The other inverse functions are found similarly; see problem 3

sinh−1y = ln y +py2+ 1
cosh−1y = ln y ±py2− 1
, y ≥ 1tanh−1y = 1

The calculus of these functions parallels that of the circular functions

d

dxsinh x =

ddx

ex− e−x

ex+ e−x

2 = cosh xSimilarly the derivative of cosh x is sinh x Note the plus sign here, not minus

Where do hyperbolic functions occur? If you have a mass in equilibrium, the total force on

it is zero If it’s in stable equilibrium then if you push it a little to one side and release it, the forcewill push it back to the center If it is unstable then when it’s a bit to one side it will be pushedfarther away from the equilibrium point In the first case, it will oscillate about the equilibriumposition and the function of time will be a circular trigonometric function — the common sines orcosines of time, A cos ωt If the point is unstable, the motion will will be described by hyperbolicfunctions of time, sinh ωt instead of sin ωt An ordinary ruler held at one end will swing backand forth, but if you try to balance it at the other end it will fall over That’s the differencebetween cos and cosh For a deeper understanding of the relation between the circular and thehyperbolic functions, see section 3.3

1.2 Parametric Differentiation

The integration techniques that appear in introductory calculus courses include a variety of ods of varying usefulness There’s one however that is for some reason not commonly done incalculus courses: parametric differentiation It’s best introduced by an example

meth-Z ∞ 0

xne−xdx

You could integrate by parts n times and that will work For example, n = 2:

= −x2e−x

∞

0

+

Z ∞ 0

2xe−xdx = 0 − 2xe−x

∞

0

+

Z ∞ 0

2e−xdx = 0 − 2e−x

... fails for negative x, you can extend the definition by using Eq (14) to define Γ fornegative arguments What is Γ(−1/2) for example?

The same procedure works for. .. and because the numerator for small x is approximately

1, you immediately have that

The integral definition, Eq (12), for the Gamma function is defined only for the case that

x... data-page="12">

To see why this is true, sketch graphs of the integrand for a few odd n.

For the integral over positive x and still for odd n, the substitution t = αx2

dt t(n−1)/2e−t