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Mathematical Tools for Physics by James Nearing Physics Department University of Miami mailto:jnearing@miami.edu Copyright 2003, James Nearing Permission to copy for individual or classroom use is granted. QA 37.2 Contents Introduction . . . . . . . . . . . . . . . . . iv Bibliography . . . . . . . . . . . . . . . . vi 1 Basic Stuff . . . . . . . . . . . . . . . . . 1 Trigonometry Parametric Differentiation Gaussian Integrals erf and Gamma Differentiating Integrals Polar Coordinates Sketching Graphs 2 Infinite Series . . . . . . . . . . . . . . . . 30 The Basics Deriving Taylor Series Convergence Series of Series Power series, two variables Stirling’s Approximation Useful Tricks Diffraction Checking Results 3 Complex Algebra . . . . . . . . . . . . . . 65 Complex Numbers Some Functions Applications of Euler’s Formula Logarithms Mapping 4 Differential Equations . . . . . . . . . . . . 83 Linear Constant-Coefficient Forced Oscillations Series Solutions Trigonometry via ODE’s Green’s Functions Separation of Variables Simultaneous Equations Simultaneous ODE’s Legendre’s Equation 5 Fourier Series . . . . . . . . . . . . . . . 118 Examples Computing Fourier Series Choice of Basis Periodically Forced ODE’s Return to Parseval Gibbs Phenomenon 6 Vector Spaces . . . . . . . . . . . . . . . 142 The Underlying Idea Axioms Examples of Vector Spaces Linear Independence Norms Scalar Product Bases and Scalar Products Gram-Schmidt Orthogonalization Cauchy-Schwartz inequality Infinite Dimensions i 7 Operators and Matrices . . . . . . . . . . 168 The Idea of an Operator Definition of an Operator Examples of Operators Matrix Multiplication Inverses Areas, Volumes, Determinants Matrices as Operators Eigenvalues and Eigenvectors Change of Basis Summation Convention Can you Diagonalize a Matrix? Eigenvalues and Google 8 Multivariable Calculus . . . . . . . . . . . 208 Partial Derivatives Differentials Chain Rule Geometric Interpretation Gradient Electrostatics Plane Polar Coordinates Cylindrical, Spherical Coordinates Vectors: Cylindrical, Spherical Bases Gradient in other Coordinates Maxima, Minima, Saddles Lagrange Multipliers Solid Angle Rainbow 3D Visualization 9 Vector Calculus 1 . . . . . . . . . . . . . 248 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 Summation Convention More Complicated Potentials 10 Partial Differential Equations . . . . . . . 283 The Heat Equation Separation of Variables Oscillating Temperatures Spatial Temperature Distributions Specified Heat Flow Electrostatics 11 Numerical Analysis . . . . . . . . . . . . 315 Interpolation Solving equations Differentiation Integration Differential Equations Fitting of Data Euclidean Fit Differentiating noisy data Partial Differential Equations 12 Tensors . . . . . . . . . . . . . . . . . . 354 Examples Components Relations between Tensors Non-Orthogonal Bases Manifolds and Fields Coordinate Systems ii Basis Change 13 Vector Calculus 2 . . . . . . . . . . . . . 396 Integrals Line Integrals Gauss’s Theorem Stokes’ Theorem Reynolds’ Transport Theorem 14 Complex Variables . . . . . . . . . . . . . 418 Differentiation Integration Power (Laurent) Series Core Properties Branch Points Cauchy’s Residue Theorem Branch Points Other Integrals Other Results 15 Fourier Analysis . . . . . . . . . . . . . . 451 Fourier Transform Convolution Theorem Time-Series Analysis Derivatives Green’s Functions Sine and Cosine Transforms Weiner-Khinchine Theorem iii Introduction I wrote this text for a one semester course at the sophomore-junior level. Our experience with studen ts 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 techn iqu es 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 e nter 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 muc h 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 d escribe 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, un til 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 e nough 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 Wi ndows, 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 studen ts 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 valu able 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 compl emen tary 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 discurs ive 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 ins ight 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 sin θ? Draw a right triangle and d esignate the tangent of θ as x, so you can draw a triangle with tan θ = x/1. x 1 θ The Pythagorean theorem says that the third side is √ 1 + x 2 . You now read the sine from the triangle as x/ √ 1 + x 2 , so sin θ = tan θ 1 + tan 2 θ Any other s uch 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? O ther? 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? R 2R s θ 2θ 1 1—Basic Stuff 2 It resul ts 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 = CR θ 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 2πR = CR 360 ◦ , and C = π 180 degree −1 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 dθ sin θ = π 180 cos θ This is the reason that the radian was invented. The radian is the unit designed so that the proportionality constant is one. 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 circle has 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 trigonometric functions. Hyperbolic Functions The circular trigonometric functions, the sines, cosines, tangents, and their reciprocals are familiar, bu t their hyperbolic counterparts are probably less so. They are related to the exponential function as cosh x = e x + e −x 2 , sinh x = e x − e −x 2 , tanh x = sinh x cosh x = e x − e −x e x + e −x (1) [...]... the numerator for small x is approximately 1, you immediately have that Γ(x) ≈ 1/x for small x (15) The integral definition, Eq (12), for the Gamma function is defined only for the case that x > 0 [The behavior of the integrand near t = 0 is approximately tx−1 Integrate this from zero to something and see how it depends on x.] Even though the original definition of the Gamma function fails for negative... extend the definition by using Eq (14) to define Γ for negative arguments What is Γ(− 1/2) for example? √ 1 − Γ(−1/2) = Γ(−(1/2) + 1) = Γ(1/2) = π, 2 so √ Γ(−1/2) = −2 π The same procedure works for other negative x, though it can take several integer steps to get to a positive value of x for which you can use the integral definition Eq (12) The reason for introducing these two functions now is not that... 2≤x 0 but does not converge for x ≤ 0 1.16 What is the Gamma function for x near to 1? near 0?... and plug in to the cubic formula, I suggest that you differentiate the whole equation with respect to x and solve for dy/dx Generalize this to finding dy/dx if f (x, y) = 0 Ans: 1/5 1.37 When flipping a coin N times, what fraction of the time will the number of heads in the run lie between − N/2 + 2 N/2 and + N/2 + 2 N/2 ? What are these numbers for N = 1000? Ans: 99.5% 1.38 For N = 4 flips of a coin, . Mathematical Tools for Physics by James Nearing Physics Department University of Miami mailto:jnearing@miami.edu Copyright 2003, James Nearing Permission. Theorem iii Introduction I wrote this text for a one semester course at the sophomore-junior level. Our experience with studen ts taking our junior physics courses is that