Notes on MathKristina BarkumeSeptember 28, 20051 Getting HelpMost of the math that is used in 108 is well explained in Mathematical Methods in the Physical Science byMary Boas. I’m working to get this book put on reserve in the Library. Another place to look is MathWorldor Eric Weisstein’s PhysicsWorld on Wolfram’s website.2 Taylor SeriesThe Taylor Series (aka Taylor Expansion) is used all the time in the physical sciences, so its best to getfamiliar with them.Taylor Series: Let f(x) be a well behaved function with no infinite derivatives. Then, f(x) about thepoint, x=a, can be written as an infinite series as follows:f(x) = f(a) + (x − a)f0(x) +12!(x − a)2f00(x) + .... +1n!(x − a)nf(n)(a) + ...Typically, a = 0 and the Taylor Series becomesf(x) = f(0) + xf0(x) +12!x2f00(x) + .... +1n!xnf(n)(a) + ...Here are a few useful series to remember:sin(x) = x −x33!+x55!−x77!+ ...... + (−1)(2n)x(2n+1)(2n + 1)!where n=0,1,2,...cos(x) = 1 −x22!+x44!−x66!+ ...... + (−1)(2n)x(2n)(2n)!where n=0,1,2,...ex= 1 + x +x22!+x33!+ ....... +xnn!where n=0,1,2...(1 + x)p= 1 + px +p(p − 1)2!x2+p(p − 1)(p − 2)3!x3+ ...... ****where p is any real number, and convergent only for |x| < 11You can use the following equations to make a number of approximations that are often used in physicalscience problems. In particular, note thatsin(x) ≈ x if x << 1tan(x) ≈ sin(x) ≈ x if x << 1(1 + x)a≈ 1 + ax if x << 1ex≈ 1 + x if x << 13 Complex NumbersComplex numbers are often used to describe wave behaviour and therefore are sometimes used for planetaryand geological purposes. Note that in engineering, i is often replaced by the symbol j, so don’t get trippedup by notation. Andy already gave some helpful equations on the class notes. Here I’ve added just a couplethat you may find useful. Boas has a good discussion of complex numbers if you need a more thoroughreminder.(eiθ)n= (cos(θ) + i sin(θ))n= cos(nθ) + i sin(nθ)If you look at the Taylor expansions for sine, cosine, and the exponential, you see where the followingequations come fromsin(θ) =eiθ− e−iθ2icos(θ) =eiθ+ e−iθ2And hence cos(θ)2+sin(θ)2=1. The formulations above can sometime s be an easier way with dealing thistrig functions, but use them with care, you don’t want to lose an i.4 Differential Equation4.1 NotationI’m not going to cover the whole of DE’s, I’ll cover the basics and focus on some of the language.Let me begin with notation. Unfortunately, there are several notation conventions for derivatives, sodon’t get hung up on this. The following are generally equivalent was of representing the first derivative off.dfdx, f0(x), f0,˙f, DfAnd the sec ond derivatives ared2fdx2, f00(x), f00,¨f, D2fYou may have to do some sluthing about to find the independent variable in the case that f0,˙f and Df,although˙f is almost exclusively used to denote the time derivative. D is called the differential operator2(notice that it’s in bold, sometimes it wears a hat too), and can be a handy way to represent the derivative,especially when talking about other linear operators. The partial derivative is always written as∂f∂xor ∂xfAlso note that there is a proper way to write the higher order derivatives using the dx/dy notation. Hereare some of the wrong ways to write itdfdx2,df2dx2,d2fd2x,It seems a bit picky, but the first case above would give you the wrong answer if you were doing the mathcorrectly. The second and third cases illustrate a very subtle difference between the meaning of d in thenumerator and denominator. In the numerator, d is acting as an operator -it does work. In the numerator,dx is a differential, i.e. a really small number, and, in the numerator, a ’d’ by itself is meaningless. Don’tget too caught up with this though because mathematicians do math operations on both the numerator anddenominator and don’t bat an eye.4.2 TerminologyOk, lets now recall what the total derivative is. Let f be a function of the variables x1, x2, ...xn, then thetotal derivative of f isdf =∂f∂x1dx1+∂f∂x2dx2+ .... +∂f∂xndxnLet me begin with some terms. Let f be our dependent variable. A differential equation is an equationthat contains at least one derivative of f. Partial differential equations are equations that contain partialderivatives. Equations with full derivatives are called ordinary differential equations. A differential equationis considered linear if all the dependent terms are first order in power, i.e.a0f + a1f0+ a2f00+ .......anf(n)= bThe following is not a linear equation (i.e. its non-linear!)a0f + a1(f0)2= bAn equation is considered homogeneous if every term contains f or a derivative of f. e.g.a0f + a1f00= 0An equation is inhomogeneous if it contains other terms, i.e.a0f + a1f00= b4.3 A Simple ODEYou’ve been using ODEs for a long time, even though you might not have realized it. Here’s an easydifferential equation from freshman physics. Recall that acceleration is the second derivative of position.That is,d2~xdt2= ~aIf you take the integral of both sides you getd~xdt= ~at + c0,3where c0is the constant of integration, since we didn’t set limits. However, we know thatd~xdtis the velocity,and at t=0, the velocity is just the initial velocity, ~v0. This means that c0= ~v0. If we integrate again, thenwe get~x(t) =12~at2+ ~v0t + c1Again, putting in 0 into the above equation tells us that c1must be the initial position, ~x0, which just getsus the familiar distance equation,~x(t) =12~at2+ ~v0t + ~x04.4 Separation of VariablesOne of the more common and easiest methods for solving ODE’s is called separation of variables. Supposeyou have an equation of the formdfdx=p(x)q(f)Then you can rewrite the equation as followsq(f)df = p(x)dxYou can then take the integral of both sides and solve the equation. This form is exemplified by the decayequation. Let the number of40K atoms at time t be denoted by N(t). The numb er of40K atoms being lostat any given time isdNdt= −λN (1)where λ is called the decay constant it tells you something of how fast particles are lost. Using separationof variables, we get that equation 1 can b e rewritten asdNN= −λdt (2)Taking the integral of equation 2 giveslog(N(t)) = −λt + c0, (3)where c0is the constant of integration. Taking the exponential of both sides of equation 3 gives a nicersolution.exp(log(N(t))) = exp(λt + c0)N(t) = e(c0)e(−λt)= N0e(−λt)if there are N0at t=0. Lets figure out what λ has to