Mathematics Self-Assessment for Physics Majors Steven Detweiler and Yoonseok Lee November 29, 2012 The lack of mathematical sophistication is a leading cause of difficulty for students in Classical Mechanics and other upper level physics courses. An official pre-requisite of PHY3321 is PHY2048, PHY2049, and the math requirements include MAC 2311, 2312 and 2313 (Vector Calculus). These math courses together cover derivatives and integrals of trig and log functions, series and sequences, analytic geometry, vectors and partial derivatives and multiple integrals. We will casually be using math from all of these subjects. None of these should be completely unfamiliar to you. Fluency in these math skills is a necessary but not sufficient condition for your success in all upper level physics courses. The following discussions and questions are grouped by subject and in approximate order of difficulty—easiest first. These are representative of the level of mathematics which is expected in this course. You should be very comfortable and fluent with mathematics at this level, at least through section F. Section D, on differential equations, is probably more difficult for you but important. The answers to the questions are not always given. If you do not know if your answer is correct, then you are not comfortable with mathematics at this level. The importance of Section G cannot be overemphasized. You will use Taylor expansion over and over again as far as you are dealing with physics. If you understand the Taylor series in section G, then you are likely to find section H, on calculators, interesting and amusing. Don’t be surprised if my discussions seem confusing at first: To understand math and physics often requires multiple, multiple readings and drills while working out algebraic details with paper and pencil in hand. Never ever try to go through math or physics problems with your eyes. You may think you understand the problem or the subject. But when a similar problem is given in an exam, you will feel that you have seen this before but cannot solve it. This is why I often hear from many students ”I studied very hard (with my eyes) but I do not perform well in exams.” Finally, Section I involves an ordinary differential equation that has an interesting application to radioactivity. A. Algebra Q1. Solve for x: f (x) = ax2 + bx + c = 0. For what value of x is f (x) a maximum or a minimum? Q2. Make a sketch of the function y(x) where y = mx + b and where m and b are constants. What are the meanings of the constants m and b in terms of your sketch? Q3. Factorize (a2 + 4ab + 4b2 ) and 1 (a2 − 9b2 ) B. Vector Algebra ~ = 1ı̂ + 2̂ + 3k̂ and B ~ = 4ı̂ + 5̂ + 6k̂. Q4. Let A ~ What is |A|? ~ · B? ~ What is A ~ × B? ~ What is A ~ and B? ~ What is the cosine of the angle between A Do you know ı̂ ̂ k̂ ~×B ~ = 1 2 3 ? A 4 5 6 ~ ·C ~ = 10 and the angle between A ~ and C ~ is 30◦ , then what is the magnitude of C? ~ Q5. If A C. Calculus Q6. If x0 , v0 and a are constants and 1 x(t) = x0 + v0 t + at2 2 then what is dx/dt? What is d2 x/dt2 ? If a < 0, does the function x(t) curve up or down? If x is negative when t = 0 and x is positive when t is very large: then for precisely which values of t is x positive? Q7. Evaluate the derivative d A cos(ωt + φ) dt where A, ω and φ are constant. Q8. If f (x) = x , cos x what is df (x) ? dx Q9. If f (x) = tan(ax2 + b), what is df (x) ? dx Q10. Plot y = 13 x3 − 2x2 + 3x + 2 without using a graphing calculator. By just looking at the functional form can you tell how many extrma at most exist in this curve? Q11. What is the value of x where the following function peaks (a is a positive constant)? f (x) = p 1 (a2 − x2 )2 + 4 2 . Can you plot the above function? Q12. Evaluate the following integrals π Z sin θ dθ, 0 k dx x2 Z where k is a constant, and Z x 1 k dx. x Is sin x an odd or even function? How about cos x, sin2 x, x + 5x5 ? If function f (x) is even, then f (x) = f (−x). If f (x) is odd, then f (x) = −f (−x). Do you see then why Z +a f (x)dx = 0 −a for any odd function f (x)? D. Differential equations These next two problems might be difficult or possibly unfamiliar to you, but take a careful look at them because these are very important in classical mechanics. Q13. Find two different functions which satisfy the differential equation d2 f (x) − λ2 f (x) = 0. dx2 Q14. Find two different functions which satisfy the differential equation d2 f (x) + ω 2 f (x) = 0. 2 dx E. Trigonometry and Geometry Euler’s identity, eiθ = cos θ + i sin θ, is probably new to you. But it provides a convenient and easy way to derive some of the basic trig identities such as 3 ei(α+β) = eiα eiβ cos(α + β) + i sin(α + β) = (cos α + i sin α) × (cos β + i sin β) or, after multiplying out the right hand side, cos(α + β) + i sin(α + β) = cos α cos β − sin α sin β + i(cos α sin β + sin α cos β) The real and the imaginary parts of this equation give the well known trig identities: cos(α ± β) = cos α cos β ∓ sin α sin β and sin(α ± β) = sin α cos β ± cos α sin β. or, after multiplying out the right hand side, cos(α + β) + i sin(α + β) = cos α cos β − sin α sin β + i(cos α sin β + sin α cos β) At this point let’s check if you are comfortable with complex numbers. Any complex number z can be represented in one of the following ways: z = a + ib (a, b ∈ R) b (θ = arctan ) a iδ i(θ−δ) iθ′ = |z|e e = Ce (C ∈ C). = |z|eiθ And, ′ z ∗ = a − ib = |z|e−iθ = C ∗ e−iθ , √ |z| = zz ∗ . Q15. A real function x(θ) is given by x(θ) = C1 eiθ + C2 e−iθ (C1 , C2 ∈ C, and θ ∈ R) = B1 cos θ + B2 sin θ (B1 , B2 ∈ R). Show that (i) C2 = C1∗ and (ii) B1 = C1 + C2 , B2 = i(C1 − C2 ). Q16. Use the Euler identity to show that sin2 θ + cos2 θ = 1. Hint: start with eiα e−iα = 1 and then use the Euler Identity. At this point, it is appropriate to introduce hyperbolic functions. e±x = cosh x ± sinh x. 4 FIG. 1: See Q18. Q17. What is the first derivative of tanh x? Q18. See the figure above. AB = BC = 2, DE = 1, and ∠(DF E) = π/2. What is ∠(F AD)? What is AF ? What is DF ? What is cos θ (θ = ∠(EDF ))? Let’s call the crossing point of AD and BE G. What is DG? F. Sums Q19. Evaluate the sum S(x) = ∞ X xn n=0 for |x| < 1 . Ans: Note that S(x) = ∞ X n x = 1+ n=0 ∞ X xn n=1 ∞ X = 1+x xn n=0 = 1 + xS(x) So we have and, finally S = 1 + xS (1 − x)S = 1 1 . S(x) = 1−x 5 By simply substituting x = −a, you have ∞ X 1 = 1 − a + a2 − a3 + · · · = (−1)n an . 1+a n=0 Q20. How about the following summation? SN (x) = N X xn . n=0 G. Taylor expansions of a function Any differentiable function f (x) may be approximated in the neighborhood of a point x0 by the Taylor expansion 2 n 1 1 df 2d f nd f + · · · + (x − x0 ) + · · · (1) + (x − x0 ) f (x) = f (x0 ) + (x − x0 ) dx x=x0 2 dx2 x=x0 n! dxn x=x0 For example, consider f (x) = 1/(1 − x), expanded about x0 = 0. Then f (x) = 1/(1 − x), df = [1/(1 − x)2 ]x=0 = 1 dx x=0 d2 f = 2[1/(1 − x)3 ]x=0 = 2 2 dx x=0 d3 f = 6[1/(1 − x)4 ]x=0 = 6 dx3 x=0 dn f = n![1/(1 − x)n+1 ]x=0 = n! dxn x=0 The Taylor expansion for 1/(1 − x) with x0 = 0 is now 1 1 1 1 = 1 + x + x2 × 2 + x3 × 6 + · · · + xn × n! + · · · 1−x 2 6 n! And this is easily seen to be ∞ X 1 = xn , 1 − x n=0 the same as our example for doing sums above! Taylor expansions of this sort are extremely useful in physics. You will have to use Taylor expansion over and over again in physics. Trust me! For example in special relativity 6 when we are interested to see how close special relativity is to Newtonian physics for small speeds v, we usually make the assumption that v/c ≪ 1. Then we make Taylor expansions of the relevant formulae, and include only the terms proportional to v/c or maybe also v 2 /c2 . Q21. Try this! You do not have to understand physics here. Just follow the mathematical procedure. The displacement z of a particle of rest mass mo , resulting from a constant force mo g along the z-axis is c2 gt 1 z = {[1 + ( )2 ] 2 − 1}, g c including relativistic effect. Find the displacement z as a power series in time t. Compare with the well-known classical result, 1 z = gt2 . 2 Here, g is the gravitational acceleration and c is the speed of light. Hint: You should realize gt/c << 1 and behave as a small parameter as ǫ in the formulae above. In the complete classical limit where the speed of light is considered infinite, you will recover the classical result. You know that you cannot just put c = ∞ in the above expression, which will give you meaningless z = 0. Common Taylor expansions give approximations such as 1 = 1 + ǫ + O(ǫ2 ), 1−ǫ (1 + ǫ)n = 1 + nǫ + O(ǫ2 ), √ 1 1 − ǫ = 1 − ǫ + O(ǫ2 ), 2 1 1 √ = 1 + ǫ + O(ǫ2 ), 2 1−ǫ 1 1 1 eǫ = 1 + ǫ + ǫ2 + ǫ3 + ǫ4 + O(ǫ5 ), 2 6 24 1 1 1 eiǫ = 1 + iǫ − ǫ2 − i ǫ3 + ǫ4 + O(ǫ5 ), 2 6 24 1 1 ln(1 + ǫ) = ǫ − ǫ2 + ǫ3 + O(ǫ4 ), 2 3 1 2 1 3 ln(1 − ǫ) = −ǫ − ǫ − ǫ + O(ǫ4 ). 2 3 The O(ǫn ) term here is standard mathematical notation to mean a function which is less than some constant times ǫn in the limit that ǫ → 0. In other words for small ǫ, O(ǫn ) is no bigger than something times ǫn . 7 We can use the Euler identity eiǫ = cos ǫ + i sin ǫ to easily pick off the purely real terms from this last expansion which give the expansion of cos ǫ for a small angle ǫ, and the purely imaginary terms, which give the expansion of sin ǫ for small ǫ: and 1 1 cos ǫ = 1 − ǫ2 + ǫ4 + O(ǫ6 ) 2 24 1 3 sin ǫ = ǫ − ǫ + O(ǫ5 ). 6 Since cos ǫ is an even function, you can only have terms of a even power such as ǫ0 = const, ǫ2,4,.. . Similarly for sin ǫ only odd power terms exist. √ Q22. Expand 1 − 2tz + t2 in powers of t assuming that it is small. What is the coefficient of the linear term, t in this √ expansion? Ans: You should reach 1 − 2tz + t2 ≈ 1 + zt + 21 (3z 2 − 1)t2 . Let’s go back to Eq.(1). If a function f (x) has a extremum (maximum or minimum) at x = xo , then df = 0. dx x=x0 This means that around x = xo , the shape of the function is parabolic bending upward 2 2 (minimum) if ddxf2 x=x0 > 0 or bending downward (maximum) if ddxf2 x=x0 < 0. An analytic function can be approximated by a quadratic function near its minimum or maximum! Q23. Show that for x << 1, tanh x ≈ x, and tanh x → ±1 as x → ±∞. Then, sketch the curve of y = tanh x on the graph. H. Calculators When solving a physics problem, think with your brain not with your calculator! Before touching your calculator, check to see that your algebraic answer has the correct units and that it has the expected behavior for various limits. It is nearly impossible to check the correctness of an answer once you touch your calculator. You might find it amusing that the number 10100 has been given the name googol, and 10googol is called googolplex—and these names were coined well before the internet was invented. But, note the difference in spelling between googol and the name of the internet search engine. As an aside: The internet was invented by physicists who wanted to exchange easily experimental data between the United States and Europe. Here are a couple examples which are relevant to one of the homework problems for this course. Let f (n) = n2 , where n is an integer. First evaluate f (102 ) − f (102 − 1) on your calculator. You should get 199. Now try to evaluate f (10100 ) − f (10100 − 1). Your calculator will choke on this problem, but your brain can easily find the answer to 100 significant digits. Note that f (n) − f (n − 1) = n2 − (n − 1)2 = n2 − (n2 − 2n + 1) = 2n − 1. 8 With n = 10100 , it is easy to see that f (n) − f (n − 1) = 2 × 10100 − 1 ≈ 2 × 10100 with 100 significant digits. Here is a second, more challenging, problem. Let f (n) = n−2 where n is an integer. Evaluate f (10100 − 1) − f (10100 ). Your calculator will also choke on this problem, but again you can easily find the answer to about 100 significant digits. Use the Taylor expansion f (n + δn) = f (n) + δn df df 2 + . . . ⇒ f (n + δn) − f (n) = δn + . . . = −δn 3 + . . . dn dn n With n = 10100 and δn = −1, we easily have f (10100 − 1) − f (10100 ) = 2 × 10−300 + . . ., where the . . . represents terms which are comparable to 1/n4 = 10−400 or smaller. Thus, the answer is correct for the first 100 digits. For a final example which reveals the limitations of your calculator, evaluate √ 1 − 1 − 3 × 10−30 The answer is not zero. Analytically, find an approximation to the answer. In this context, the word “analytically” means that you should use algebra and calculus to find the answer. And you shouldn’t touch a calculator or computer. Hint: use a Taylor expansion. I. Radioactivity and a simple differential equation The radioactive nucleus 14 C spontaneously decays into 14 N a β − and a ν̄e . That is to say, carbon–14 decays into nitrogen–14, a beta particle (also known as an electron), and an antielectron-neutrino, which is generically described as just a neutrino. If you start with a glass full of 14 C today, then in 5730 years you will only have half a glass of 14 C. After a total of 11460 years only a quarter of a glass will remain. And so forth. We say that the half-life of 14 C is 5730 years. In general for any radioactive particle, if we start at t = 0 with N0 particles, then after a time t the number remaining is 1 t/t1/2 N (t) = N0 2 where t1/2 is the half-life. Radioactive decay gives one example of a number N (t) whose rate-of-change in time dN (t)/dt is proportional to the number N (t) itself. In other words, dN (t) ∝ N (t). dt For definiteness assume that dN (t) = −λN (t), dt where λ is a constant. We solve this differential equation by rewriting it as dN (t) = −λ dt N 9 and integrating both sides Z dN (t) = − N Z λ dt ln N = −λt + constant or N (t) = No e−λt where ln(N0 ) = constant is a constant of integration determined by the initial conditions. The last line follows by taking the logarithm of both side of the previous equation. With radioactivity, we often define the “e-folding time” τ ≡ 1/λ, which also happens to be the “mean-lifetime” of the particle, so that N (t) = N0 e−t/τ . τ is called the e-folding time because the number of particles decreases by a factor of e after a time τ . It is easy to see the relationship between t1/2 and τ by starting with N0 e−t/τ = N0 1 t/t1/2 2 . Now divide out the N0 , and take the natural logarithm of both sides 1 t t ln . (We are using ln(AB ) = B ln(A) and ln e = 1) − = τ t1/2 2 Finally, cancel the t, invert each side of the equation, and use the fact that ln(1/2) = − ln 2. The result is t1/2 = τ ln 2. Note that τ (14 C) = 5370yr/ ln 2 = 7750 yr is the e-folding time of 14 C. The mean-lifetime (e-folding time) of a muon is about 2µs. So, the half-life of a muon is about t1/2 (muon) = ln 2 × 2 µs ≈ 1.4 µs. 10

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