Friday, December 29, 2017

Used Math

Used Math, by Clifford Swartz, superimposed on Intermediate Physics for Medicine and Biology.
Used Math,
by Clifford Swartz.
How much mathematics is needed when taking a class based on Intermediate Physics for Medicine and Biology? Students come to me all the time and say “I am interested in your class, but I don’t know if I have enough math background.” I wish I had a small book that reviewed the math needed for a class based on IPMB. Guess what? Used Math by Clifford Swartz is just what I need. In the preface, Swartz writes:
In this book, which is part reference and part reminder, we are concerned with how to use math. We concentrate on those features that are most needed in the first two years of college science courses. That range is not rigorously defined, of course. A sophomore physics major at M.I.T. or Cal. Tech. must use differential equations routinely, while a general science major at some other place may still be troubled by logarithms. It is possible that even the Tech student has never really understood certain things about simple math. What, for instance, is natural about the natural logs? We have tried to cover a broad range to topics—all the things that a science student might want to know about math but has never dared ask.
Students and instructors might benefit if I went through Used Math chapter by chapter, assessing what math is needed, and what is math not needed, when studying from IPMB. Also, what math is needed but is not included in Used Math.

Chapter 1: Reporting and Analyzing Uncertainty

Russ Hobbie and I assume our readers know about scientific notation and significant figures. The best time to teach significant figures is during laboratory. (Wait! Is there is a lab that goes along with IPMB? No. At least not that I know of. But perhaps there should be.) In my Biological Physics class, students often answer homework using too many significant figures. I don’t take off points, but I write annoying notes in red ink.

Chapter 2: Units and Dimensions

Russ and I do not review how to convert between units. My students usually don’t have trouble with this. Often, however, they will do algebra and derive an equation that is dimensionally wrong (for example, containing “a + a2” where a has units of length). I take off extra points for such mistakes, and I harp about them in class.

Chapter 3: Graphs

We assume students can plot a simple graph of y(x) versus x. In class, when we derive a result such as y(x) = x/(x2 + a2), I ask the students what a sketch of this function looks like. Often they have trouble drawing it. Our homework problems routinely ask students to plot their result. I deduct points if these plots are not qualitatively correct. IPMB discusses semilog and log-log plots in Chapter 2.

Chapter 4: The Simple Functions of Applied Math

Students should be familiar with powers, roots, trigonometric functions, and the exponential function before taking a class based on IPMB. Chapter 2 is devoted to the exponential, and Appendix C lists properties of exponents and logarithms. We define the hyperbolic functions sinh and cosh upon first use (Eq. 6.98). I don’t give placement quizzes at the first class meeting, but if I did I would have the students sketch plots of x2, √x, sin(x), cos(x), tan(x), ex, log(x), sinh(x), cosh(x), and tanh(x). If you can’t do that, you will never be able to translate mathematical results into physical insight.

Chapter 5: Statistics

I discussed the statistics used in IPMB before in this blog. We analyze probability distributions in Chapter 3 on thermodynamics and Chapter 4 on diffusion, and go into more detail in Appendix G (mean and standard deviation), Appendix H (the binomial distribution), Appendix I (the Gaussian distribution), and Appendix J (the Poisson distribution). We don’t discuss analyzing data, such as testing a hypothesis using a student t-test. One topic missing from Used Math is simple concepts from probability; for example, when you role two dice what is the probability that they add to five? When I taught quantum mechanics (a subject in which probability is central), I spent an entire class calculating the odds of winning at craps. You will understand probability by the time you finish that calculation.

Chapter 6: Quadratic and Higher Power Equations

Russ and I use the quadratic equation without review. We don’t solve any higher order equations in IPMB, and we never ask the student to factor a polynomial using a procedure similar to long division (yuk!).

Chapter 7: Simultaneous Equations

Students should know how to solve systems of linear equations. I often solve small systems (two or three equations) in class. Sometimes when teaching I derive the equations and then say “the rest is just math” and state the solution. This happens often when doing a least-squares fit at the start of Chapter 11. I don’t ask students to solve a system of many (say, five) equations.

Chapter 8: Determinants

IPMB does not stress linear algebra and we never require that students calculate the determinant of a matrix. However, we do occasionally require the student to calculate a cross product using a method similar to taking a determinant (Eq. 1.9), so students need to know the rules for evaluating 2 × 2 and 3 × 3 determinants.

Chapter 9: Geometry

Used Math goes into more detail about analytical geometry (conic sections, orbits, and special curves like the catenary) than is needed in IPMB. The words ellipse and hyperbola never appear in our book (parabola does.) We discuss cylindrical and spherical coordinates in Appendix L. Students should know how to find the surface area and volume of simple objects like a cube, cylinder, or sphere.

Chapter 10: Vectors

Russ and I use vectors throughout IPMB. They are reviewed in Appendix B. We define the dot and cross product of two vectors when they are first encountered.

Chapter 11: Complex Numbers

We avoid complex numbers. I hate them. One exception: we introduce complex exponentials when discussing Fourier methods, where we present them as an alternative to sines and cosines that is harder to understand intuitively but easier to handle algebraically. You could easily skip the sections using complex exponentials, thereby banishing complex numbers from the class.

Chapter 12: Calculus—Differentiation

Students must know the definition of a derivative. In class I derive a differential equation for pressure by adding the forces acting on a small cube of fluid and then taking the limit as the size of the cube shrinks to zero. If students don’t realize that this process is equivalent to taking a derivative, they will be lost. Also, they should know that a derivative gives the slope of a curve or a rate of change. What functions should students be able to differentiate? Certainly powers, sines and cosines, exponentials, and logarithms. Plus, students must know the chain rule and the product rule. They should be able to maximize a function by setting its derivative to zero, and they should realize that a partial derivative is just a derivative with respect to one variable while the other variables are held constant (Appendix N).

Chapter 13: Integration

Students must be able to integrate simple functions like powers, sines and cosines, and exponentials. They should know the difference between a definite and indefinite integral, and they should understand that an integral corresponds to the area under a curve. Complicated integrals are provided to the student (for example, Appendix K explains how to evaluate integrals of e-x2) or a student must consult a table of integrals. In my class, I always use the “guess and check” method for solving a differential equation: guess a solution containing some unknown parameters, plug it into the differential equation, and determine what parameters satisfy the equation; no integration is needed. One calculation that some students have problems with is integrating a function over a circle. In class, I carefully explain in how the area element becomes rdrdĪø. At first the students look bewildered, but most eventually master it. I avoid integration by parts (which I dislike), but it is needed when calculating the electrical potential of a dipole. Perhaps you can devise a way to eliminate integration by parts altogether?

Chapter 14: Series and Approximations

Appendix D of IPMB is about Taylor series. If you remember only that ex is approximately 1+x, you will know 90% of what you need. The expansions of sin(x) and cos(x) are handy, but not essential. When deriving the dipole approximation, I use the Taylor series of 1/(1-x). (The day I discuss the dipole is one of the most mathematical of the semester.) Student’s never need to derive a Taylor series, and they rarely require more than the first two terms of the expansion. The geometric series (1+x+x2+…) appears in Homework Problem 28 of Chapter 8, but the sum of the series is given. In IPMB, we never worry about convergence of an infinite series. Fourier series is central to imaging. In Medical Physics (PHY 3260), I spend a couple weeks discussing Fourier series and Fourier transforms, the most mathematically intensive part of IPMB. If students can handle Chapters 11 and 12, they can handle any math in the book.

Chapter 15: Some Common Differential Equations

I always tell my class “if you can solve only one differential equation, let it be dy/dx = by” (in case you are wondering, the solution is y = ebx). As I mentioned earlier, I preferred to solve differential equations by guess and check. In IPMB, you can get away with guesses that involve powers, trig functions, and exponentials. Some students claim that a course in differential equations is needed before taking a class using IPMB. I disagree. We don’t need advanced methods (e.g., exact differential equations) and we never analyze existence and uniqueness of solutions. We just guess and check. Appendix F discusses differential equations in general, but my students rarely need to consult it. I emphasize understanding differential equations from a physical point-of-view. I expect my students to be able to translate a physical statement of a problem into a differential equation. Yes, I put such questions on my exams. To me, that is a crucial skill.

Chapter 16: Differential Operators

What Used Math calls differential operators, I call vector calculus: divergence, gradient, and curl. Russ and I use vector calculus occasionally. I expect students to be able to do homework problems using it, but I don’t expect them to do such calculations on exams. Mostly, vector calculus appears when talking about electricity and magnetism in Chapters 6-8. I think an instructor could easily design the class to avoid vector calculus altogether. Whenever Russ and I use vector calculus, we typically cite Div, Grad, Curl and All That, which is my favorite introduction to these concepts.

That sums up of the topics in Used Math. Is there any other math in IPMB? Special functions sometimes pop up, such as Bessel functions, the error function, and Legendre polynomials. Usually these appear in homework problems that you don’t have to assign. We occasionally ask students to solve differential equations numerically (see Sec. 6.14), usually in the homework. I skip these problems when I teach from IPMB; there is not enough time for everything. In some feedback problems in Chapter 10 (for example, Problems 10.12 and 10.17) the operating point must be evaluated numerically. I do assign these problems, and I tell students to find the solution by trial and error. We don’t spend time developing fancy methods for solving nonlinear equations, but I want students to realize they can solve equations such as xex = 1 numerically (the solution is approximately x = 0.57).

In summary, Used Math contains almost all the mathematics you need when taking a class from IPMB. It would be an excellent supplementary reference for students. From now on, when students ask me how much math they need to know for my Biological Physics or Medical Physics class, I will tell them all they need is in Used Math.

No comments:

Post a Comment