Calculus you need for Physics

 
Updated: 17 July 2006

Physics 2048:
This course makes minimal use of the calculus for homework or exam problems, but I do use the language of the calculus at an early stage (i.e. before you will cover it formally in Calculus I). By the middle of the semester I will assume that you can do the simple derivatives covered in Calculus I up to that point, and I will assume you can do the simple integrals by the end of the semester. Calculus-based problems may appear on the Midterm (derivatives) and are very likely on the Final Exam.

In simple terms, I will assume you are passing Calculus I, not just "taking" it. Similarly, I will assume you remember what you knew when you took Calculus I if you have already passed it, no matter when you took the class. (The assumption in college is that "passed the class" equals "still knows the basic material for the next several years".)

In a few words, what I expect you to learn very early in my physics class are: the definition of the derivative (slope of a continuous function at a point), the definition of the integral (area under a continuous function between two points), that either of these might be a function rather than a number, and the notation for each of these. (You can find some of this in Appendix D of our textbook.) This is the mathematical language we will use for kinematical definitions in Chapter 2 and the definition of work in Chapter 6, among other things.

Since we will not require you to actually do much calculus, you might guess that what you really need to be able to do to pass this course is apply algebra and trig. That would be a correct guess.

Mathematics is the language of physics. Learn it well.

Calculus learning tools on the web:

These may help you visualize what is going on when you take limits to define the derivative. There are also illustrations of other common calculus problems, including some tricky volume integrals (one from calc III as I recall). Check it out.

Physics 2049:
This course makes extensive use of the calculus, although most exam problems only assume you know the basics of integration and differentiation and/or how to set up the integral for an "application" problem. Only a few problems require that you do an integral or take a derivative, and most (but not all) of those depend only on what you were supposed to learn in Calculus I. (Note that this includes the chain rule, definitions of the derivative and integral, and the Fundamental Theorem.) As in 2048, you will be expected to be able to apply trig and geometry to problems such as finding vector components, but you will also work with 3-dimensional geometric objects such as are encountered in the typical analytic geometry and trig class taught in HS 30 to 40 years ago. I will also use complex numbers. (You can read about the main results in the Appendix to our textbook and your calculus text.)

Two things will be used in PHY2049 that the TCC math curriculum only covers in Calculus II: the derivative and integral of the exponential function (its graph and other properties are assumed known from algebra and pre-calc), and the "log" integral. These are easy (see Appendix B of our textbook for the formulas), so it should not be a problem for you to pick them up if you mistakenly put off Calculus II this semester. The exponential function will appear in many examples for some topics we cover (particularly in AC circuits), and you will be expected to know all of its properties on the tests.

[ A side comment on the treatment of exponential and log problems in the calculus curriculum: TCC uses the "late transcendental" approach, which groups log and exponential functions with the inverse trig and hyperbolic functions, while other curricula group them with calc I application problems. Physics texts are agnostic on this subject, expecting you to know these functions and how to do simple applications like work integrals when you complete calc I. Since I think it is easier to pick up the exponential integral than to do "application" problems, the emphasis on the latter in calc I is probably the better one for physics students given current math textbooks. My view of those textbooks, and the ones available for physics, is another matter.]

In PHY2049, I will ignore the distinction between integrals over one variable (Calc I) and integrals over several variables (Calc III), treating volume or area as if they were single variables. They are single variables (for all practical purposes) in the problems we cover on exams, assuming you remember how to apply the definitions and the fundamental theorem of the calculus. This approach enables me to use the correct notation for Maxwell's equations rather than some bogus approximate form. The exam problems arising from Maxwell's equations can be done by just writing down the result of an integral based on definitions and elementary geometry.

Most semesters I will do at least one lecture example of the use of Calc III methods, primarily as an indication of why some engineering majors use (not just require) vector calculus. Don't worry; those kinds of problems will not be on any test, quiz, or homework ... in PHY2049.

 
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