This math text is intended to be used to support the two-semester series of courses teaching introductory physics at the college level. Students who hope to succeed in learning physics, from my two online textbooks that teach it or elsewhere, need as a prerequisite a solid grasp of a certain amount of mathematics.
I usually recommend that all students have mastered mathematics at least through single-variable differential calculus (typified by the AB advanced placement test or a first-semester college calculus course) before tackling either semester of physics: Mechanics or Electricity and Magnetism. Students should also have completed single variable integral calculus, typified by the BC advanced placement test or a second-semester college calculus course, before taking the second semester course in Electricity and Magnetism. It is usually OK to be taking the second semester course in integral calculus at the same time you are taking the first semester course in physics (Mechanics); that way you are finished in time to start the second semester of physics with all the math you need fresh in your mind.
In my (and most college level) textbooks it is presumed that students are competent in geometry, trigonometry, algebra, and single variable differential and integral calculus; more advanced multivariate calculus is used in a number of places but it is taught in context as it is needed and is always ``separable'' into two or three independent one-dimensional integrals of the sort you learn to do in single variable integral calculus. Concepts such as coordinate systems, vectors algebra, the algebra of complex numbers, and at least a couple of series expansions help tremendously - they are taught to some extent in context in the course, but if a student has never seen them before they will probably struggle.
This book (in which you are reading these words) is not really intended to be a ``textbook'' in math. It is rather a review guide, one that presumes that students have already had a ``real'' course in most of the math it covers but that perhaps it was some years ago when they took it (or then never did terribly well in it) and need some help relearning the stuff they really, truly need to know in order to be able to learn physics. It is strongly suggested that all physics students that are directed here for help or review skim read the entire text right away, reading it just carefully enough that they can see what is there and sort it out into stuff they know and things that maybe they don't know. If you do this well, it won't take very long (a few hours, perhaps a half a day) and afterwords you can use it as a working reference as needed while working on the actual course material.
This isn't really a math textbook, but math is an extremely important part of physics. Physics textbooks usually at least attempt to include math support for key ideas, reviewing e.g. how to do a cross product. The problem with this is that this topical review tends to be scattered throughout the text or collected in an appendix that students rarely find when they most need it (either way).
I don't really like either of these solutions. This is my own solution to the same problem: a very short math review textbook that contains just precisely what is needed in order to really get going with physics in the introductory classes one takes as a freshman physics major or later, perhaps as a pre-medical student or math major.
This math is not horrible difficult, but it often (and quite reasonably) is challenging for students of introductory physics. It is often the first time they are called upon to actually use a lot of the math they took over several years of instruction in high school and college. To my experience, most introductory physics students struggle with simple things like decomposing vectors into components and adding them componentwise, with the quadratic formula, with complex numbers, with simple calculus techniques, sometimes even with basic algebra.
College level math textbooks tend to be useless to help buff up one's skills in this kind of thing at the level needed to support the physics, alas. One would need a bunch of them - one for vectors, coordinate systems, and trig, one for basic calculus, one to review high school algebra, one for numbers in general and complex numbers in particular, one for basic geometry. It is rare to find a single book that treats all of this and does so simply and concisely and without giving the student a dozen examples or exercises per equation or relation covered in the book. What is needed is a comprehensive review of material, one that is shallow and fast enough to let a student quickly recall it if they've seen it before well enough to use, yet deep and complete enough that they can get to where they can work with the math even if they have not had a full course in it, or if they can't remember three words about e.g. complex variables from the two weeks in an algebra class three years ago when they covered them - in high school.
Hence this book. I recommend skimming it quickly right now to learn what it contains, then making a slightly slower pass to review it, then go ahead and move on the the physics and come back anytime you are stumped by not remembering how to integrate something like (for example):
&int#int;_0^&infin#infty;x^2 e^-ax dx
Here are some of the things you should be able to find help for in this book. Note well that this is a work in progress, and not all of them may be in place. Feel free to bug me at rgb at phy dot duke dot edu if something you need or think should be here isn't here. I'm dividing my time between the writing and development of several textbooks (including the two semester physics textbooks this short review was once a part of and is now intended to support) but squeaky wheels get the oil, I always say.
This book leverages existing online resources for learning or reviewing math to the extent possible, especially Wikipedia. If you bought a paper copy of this book to help support the author, Thank You! However, I would still recommend that you read through the book in a computer browser from time to time, especially one that supports active links. Most of the footnotes that contain wikipedia pages will pipe you straight through to the referenced pages if clicked!
Integers, real numbers, complex numbers, prime numbers, important numbers, the algebraic representation of numbers. Physics is all about numbers.
Algebra is the symbolic manipulation of numbers according to certain rules to (for example) solve for a particular desired physical quantity in terms of others. We also review various well-known functions and certain expansions.
Cartesian, Cylindrical and Spherical coordinate systems in 2 and 3 dimensions, vectors, vector addition, subtraction, inner (dot) product of vectors, outer (cross) product of vectors.
There is a beautiful relationship between the complex numbers and trig functions such as sine, cosine and tangent. This relationship is encoded in the ``complex exponential'' , which turns out to be a very important and useful relationship. We review this in a way that hopefully will make working with these complex numbers and trig functions both easy.
We quickly review what differentiation is, and then present, sometimes with a quick proof, a table of derivatives of functions that you should know to make learning physics at this level straightforward.
Integration is basically antidifferentiation or summation. Since many physical relations involve summing, or integrating, over extended distributions of mass, of charge, of current, of fields, we present a table of integrals (some of them worked out for you in detail so you can see how it goes).
Natural, or Counting Numbers
is the set of numbersNumber that is pretty much the first piece of mathematics most ordinary human beings (and possibly a few extraordinary dogshttp://www.sciencedaily.com/releases/2009/08/090810025241.htm) learns. They are used to count, initially to count things: concrete objects such as pennies or marbles. This is in some respects surprising, since pennies and marbles are never really identical, but the mind is very good at classifying things based on their similarities and glossing over their differences. In physics, however, one encounters particles that are identical as far as we can tell even with careful observations and measurements - electrons, for example, differ only in their position or orientation.
The natural numbers are usually defined along with a set of operations known as arithmeticArithmetic. The well-known operations of ordinary arithmetic are addition, subtraction, multiplication, and division. One rapidly sees that the set of natural/counting numbers is not closed with respect to all of them. That just means that if one subtracts two natural numbers, say 7 from 5, one does not necessarily get a natural number. More concretely, one cannot take seven cows away from a field containing five cows, one cannot remove seven pennies from a row of five pennies.
This helps us understand the term closure in mathematics. A set (of, say, numbers) is closed with respect to some binary operation (say, addition, or subtraction) if any two members of the set combined with the operation produce a member of the set. The natural numbers are closed with respect to addition (the sum of any two natural numbers is a natural number) and multiplication (the product of any two natural numbers is a natural number) but not, if you think about it, subtraction or division. More on this later.
Natural numbers greater than 1 in general can be factored into a representation in prime numbersPrime Number. For example: 45 = 2^0 3^2 5^1 7^0... or 56 = 2^3 3^0 5^0 7^1 11^0... This sort of factorization can sometimes be very useful, but not so much in introductory physics.
It is easy to see that there is no largest natural number. Suppose there was one, call it . Now add one to it, forming . We know that , contradicting our assertion that was the largest. This lack of a largest object, lack of a boundary, lack of termination in series, is of enormous importance in mathematics and physics. If there is no largest number, if there is no ``edge'' to space or time, then it in some sense they run on forever, without termination.
In spite of the fact that there is no actual largest natural number, we have learned that it is highly advantageous in many context to invent a pretend one. This pretend number doesn't actually exist as a number, but rather stands for a certain reasoning process.
In fact, there are a number of properties of numbers (and formulas, and integrals) that we can only understand or evaluate if we imagine a very large number used as a boundary or limit in some computation, and then let that number mentally increase without bound. Note well that this is a mental trick, no more, encoding the observation that there is no largest number and so we can increase a number parameter without bound. However, mathematicians and physicists use this mental trick all of the time - it becomes a way for our finite minds to encompass the idea of the infinite, of unboundedness. To facilitate this process, we invent a symbol for this unreachable limit to the counting process and give it a name.
We call this unboundedness infinityInfinity - the lack of a finite boundary - and give it the symbol in mathematics.
In many contexts we will treat
as a number in all of the
number systems mentioned below. We will talk blithely about ``infinite
numbers of digits'' in number representations, which means that the
digits specifying some number simply keep on going without bound.
However, it is very important to bear in mind that &infin#infty;
is not a number, it is a concept! Or at the very least, it is a
highly special number, one that doesn't satisfy the axioms or
participate in the usual operations of ordinary arithmetic. For
any finite number:
&infin#infty;+ &infin#infty;& = & &infin#infty;
&infin#infty;+ N & = & &infin#infty;
&infin#infty;- &infin#infty;& = & undefined
&infin#infty;* N & = & &infin#infty;
N / &infin#infty;& = & 0 (But 0*&infin#infty; is not equal to N !)
&infin#infty;/ &infin#infty;& = & undefined These are certainly ``odd'' rules compared to ordinary arithmetic! They all make sense, though, if you replace the symbol with ``something (infinitely) bigger than any specific number you can imagine''.
For a bit longer than a century now (since Cantor organized set theory and discussed the various ways sets could become infinite and set theory was subsequently axiomatized) there has been an axiom of infinity in mathematics postulating its formal existence as a ``number'' with these and other odd properties.
Our principal use for infinity will be as a limit in calculus and in series expansions. We will use it to describe both the very large (but never the largest) and reciprocally, the very small (but never quite zero). We will use infinity to name the process of taking a small quantity and making it ``infinitely small'' (but nonzero) - the idea of the infinitesimal, or the complementary operation of taking a large (finite) quantity (such as a limit in a finite sum) and making it ``infinitely large''. These operations do not always make arithmetical sense - consider the infinite sum of the natural numbers, for example - but when they do they are extremely valuable.
To achieve closure in addition, subtraction, and multiplication one introduces negative whole numbers and zero to construct the set of integers:
Today we take these things for granted, but in fact the idea of negative numbers in particular is quite recent. Although they were used earlier, mathematicians only accepted the idea that negative numbers were legitimate numbers by the latter 19th century! After all, if you are counting cows, how can you add negative cows to an already empty field? How can you remove seven pennies from a row containing only five? Numbers in Western society were thought of as being concrete properties of things, tools for bookkeeping, rather than strictly abstract entities about which one could axiomatically reason until well into the Enlightenment1.
In physics, integers or natural numbers are often represented by the letters , although of course in algebra one does have a range of choice in letters used, and some of these symbols are ``overloaded'' (used for more than one thing) in different formulas.
Integers can in general also be factored into primes, but problems begin to emerge when one does this. First, negative integers will always carry a factor of -1 times the prime factorization of its absolute value. But the introduction of a form of ``1'' into the factorization means that one has to deal with the fact that and . This possibility of permuting negative factors through all of the positive and negative halves of the integers has to be generally ignored because there is a complete symmetry between the positive and negative half-number line; one simply prepends a single -1 to the prime factorization to serve as a reminder of the sign. Second, 0 times anything is 0, so it (and the numbers ) are generally excluded from the factorization process.
Integer arithmetic is associative, commutative, is closed under addition, subtraction and multiplication, and has lots of nice properties you can learn about on e.g. Wikipedia. However, it is still not closed under division! If one divides two integers, one gets a number that is not, in general, an integer!
This forming of the ratio between two integer quantities leads to the next logical extension of our growing system of numbers: The rational numbers.
If one takes two integers and and divides by to form , the result will often not be an integer. For example, is not an integer (although is!), nor is , nor and so on. These numbers are all the ratios of two integers and are hence called rational numbersrational number.
Rational numbers when expressed in a base2 e.g. base 10 have an interesting property. Dividing one out produces a finite number of non-repeating digits, followed by a finite sequence of digits that repeats cyclically forever. For example: 13 = 0.3333... or 117 = 1.571428 571428 571428... (where a small space has been inserted to help you see the pattern).
Note that finite precision decimal numbers are precisely those that are terminated with an infinite string of the digit 0 , and hence are all rational. That is, if we keep numbers only to the hundredths place, e.g. 4.17, -17.01, 3.14, the assumption is that all the rest of the digits in the number are 0 - 3.14000..., which is rational.
It may not be the case that those digits really are zero. We will often be multiplying by to get an approximate answer to all of the precision we need in a problem. In any event, we generally cannot handle an infinite number of nonzero digits in some base, repeating or not, in our arithmetical operations, so truncated base two or base ten, rational numbers are the special class of numbers over which we do much of our arithmetic, whether it be done with paper and pencil, slide rule, calculator, or computer.
If all rational numbers have digit strings that eventually cyclically repeat, what about all numbers whose digit strings do not cyclically repeat? These numbers are not rational.
An irrational numberirrational number is one that cannot be written as a ratio of two integers e.g. . It is not immediately obvious that numbers like this exist at all. When rational numbers were discovered (or invented, as you prefer) by the Pythagoreans, they were thought to have nearly mystical properties - the Pythagoreans quite literally worshipped numbers and thought that everything in the Universe could be understood in terms of the ratios of natural numbers. Then Hippasus, one of their members, demonstrated that for an isosceles right triangle, if one assumes that the hypotenuse and arm are commensurable (one can be expressed as an integer ratio of the other) that the hypotenuse had to be even, but the legs had to be both even and odd, a contradiction. Consequently, it was certain that they could not be placed in a commensurable ratio - the lengths are related by an irrational number.
According to the (possibly apocryphal) story, Hippasus made this discovery on a long sea voyage accompanied by a group of fellow Pythagoreans, and they were so annoyed at his blasphemous discovery that their religious beliefs in the rationality of the Universe (so to speak) were false that they threw him overboard to drown! Folks took their mathematics quite seriously back then!
As we've seen, all digital representation of finite precision or digital representations where the digits eventually cycle correspond to rational numbers. Consequently its digits in a decimal representation of an irrational number never reach a point where they cyclically repeat or truncate (are terminated by an infinite sequence of 0 's).
Many numbers that are of great importance in physics, especially and are irrational, and we'll spend some time discussing both of them below. When working in coordinate systems, many of the trigonometric ratios for ``simple'' right triangles (such as an isoceles right triangle) involve numbers such as , which are also irrational - this was the basis for the earliest proofs of the existence of irrational numbers, and was arguably the first irrational number discovered.
Whenever we compute a number answer we must use rational numbers to do it, most generally a finite-precision decimal representation. For example, 3.14159 may look like , an irrational number, but it is really , a rational number that approximates to six significant figures.
Because we cannot precisely represent them in digital form, in physics (and mathematics and other disciplines where precision matters) we will often carry important irrationals along with us in computations as symbols and only evaluate them numerically at the end. It is important to do this because we work quite often with functions that yield a rational number or even an integer when an irrational number is used as an argument, e.g. . If we did finite-precision arithmetic prematurely (on computer or calculator) we might well end up with an approximation of -1, such as -0.999998 and could not be sure if it was supposed to be -1 or really was supposed to be a bit more.
There are lots of nifty truths regarding rational and irrational numbers. For example, in between any two rational numbers lie an infinite number of irrational numbers. This is a ``bigger infinity''infinityThere are (at least) two different kinds of infinity - countable and uncountable. Countable doesn't mean that one can count to infinity - it means one can create a one-to-one map between the (countably infinite) counting numbers and the countably infinite set in question. Uncountable means that one cannot make this mapping. The set of all real numbers in any finite interval form a continuum and is an example of an uncountably infinite set. than just the countably infinite number of integers or rational numbers, which actually has some important consequences in physics - it is one of the origins of the theory of deterministic chaos.
The union of the irrational and rational numbers forms the real number line.real line Real numbers are of great importance in physics. They are closed under the arithmetical operations of addition, subtraction, multiplication and division, where one must exclude only division by zero3. Real exponential functions such as or (where are all presumed to be real) will have real values, as will algebraic functions such as where is an integer.
However, as before we can discover arithmetical operations, such as the power operation (for example, the square root, for some real number ) that lead to problems with closure. For positive real arguments , is real, but probably irrational (irrational for most possible values of ). But what happens when we try to form the square root of negative real numbers? In fact, what happens when we try to form the square root of ?
This is a bit of a problem. All real numbers, squared or taken to any even integer power, are positive. There therefore is no real number that can be squared to make . All we can do is imagine such a number, , and then make our system of numbers bigger still to accomodate it. This process leads us to the imaginary unit such that , to all possible products and sums of this number and our already known real numbers and thereby to numbers with both real (no necessary factor of ) and imaginary (a necessary factor of ) parts. Such a number might be represented in terms of real numbers like:
Whew! A number that is now the sum of two very different kinds of number. That's complicated! Let's call these new beasts complex numbers.
At this point you should begin to have the feeling that this process of generating supersets of the numbers we already have figured out that close under additional operations or have some desired additional properties will never end. You would be right, and some of the extensions (division algebras that we will not cover here such as quaternionsQuaternions or more generally, geometric algebrasGeometric Algebra) are actually very useful in more advanced physics. However, we have a finite amount of time to review numbers here, and complex numbers are the most we will need in this course (or ``most'' undergraduate physics courses even at a somewhat more advanced level). They are important enough that we'll spend a whole section discussing them below; for the moment we'll just define them.
We start with the unit imaginary numberimaginary unit, . You might be familiar with the naive definition of as the square root of : i = +-1 This definition is common but slightly unfortunate. If we adopt it, we have to be careful using this definition in algebra or we will end up proving any of the many variants of the following: -1 = i ·i = -1·-1 = -1 ·-1 = 1 = 1
A better definition for
that it is just the algebraic number such that:
i^2 = -1
and to leave the square root bit out. Thus we have the following
i^0 & = & 1
i^1 & = & i
i^2 & = & -1
i^3 & = & (i^2)i = -1 ·i = -i
i^4 & = & (i^2)(i^2) = -1 ·-1 = 1
i^5 & = & (i^4)i = i
...& & where we can use these rules to do the following sort of simplification: +- &pi#pi;b = +i^2 &pi#pi;b = +i&pi#pi;b but where we never actually write .
We can make all the imaginary numbers by simply scaling with a real number. For example, is a purely imaginary number of magnitude . is a purely imaginary number of magnitude . All the purely imaginary numbers therefore form the imaginary line that is basically the real line, times . Note well that this line contains the real number zero - 0 is in fact the intersection of the imaginary line and the real line.
With this definition, we can define an arbitrary complex number z as the sum of an arbitrary real number plus an arbitrary imaginary number: z = x + iy where and are both real numbers. It can be shown that the roots of any polynomial function can always be written as complex numbers, making complex numbers of great importance in physics. However, their real power in physics comes from their relation to exponential functions and trigonometric functions.
Complex numbers (like real numbers) form a division algebradivision algebra - that is, they are closed under addition, subtraction, multiplication, and division. Division algebras permit the factorization of expressions, something that is obviously very important if you wish to algebraically solve for quantities.
Hmmmm, seems like we ought to look at this ``algebra'' thing. Just what is an algebra? How does algebra work?
Algebraalgebra is a reasoning process that is one of the fundamental cornerstones of mathematical reasoning. As far as we are concerned, it consists of two things:
Note well that it isn't always a matter of solving for some unknown but determined variable in terms of known variables, although this is certainly a useful thing to be able to do. Algebra is just as often used to derive relationships and hence gain conceptual insight into a system being studied, possibly expressed as a derived law. Algebra is thus in some sense the conceptual language of physics as well as the set of tools we use to solve problems within the context of that language. English (or other spoken/written human languages) is too imprecise, too multivalent, too illogical and inconsistent to serve as a good language for this purpose, but algebra (and related geometries) are just perfect.
The transformations of algebra applied to equalities (the most common case) can be summarized as follows (non-exhaustively). If one is given one or more equations involving a set of variables one can:
In the sections below, we'll give examples of each of these points and demonstrate some of the key algebraic methods used in physics problems.