Feynman's Tips on Physics

1-1Introduction to the review lectures

These three optional lectures are going to be dull: they go over the same material that we went over before, adding absolutely nothing. So I’m very surprised to see so many people here. Frankly, I had rather hoped there would be fewer of you, and that these lectures wouldn’t be necessary.

The purpose of relaxing at this time is to give you time to think about things, to piddle around with the things that you heard about. That’s by all odds the most effective way of learning the physics: it’s not a good idea to come in and listen to some review; it’s better to make up the review for yourself. So I’d advise you—if you’re not too far lost, completely befuddled and confused—that you forget about these lectures and piddle around by yourself, and try to find out what’s interesting without grinding down some particular track. You’ll learn infinitely better and easier and more completely by picking a problem for yourself that you find interesting to fiddle around with—some kind of a thing that you heard that you don’t understand, or you want to analyze further, or want to do some kind of a trick with—that’s the best way to learn something.

The lectures that we have been giving so far are a new course, and have been designed to answer a problem we presumed existed: nobody knows how to teach physics, or to educate people—that’s a fact, and if you don’t like the way it’s being done, that’s perfectly natural. It’s impossible to teach satisfactorily: for hundreds of years, even more, people have been trying to figure out how to teach, and nobody has ever figured it out. So if this new course is not satisfactory, that’s not unique.

At Caltech we are always changing the courses in the hope of improving them, and this year we changed the physics course again. One of the complaints in the past was that the students who are nearer the top find the whole subject of mechanics dull: they would find themselves grinding along, doing problems, studying reviews, and doing examinations, and there was no time to think about anything; there was no excitement in it; there was no description of its relation to modern physics, or anything like that. And so this set of lectures was designed to be better that way, to a certain extent, to help out those fellows, and to make the subject more interesting, if possible, by connecting it to the rest of the universe.

On the other hand, this approach has the disadvantage that it confuses many people, because they don’t know what it is they’re supposed to learn—or, rather, that there’s so much stuff that they can’t learn all of it, and they haven’t got enough intelligence to figure out what is interesting to them, and to pay attention only to that.

Therefore, I’m addressing myself to those people who have found the lectures very confusing, very annoying, and irritating, in the sense that they don’t know what to study, and they’re kind of lost. The other people, who don’t feel as lost, shouldn’t be here, so I now give you the opportunity to go out . . .¹

I see nobody has the nerve. Or I guess I’m a great failure, then, if I got everybody lost! (Maybe you’re just here for entertainment.)

1-2Caltech from the bottom

Now, I am therefore imagining that one of you has come into my office and said, “Feynman, I listened to all the lectures, and I took that midterm exam, and I’m trying to do the problems, and I can’t do anything, and I think I’m in the bottom of the class, and I don’t know what to do.”

What would I say to you?

The first thing I would point out is this: to come to Caltech is an advantage in certain ways, and in other ways a disadvantage. Some of the ways that it’s an advantage you probably once knew, but now forget, and they have to do with the fact that the school has an excellent reputation, and the reputation is well deserved. There are pretty good courses. (I don’t know about this particular physics course; of course I have my own opinion about it.) The people who have come out the other end of Caltech, when they go into industry, or go to do work in research, and so forth, always say that they got a very good education here, and when they compare themselves with people who have gone to other schools (although many other schools are also very good) they never find themselves behind and missing something; they always feel they went to the best school of them all. So that’s an advantage.

But there is also a certain disadvantage: because Caltech has such a good reputation, almost everybody who’s the first or second in his high school class applies here. There are lots of high schools, and all the very best men² apply. Now, we have tried to figure out a system of selection, with all kinds of tests, so that we get the best of the best. And so you guys have been very carefully picked out from all these schools to come here. But we’re still working on it, because we’ve found a very serious problem: no matter how carefully we select the men, no matter how patiently we make the analysis, when they get here something happens: it always turns out that approximately half of them are below average!

Of course you laugh at this because it’s self-evident to the rational mind, but not to the emotional mind—the emotional mind can’t laugh at this. When you’ve lived all the time as number one or number two (or even possibly number three) in high school science, and when you know that everybody who’s below average in the science courses where you came from is a complete idiot, and now you suddenly discover that you are below average—and half of you guys are—it’s a terrible blow, because you imagine that it means you’re as dumb as those guys used to be in high school, relatively. That’s the great disadvantage of Caltech: that this psychological blow is so difficult to take. Of course, I’m not a psychologist; I’m imagining all this. I don’t know how it would really be, of course!

The question is what to do if you find you’re below average. There are two possibilities. In the first place, you could find that it’s so difficult and annoying that you have to get out—that’s an emotional problem. You can apply your rational mind to that and point out to yourself what I just pointed out to you: that half of the guys in this place are going to be below average, even though they’re all tops, so it doesn’t mean anything. You see, if you can stick out that nonsense, that funny feeling, for four years, then you’ll go out into the world again, and you’ll discover that the world is just like it used to be—that when, for example, you get a job somewhere, you’ll find you’re Number One Man again, and you’ll get the great pleasure of being the expert they all come running to in this particular plant whenever they can’t figure out how to convert inches to centimeters! It’s true: the men who go out into industry, or go to a small school that doesn’t have an excellent reputation in physics, even if they’ve been in the bottom third, the bottom fifth, the bottom tenth of the class—if they don’t try to drive themselves (and I’ll explain that in a minute), then they’ll find themselves very much in demand, that what they learned here is very useful, and they’re back where they were before: happy, Number One.

On the other hand you can make a mistake: some people may drive themselves to a point where they insist they have to become Number One, and in spite of everything they want to go to graduate school and they want to become the best Ph.D. in the best school, even though they’re starting out at the bottom of the class here. Well, they are likely to be disappointed and to make themselves miserable for the rest of their lives being always at the bottom of a very first-rate group, because they picked that group. That’s a problem, and that’s up to you—it depends on your personality. (Remember, I’m talking to the guy who came into my office because he’s in the lowest tenth; I’m not talking to the other fellows who are happy because they happen to be in the upper tenth—that’s a minority anyway!)

So, if you can take this psychological blow—if you can say to yourself, “I’m in the lower third of the class, but a third of the guys are in the lower third of the class, because it’s got to be that way! I was the top guy in high school, and I’m still a smart son-of-a-gun. We need scientists in the country, and I’m gonna be a scientist, and when I get out of this school I’ll be all right, damn it! And I’ll be a good scientist!”—then it’ll be true: you will be a good scientist. The only thing is whether you can take the funny feelings during these four years, in spite of the rational arguments. If you find you can’t take the funny feelings, I suppose the best thing to do is to try to go somewhere else. It’s not a point of failure; it’s simply an emotional thing.

Even if you’re one of the last couple of guys in the class, it doesn’t mean you’re not any good. You just have to compare yourself to a reasonable group, instead of to this insane collection that we’ve got here at Caltech. Therefore, I am making this review purposely for the people who are lost, so that they have still a chance to stay here a little longer to find out whether or not they can take it, okay?

I make now one more point: that this is not a preparation for an examination, or anything like that. I don’t know anything about the examinations—I mean, I have nothing to do with making them up, and I don’t know what’s going to be on them, so there’s no guarantee whatsoever that what’s on the examination is only going to deal with the stuff reviewed in these lectures, or any nonsense of that kind.

1-3Mathematics for physics

So, this guy comes into my office and asks me to try to make everything straight that I taught him, and this is the best I can do. The problem is to try to explain the stuff that was being taught. So I start, now, with the review.

I would tell this guy, “The first thing you must learn is the mathematics. And that involves, first, calculus. And in calculus, differentiation.”

Now, mathematics is a beautiful subject, and has its ins and outs, too, but we’re trying to figure out what the minimum amount we have to learn for physics purposes are. So the attitude that’s taken here is a “disrespectful” one towards the mathematics, for sheer efficiency only; I’m not trying to undo mathematics.

What we have to do is to learn to differentiate like we know how much is 3 and 5, or how much is 5 times 7, because that kind of work is involved so often that it’s good not to be confounded by it. When you write something down, you should be able to immediately differentiate it without even thinking about it, and without making any mistakes. You’ll find you need to do this operation all the time—not only in physics, but in all the sciences. Therefore differentiation is like the arithmetic you had to learn before you could learn algebra.

Incidentally, the same goes for algebra: there’s a lot of algebra. We are assuming that you can do algebra in your sleep, upside down, without making a mistake. We know it isn’t true, so you should also practice algebra: write yourself a lot of expressions, practice them, and don’t make any errors.

Errors in algebra, differentiation, and integration are only nonsense; they’re things that just annoy the physics, and annoy your mind while you’re trying to analyze something. You should be able to do calculations as quickly as possible, and with a minimum of errors. That requires nothing but rote practice—that’s the only way to do it. It’s like making yourself a multiplication table, like you did in elementary school: they’d put a bunch of numbers on the board, and you’d go: “This times that, this times that,” and so on—Bing! Bing! Bing!

1-4Differentiation

In the same way you must learn differentiation. Make a card, and on the card write a number of expressions of the following general type: for example,

and so on. Write, say, a dozen of these expressions. Then, every once in a while, just take the card out of your pocket, put your finger on an expression, and read out the derivative.

In other words, you should be able to see right away:

See? So the first thing to do is to memorize how to do derivatives—cold. That’s a necessary practice.

Now, for differentiating more complicated expressions, the derivative of a sum is easy: it’s simply the sum of the derivatives of each separate summand. It isn’t necessary at this stage in our physics course to know how to differentiate expressions any more complicated than those above, or sums of them, so that in the spirit of this review, I shouldn’t tell you any more. But there is a formula for differentiating complicated expressions, which is usually not given in calculus class in the form that I’m going to give it to you, and it turns out to be very useful. You won’t learn it later, because nobody will ever tell it to you, but it’s a good thing to know how to do.

Suppose I want to differentiate the following:

Now, the question is how to do it with dispatch. Here’s how you do it with dispatch. (These are just rules; it’s the level to which I’ve reduced the mathematics, because we’re working with the guys who can barely hold on.) Watch!

You write the expression down again, and after each summand you put a bracket:

Next, you’re going to write something inside the brackets, such that when you’re all finished, you’ll have the derivative of the original expression. (That’s why you write the expression down again, in case you don’t want to lose it.)

Now, you look at each term and you draw a bar—a divider—and you put the term in the denominator: The first term is 1 + 2t²; that goes in the denominator. The power of the term goes in front (it’s the first power, 1), and the derivative of the term (by our practice game), 4t, goes in the numerator. That’s one term:

(What about the 6? Forget it! Any number in front doesn’t make any difference: if you wanted to, you could start out, “6 goes in the denominator; its power, 1, goes in front; and its derivative, 0, goes in the numerator.”)

Next term: t³ − t goes in the denominator; the power, +2, goes in front; the derivative, 3t² − 1, goes in the numerator. The next term, t + 5t², goes in the denominator; the power, −1/2 (the inverse square root is a negative half power), goes in front; the derivative, 1 + 10t, goes in the numerator. The next term, 4t, goes in the denominator; its power, −3/2, goes in front; its derivative, 4, goes in the numerator. Close the bracket. That’s one summand:

Next summand, first term: the power is +1/2. The object whose power we’re taking is 1 + 2t; the derivative is 2. The power of the next term, , is −1. (You see, it’s a reciprocal.) The term goes in the denominator, and its derivative (this is the only hard one, relatively) has two pieces, because it’s a sum: . Close the bracket:

That’s the derivative of the original expression. So, you see, that by memorizing this technique, you can differentiate anything—except sines, cosines, logs, and so on, but you can learn the rules for those easily; they’re very simple. And then you can use this technique even when the terms include tangents and everything else.

I noticed when I wrote it down you were worried that it was such a complicated expression, but I think you can appreciate now that this is a really powerful method of differentiation because it gives the answer—boom—without any delay, no matter how complicated.

The idea here is that the derivative of a function f = k · u^a · v^b · w^c . . . with respect to t is

(where k and a, b, c . . . are constants).

However, in this physics course, I doubt any of the problems will be that complicated, so we probably won’t have any opportunity to use this. Anyway, that’s the way I differentiate, and I’m pretty good at it now, so there we are.

1-5Integration

Now, the opposite process is integration. You should equally well learn to integrate as rapidly as possible. Integration is not as easy as differentiation, but you should be able to integrate simple expressions in your head. It isn’t necessary to be able to integrate every expression; for example, (t + 7t²)^1/3 is not possible to integrate in an easy fashion, but the others below are. So, when you choose expressions to practice integration, be careful that they can be done easily:

I have nothing more to tell you about calculus. The rest is up to you: you have to practice differentiation and integration—and, of course, the algebra required to reduce horrors like Eq. (1.7). Practicing algebra and calculus in this dull way—that’s the first thing.

1-6Vectors

The other branch of the mathematics that we’re involved in as a pure mathematical subject is vectors. You first have to know what vectors are, and if you haven’t got a feel for it, I don’t know what to do: we’d have to talk back and forth a while for me to appreciate your difficulty—otherwise I couldn’t explain. A vector is like a push that has a certain direction, or a speed that has a certain direction, or a movement that has a certain direction—and it’s represented on a piece of paper by an arrow in the direction of the thing. For instance, we represent a force on something by an arrow that is pointing in the direction of the force, and the length of the arrow is a measure of the magnitude of the force in some arbitrary scale—a scale, however, which must be maintained for all the forces in the problem. If you make another force twice as strong, you represent that by an arrow twice as long. (See Fig. 1-1.)

Now, there are operations that can be done with these vectors. That is, if there are two forces acting at the same time on an object—say, two people are pushing on a thing—then the two forces can be represented by two arrows F and F′. When we draw a diagram of something like this, it is often convenient to place the tails of the arrows where the forces are applied, even though in general there’s no meaning to the location of vectors. (See Fig. 1-2.)

FIGURE 1-1Two vectors, represented by arrows.

FIGURE 1-2Representation of two forces applied at the same point.

If we want to know the net resultant force, or total force, that corresponds to adding the vectors, and we can draw this by moving the tail of one onto the head of the other. (They’re still the same vectors after you move them because they have the same direction and the same length.) Then F + F′ is the vector drawn from the tail of F to the head of F′ (or from the tail of F′ to the head of F), as shown in Figure 1-3. This way of adding vectors is sometimes called the “parallelogram method.”

On the other hand, suppose there are two forces acting on an object, but we only know one of them is F′; the other one, which we don’t know, we’ll call X. Then, if the total force on the object is known to be F, we have F′ + X = F. And so, X = F − F′. Thus to find X you have to take the difference of two vectors, and you can do that in either of two ways: you can take −F′, which is a vector in the opposite direction as F′, and add it to F. (See Fig. 1-4.)

FIGURE 1-3Vector addition by the “parallelogram method.”

Otherwise, F − F′ is simply the vector drawn from the head of F′ to the head of F.

Now, the disadvantage of the second method is that you may have a tendency to draw the arrow as shown in Figure 1-5; although the direction and length of the difference is right, the application of the force is not located at the tail of the arrow—so watch out. In case you’re nervous about it, or there’s any confusion, use the first method. (See Fig. 1-6.)

We can also project vectors in certain directions. For example, if we would like to know what the force is in the ‘x’ direction (called the component of the force in that direction) it’s easy: we just project F down with a right angle onto the x axis, and that gives the component of the force in that direction, which we call F_x. Mathematically, F_x is the magnitude of F (which I’ll write |F|) times the cosine of the angle that F makes with the x axis; this comes from the properties of the right triangle. (See Fig. 1-7.)

FIGURE 1-4Vector subtraction, first method.

FIGURE 1-5Vector subtraction, second method.

FIGURE 1-6Subtraction of two forces applied at the same point.

Now, if A and B are added to make C, then the projections that are brought down to form a right angle in a given direction ‘x’, evidently add. So the components of the vector sum are the sum of the vector components, and that’s true of components in any direction. (See Fig. 1-8.)

FIGURE 1-7The component of vector F in direction x.

FIGURE 1-8A component of a vector sum equals the sum of the corresponding vector components.

Particularly convenient is the description of vectors in terms of their components on perpendicular axes, x and y (and z—there’s three dimensions in the world; I keep forgetting that, because I’m always drawing on a blackboard!). If we have a vector F that is in the x-y plane, and we know its component in the x direction, that doesn’t completely define F, because there are many vectors in the x-y plane that have the same component in the x direction. But if we also know F’s component in the y direction, then F is completely specified. (See Fig. 1-9.)

The components of F along the x, y, and z axes can be written as F_x, F_y, and F_z; summing vectors is equivalent to summing their components, so if the components of another vector F′ are , and , then F + F′ has the components , and

FIGURE 1-9A vector in the x-y plane is completely specified by two components.

That’s the easy part; now it gets a bit more difficult. There’s a way of multiplying two vectors to produce a scalar