The Theoretical Minimum: What You Need to Know to Start Doing Physics

by Leonard Susskind

We define the functions sine (sin), cosine (cos), and tangent (tan), as ratios of the various sides according to the following relationships:

There are a couple of useful things to know about the trigonometric functions. The first is that we can draw a triangle within a circle, with the center of the circle located at the origin of a Cartesian coordinate system, as in Figure 11

Here the line connecting the center of the circle to any point along its circumference forms the hypotenuse of a right triangle, and the horizontal and vertical components of the point are the base and altitude of that triangle.

Suppose a certain angle θ is the sum or difference of two other angles using the greek letters alpha, α, and beta, β, we can write this angle, θ, as α ± β.

This equation is the Pythagorean theorem in disguise. If we choose the radius of the circle in Figure 11 to be 1, then the sides a and b are the sine and cosine of θ, and the hypotenuse is 1. Equation (1) is the familiar relation among the three sides of a right triangle: a2 + b2 = c2.

A vector can be thought of as an object that has both a length (or magnitude) and a direction in space.

If an object is moved from some particular starting location, it is not enough to know how far it is moved in order to know where it winds up. One also has to know the direction of the displacement.

Symbolically vectors are represented by placing arrows over them. Thus the symbol for displacement is . The magnitude, or length, of a vector is expressed in absolute-value notation. Thus the length of is denoted

First of all, you can multiply them by ordinary real numbers. When dealing with vectors you will often see such real numbers given the special name scalar. Multiplying by a positive number just multiplies the length of the vector by that number.

you can also multiply by a negative number, which reverses the direction of the vector.

Vectors may be added. To add and , place them as shown in Figure 13 to form a quadrilateral (this way the directions of the vectors are preserved). The sum of the vectors is the length and angle of the diagonal.

define three unit vectors that lie along these axes and have unit length. The unit vectors along the coordinate axes are called basis vectors.

The basis vectors are useful because any vector can be written in terms of them in the following way:

The quantities Vx, Vy, and Vz are numerical coefficients that are needed to add up the basis vectors to give . They are also called the components of .

We can say that Eq. (2) is a linear combination of basis vectors. This is a fancy way of saying that we add the basis vectors along with any relevant factors.

We can also write a vector as a list of its components—in this case (Vx, Vy, Vz). The magnitude of a vector can be given in terms of its components by applyin...

Can we multiply vectors? Yes, and there is more than one way. One type of product—the cross product—gives another vector. For now, we will not worry about the cross product and only consider the other method, the dot product. The dot product of two vectors is an ordinary number, a scalar.

In ordinary language, the dot product is the product of the magnitudes of the two vectors and the cosine of the angle between them.

Most of classical mechanics deals with things that change smoothly—continuously is the mathematical term—as time changes continuously.

To cope, mathematically, with continuous changes, we use the mathematics of calculus. Calculus is about limits,

Suppose we have a sequence of numbers, l1, l2, l3, . . ., that get closer and closer to some value L. Here is an example: 0.9, 0.99, 0.999, 0.9999, . . . . The limit of this sequence is 1. None of the entries is equal to 1, but they get closer and closer to that value.

Suppose we have a function, f(t), and we want to describe how it varies as t gets closer and closer to some value, say a. If f(t) gets arbitrarily close to L as t tends to a, then we say that the limit of f(t) as t approaches a is the number L.

Differential calculus deals with the rate of change of such functions. The idea is to start with f(t) at some instant, and then to change the time by a little bit and see how much f(t) changes.

To define the rate of change precisely at time t, we must let Δt shrink to zero. Of course, when we do that Δf also shrinks to zero, but if we divide Δf by Δt, the ratio will tend to a limit.

Terms of higher order in Δt can be ignored when you calculate derivatives.

How long does the expression go on? If n is an integer, it eventually terminates after n+1 terms. But the binomial theorem is more general than that; in fact, n can be any real or complex number.

One important point is that this relation holds even if n is not an integer; n can be any real or complex number.

If n = 0, then f(t) is just the number 1. The derivative is zero—this is the case for any function that doesn’t change. If n = 1, then f(t) = t and the derivative is 1—this is always true when you take the derivative of something with respect to itself.

The meaning of et is pretty clear if t is an integer. For example, e3 = e × e × e. Its meaning for non-integers is not obvious. Basically, et is defined by the property that its derivative is equal to itself. So the third formula is really a definition.

There are a few useful rules to remember about derivatives. You can prove them all if you want a challenging exercise. The first is the fact that the derivative of a constant is always 0.

Suppose we have two functions, f(t) and g(t). Their sum is also a function and its derivative is given by This is called the sum rule.

Their product of two functions is another function, and its derivative is Not surprisingly, this is called the product rule.

Next, suppose that g(t) is a function of t, and f(g) is a function of g. That makes f an implicit function of t. If you want to know what f is for some t, you first compute g(t). Then, knowing g, you compute f(g). It’s easy to ca...

The concept of a point particle is an idealization. No object is so small that it is a point—not even an electron.

in many situations we can ignore the extended structure of objects and treat them as points. For example, the planet Earth is obviously not a point, but in calculating its orbit around the Sun, we can ignore the size of Earth to a high degree of accuracy.

Next to its position, the most important thing about a particle is its velocity. Velocity is also a vector. To define it we need some calculus.

Placing a dot over a quantity is standard shorthand for taking the time derivative. This convention can be used to denote the time derivative of anything, not just the position of a particle. For example, if T stands for the temperature of a tub of hot water, then Ṫ will represent the rate of change of the temperature with time.

The velocity vector has a magnitude , this represents how fast the particle is moving, without regard to the direction. The magnitude is called speed.

Acceleration is the quantity that tells you how the velocity is changing. If an object is moving with a constant velocity vector, it experiences no acceleration.

A constant velocity vector implies not only a constant speed but also a constant direction. You feel acceleration only when your velocity vector c...

Next let’s consider an oscillating particle that moves back and forth along the x axis. Because there is no motion in the other two directions, we will ignore them.

Whenever the position x is at its maximum or minimum, the velocity is zero. The opposite is also true: When the position is at x = 0, then velocity is either a maximum or a minimum.

That minus sign says that whenever x is positive (negative), the acceleration is negative (positive). In other words, wherever the particle is, it is being accelerated back toward the origin.

This shows an interesting property of circular motion that Newton used in analyzing the motion of the moon: The acceleration of a circular orbit is parallel to the position vector, but it is oppositely directed. In other words, the acceleration vector points radially inward toward the origin.

Differential calculus has to do with rates of change. Integral calculus has to do with sums of many tiny incremental quantities. It’s not immediately obvious that these have anything to do with each other, but they do.

The central problem of integral calculus is to calculate the area under the curve defined by f(t). To make the problem well defined, we consider the function between two values that we call limits of integration, t = a and t = b.

Of course this involves an approximation, but it becomes accurate if we let the width of the rectangles tend to zero.

the uppercase greek letter sigma, Σ, indicates a sum of successive values defined by i.

The function F(T) is called the indefinite integral of f(t). It is indefinite because instead of integrating from a fixed value to a fixed value, we integrate to a variable.

The fundamental theorem of calculus is one of the simplest and most beautiful results in mathematics. It asserts a deep connection between integrals and derivatives.