Feynman Lectures Simplified 4A: Math for Physicists

Chapter 1

Review of Basic Math

For those who have previously studied geometry, trigonometry, and basic algebra, the first few chapters of this book provide a quick review of those topics and definitions of key terms.

§1.1 Primary Symbols & Functions

Equality and Inequality Signs:

    A=B, means A equals B.

    A

    A>B, means A is greater than B

    A<

    A>>B, means A is much greater than B.

    A<=B, means A is less than or equal to B.

    A>=B, means A is greater than or equal to B.

~ is the Proportionality symbol. Two variables X and Y are proportional to one another, written X ~ Y, if the ratio (X / Y) is a constant.

√ is the Square Root. If y² = x, then y = √(x). Note that +y and –y are equally valid square roots of y².

|x| denotes the Absolute Value of x, it is the unsigned magnitude of x. See Section §4.1 and §11.1 for vectors.

Σ is the Summation sign denoting the sum of a series of quantities; see Section §6.2.

∫  is the Integration symbol denoting a continuous summation; see Section §13.1.

df/dx symbolizes a single entity, the Derivative. See Section §12.3.

§1.2 Geometry & Trigonometry

Let’s begin by reviewing some basic shapes of Euclidean geometry and their key properties. Figure 1-1 shows five two-dimensional shapes.

In the upper left is a triangle, a figure bounded by three straight line segments; its three internal angles sum to π radians (180 degrees), and its area equals hw/2, where h is its height, and w is its width; h and w are mutually perpendicular.

In the upper right is a square, a figure bounded by four line segments of equal length, with four internal angles that are each 90 degrees (π/2 radians); its area equals h² (h=w).

Figure 1-1 Two-Dimensional Shapes

In the middle is a rectangle, a figure bounded by four line segments with opposite sides of equal length, and four internal angles that are each 90 degrees; its area equals hw. (All squares are rectangles, but not all rectangles are squares.)

In the lower left is a circle of radius R and diameter D=2R, which is the locus (collection) of all points that are a distance R from the circle’s center. The circle’s circumference (length of its perimeter) equals πD, and its area equals πR². The length of the bolded arc that subtends angle θ equals θR, when θ is measured in radians. This makes sense: for θ=2π, the arc becomes the circle’s circumference whose length is 2πR. (This is why God invented radians.)

The area enclosed by a circle is called a disk.

Lastly, in the lower right is a parallelogram, a figure bounded by four line segments with opposite sides of equal length, and opposite angles equal; its area equals hw, the product of its height and width. (All rectangles are parallelograms, but not all parallelograms are rectangles.)

Moving to three dimensions, Figure 1-2 shows three common shapes.

In the upper right is a cuboid bounded by six rectangles, with all internal angles being equal. Its height h, width w, and length L may be different. Its enclosed volume equals hwL, and its surface area equals 2(hw+hL+wL). In the upper left is a cube, a cuboid in which h=w=L; its enclosed volume equals h³ and its surface area equals 6h², where h is the length of any side. (All cubes are cuboids, but not all cuboids are cubes.)

Figure 1-2 Three-dimensional Shapes

Lastly, in the lower image, is a sphere, the locus of all points that are a distance R from the sphere’s center. The sphere’s area equals 4πR², and its enclosed volume equals 4πR³/3. Proper mathematical terminology defines a sphere as the 2-D surface that encloses a 3-D volume called a ball.

We wish to examine some quantitative relationships established by trigonometry. But first, we must discuss functions.

§1.3 Functions & Fields

In mathematics, functions define relationships among variables. Since physics is all about relationships, functions are the bread and butter of mathematical physics.

Variables are quantities whose values change; they can change with location, change over time, or change for some other reason. Temperature is a variable that changes with both location and time. We can describe how temperature T varies with location x and time t by using the function f:

T = f(x,t)

Here, x and t are called independent variables, and T is called a dependent variable. Functions can have one or more independent variables, but they must have exactly one dependent variable. In this case, T is a function of both x and t.

As the terms suggest, we are free to choose the values of x and t, and those values uniquely determine the value of T. Some prefer to think of functions as being black boxes: when x and t are input into f, f outputs T. A more elegant mathematical description is: f maps (x,t) to T.

The essential characteristic of functions is that for each combination of independent variables there is one and only one value of the dependent variable.

In general, there may be more than one combination of independent variables that produce the same value of the dependent variable. For example, the temperature in Fairbanks, Alaska in mid-August might be the same as the temperature in Miami, Florida in mid-February. We can describe this mathematically by saying: there is a one-to-one mapping from (x,t) to T, but there is not a one-to-one mapping from T to (x,t).

Physicists often use the terms scalar field and vector field when describing functions whose independent variables are the dimensions of space, or space and time. A scalar field is a function whose value is always a simple number. A vector field is a function whose value is always a vector. In Earth’s atmosphere, at each location and moment in time, the temperature is a single number, a scalar, but the wind has a certain velocity, which is a vector with magnitude and direction.

§1.4 Graphing Functions

Graphs are visual representations that can be extremely helpful in understanding the key properties of functions. Graphs typically plot a function’s dependent variable vertically, and the function’s independent variable horizontally.

We will discuss sine functions shortly and exponential functions in Chapter 7, but for now suffice it to say that both are very important functions in physics. Here, we will discuss graphs of these two functions.

The upper graph in Figure 1-3 plots the value of Y that corresponds to each value of X, as defined by the exponential function:

Y = A + B eX

Here, A and B are constants.

Figure 1-3 Exponential & Sine Functions

The lower graph plots the value of Y that corresponds to each value of X, as defined by the sine function:

Y = A sin(X)

In the lower graph, the 5 black dots along the dotted horizontal line indicate 5 values of X for which sin(X) has the same value of Y. Like the prior example of the temperature in Fairbanks and Miami, Y=Asin(X) provides a one-to-one mapping from X to Y, but not a one-to-one mapping from Y to X.

Conversely, in the upper graph, there is only one black dot along the dotted line. In fact, for any Y value there is only one value of X for which Y=A+BeX. This means exponentials provide one-to-one mappings from X to Y and from Y to X. Any function f with this special property has an inverse function g, such that:

if y = f(x)

then g(y) = x

and g( f(x) ) = x

Again, the key property of such functions is that the mapping and the inverse mapping are both one-to-one.

§1.5 Trig Functions

Trigonometry quantifies the geometric relationships among angles and distances, and is most often employed in analyzing triangles. Let’s see how trig functions are used.

Figure 1-4 shows a triangle whose longest side has length r, whose vertical side has length y, and whose horizontal side has length x. Because the vertical and horizontal sides are orthogonal (perpendicular to one another), this is a right triangle and the longest side is the hypotenuse.

Figure 1-4 Angles & Sides of a Right Triangle

The angle β is a right angle, equal to 90 degrees (π/2 radians). Angles θ and ø can have any values that sum to 90 degrees.

The three primary trig functions are listed below with their English names, mathematical notations, and defining equations.

sine: sin(θ) = y / r

cosine: cos(θ) = x / r

tangent: tan(θ) = y / x

As we learned above, the sine function does not have a well-defined inverse function throughout the entire range of all possible angles. Indeed, this applies to all trig functions, because all are periodic, meaning that they all repeat exactly at regular intervals. More precisely, for any integer n:

sin(2nπ+θ) = sin(2nπ+π–θ) = sin(θ)

cos(2nπ+θ) = cos(2nπ–θ) = cos(θ)

tan(nπ+θ) = tan(θ)

Well-defined inverse functions do exist if we restrict the range of θ. The conventional allowed ranges, English names, mathematical notations, and defining equations of the inverse trig functions are:

–π/2<θ≤+π/2: arc sine: arcsin(y/r) = θ

 +0 ≤ θ < +π: arc cosine: arccos(x/r) = θ

–π/2<θ≤+π/2: arc tangent: arctan(y/x) = θ

These inverse functions are sometimes written:

arcsine: sin–1(y/r) = θ

arccosine: cos–1(x/r) = θ

arctangent: tan–1(y/x) = θ

However, this notation can be confusing: is sin–1 the arcsin or the reciprocal 1/sin? Context often resolves this ambiguity: the argument of arcsin is a ratio of lengths, while the argument of sin is an angle. But since both arguments are dimensionless numbers, it may be better to avoid this ambiguity entirely. I will use sin–1 only to reduce clutter in very messy equations, and then only (I hope) to represent 1/sin.

The following reciprocal functions are less commonly used:

cotangent: cot(θ) = 1/tan(θ) = x / y

secant: sec(θ) = 1/cos(θ) = r / x

cosecant: csc(θ) = 1/sin(θ) = r / y

In all equations, angles must be in units of radians, with 2π radians equal to 360 degrees.

For the triangle in Figure 1-4, the Pythagorean theorem states:

r² = x² + y²

With the above definitions, we replace x with rcos(θ) and y with rsin(θ), yielding the very important equation:

1 = cos²θ + sin²θ

§1.6 Laws of Sines & Cosines

Figure 1-5 shows a triangle whose sides have lengths A, B, and C, and whose opposite angles are a, b, and c, respectively.

Figure 1-5 Triangle with Sides A, B, C

For any triangle:

a + b + c = π radians = 180 degrees

Law of Sines: A/sin(a) = B/sin(b) = C/sin(c)

Law of Cosines: A² = B² +C² –2BC cos(a)

Chapter 2

Coordinate Systems

Coordinates systems are human conventions that facilitate quantifying angles and distances. Nature has no inherent coordinate system. Hence we are free to choose whatever coordinates seem most convenient. Often the best choice in any particular situation is one that matches a natural symmetry.

§2.1 Coordinates in 1-D & 2-D

The simplest coordinate system has only one dimension. For example, an object falling straight down in Earth’s gravity can be described with only one dimension: let’s call it  height h. Figure 2-1 shows the h-axis pointing straight upward, with tic-marks indicating various values of h. A black ball is shown at h=3.

Figure 2-1 Ball at h=3 on h-axis

With this coordinate system, we can follow the ball’s motion as h changes over time.

Now imagine a basketball player throwing a ball, hoping it goes through the hoop. Here, the motion is two-dimensional. As shown in Figure 2-2, Y is the vertical axis and X is the horizontal axis. This is called a rectilinear coordinate system because the axes are orthogonal to one another.

Figure 2-2 (X,Y) Coordinates in 2-D

Any point P in this 2-D space is uniquely specified by how far up P is in the Y-direction, and how far to the right P is in the X-direction. We can choose the origin of our coordinate system, the point with coordinates (0,0), to be anywhere we wish. Here, we choose the origin to be the point at which the ball was released. At any particular instant in time, the ball has height Y, horizontal distance X, and coordinates (X,Y).

§2.2 Polar Coordinates

A quite different situation is the motion of a lone planet around a star. This occurs in three dimensions of course, but due to spherical symmetry, the planet orbits entirely within a single plane. This allows us to analyze its motion in two dimensions. The most convenient approach employs polar coordinates, as illustrated in Figure 2-3.

Figure 2-3 Polar Coordinates (r,θ)

Here, r is the planet’s distance from the origin, the length of the radial line from the origin (the center of the star) to the planet, and θ is the angle between the radial line and a chosen reference direction. Typically, θ ranges from 0 to 2π radians, although in verbal descriptions we often say 0 to 360 degrees. The most common reference direction is the horizontal axis, as shown in Figure 2-3, and the most common convention is to measure the angle in the counterclockwise direction. Those choices are arbitrary, as is the choice of the origin.

If we know a planet’s location (r,θ) in 2-D polar coordinates, we can calculate its location (x,y) in 2-D rectilinear coordinates, or we can do the reverse. The conversion equations are:

x = r cosθ

y = r sinθ

r = √(x² + y²)

tanθ = y / x

§2.3 Becoming Three-Dimensional

Computer screens and book pages are two-dimensional, with points defined by their horizontal and vertical positions. A three-dimensional coordinate system adds depth — it has three independent directions. In physics, we most commonly employ a 3-D rectilinear system with three mutually orthogonal coordinate directions labeled x, y, and z, as illustrated in Figure 2-4. Here x is the horizontal right-left direction; y the vertical up-down direction, and z is the in-out direction perpendicular to the page. This is often called a Cartesian coordinate system.

Figure 2-4 Rectilinear 3-D Coordinates

The coordinates of the black ball are its distance from three plane surfaces: x is the distance to the right of the x=0 plane; y is the distance above the y=0 plane; and z is the distance out from the z=0 plane (the page). In this case, counting tic-marks along each axis, we have: x=5, y=4, and z=3.

Figure 2-5 shows a 3-D cylindrical coordinate system, which might be useful in analyzing the tip of a corkscrew as it cuts into a cork. The three coordinates are: height z; radial distance from centerline r; and azimuthal angle ø.

Figure 2-5 3-D Cylindrical Coordinates

The tip of the corkscrew has a constant r, an increasing ø, and a decreasing