Electrodynamics Tutorial Series/Math Background

Topics Covered

• Everything in Introduction to Electrodynamics/Chapter 1
• Vector Algebra
• Multivariable Calculus
• Nabla

Script

Before we can even begin talking about electrodynamics, we need to come to a mutual understanding of notation and meaning. That means math. We need to know what vectors are, how they add, subtract, and multiple. We also need to know how to do calculus with multi-variable functions, including vectors of functions.

Caution

A word of caution: I'm going to go really fast. I'm going to use a notation that may be slightly different than what you're used to. And I'm not going to repeat myself. (I think i just did!) Go back, review the videos, and go look for external resources if you get lost. Don't forget you can ask questions in the comments below.

Vectors

You know about numbers, 1, 5, -7, pi, etc... We call these 'scalars' because they are sort of like 1-dimensional vectors.

What's a vector? It's hard to nail down exactly. A vector has a length, which we sometimes call the magnitude. It also has a direction. Vectors can exist in 1 dimension, 2-dimensions, 3-dimensions, any dimension, really.

Vectors can be written on the board, or I can use my fingers to show you in 3D space where the tip and the tail is. They can also be described in words.

An important point is that vectors can be moved around. As long as their length and direction don't change, they don't change. This is no different with numbers: a 3 over here is the same as a 3 over there.

Vectors can also be written in component form. This requires that you have some pre-defined "unit vectors" lying around to use. Once you know how to add and multiply vectors, component form becomes obvious.

Vectors can add together. You do this by taking the tip of one and putting it on the tail of the other. The vector from the tail to the tip is the sum of the two vectors.

You can make vectors longer or shorter by multiplying them with a scalar. Multiplying by a negative number flips the direction.

So subtracting is just adding the negative. So take the tip of the subtracted vector and line it up with the tip of the original vector. The result is the vector from the first's tail to the second's tail.

Now we can represent vectors as the sum of the unit vector's you've chosen to use. Here's a few examples. In 3D, we use i-hat, j-hat, and k-hat as unit vectors.

This funny carat on the top of a vector tells us that we're talking about the unit vector of the same direction but length 1. You can find the unit vector of any vector except the zero vector by dividing by its length.

What else can we do with vectors? We can multiply them. In 3D, there are two ways to multiple two vectors together.

One way is the dot product. This is represented with a dot. The result is the square of the length of the "shadow" of one vector on the other---a scalar.

If you take the dot product of a vector with itself, you get the square of its length. Take the square root, and voila! The length.

A dot product of two vectors can be found by this simple formula: a b cos(theta) That is, take the length of the one vector, times it by the length of the other vector, and then take the cosine of the angle between them. Simple, huh?

Two vectors that are perpendicular to one another have a dot product of 0.

If you have both vectors in the same units, you can take their components, the factors of each unit vector, multiply them across-wise, and you get the same answer. It's not trivial to show this, but it's something you can probably prove for yourself. Remember that with orthogonal unit vectors, any unit vector dotted with any other unit vector will give you 0, since they are perpendicular.

Cross Product

The dot product tells us how much two vectors "agree". The cross product tell us how much they do not.

We use this x symbol you learned about in 1st grade. The result is a vector that is perpendicular to both vectors. It's magnitude increases the longer the vectors are and the more perpendicular the vectors are.

Find the cross product by taking your RIGHT hand (not your left), pointing your fingers along the first vector, then curling them in the direction of the second. Stick your thumb straight out, and that's the direction of the cross product.

The length of the vector is <math>a b \sin(\theta)</math>, where a and b are the respective lengths, and theta is the angle between them. Note that if the two vectors are parallel or anti-parallel, then sin is 0, and the cross product is 0.

In 3D, you can also find the cross product vector by using a bit of linear algebra.

<math>\mathbf{a} \times \mathbf{b} = \begin{vmatrix} \hat \mathbf{i} & \hat \mathbf{j} & \hat \mathbf{k} \\ a_x & a_y & a_z \\ b_x & b_y & b_z \\ \end{vmatrix} </math>

(Script: To find the determinant, start in the UL corner. Multiply this by the determinant of the box to the BR of that corner. Then go to the next row on the top. Then multiply by the BR determinant. Fill the empty cells by wrapping around, so you're finding the determinant of az, ax, bz, and bx. Finally, do the last one.

If you have some free time, you can also memorize this not-so-simple formula.

<math>\mathbf{a} \times \mathbf{b} = (a_y b_z - b_y a_z) \hat \mathbf{i} + (b_x a_z - a_x b_z) \hat \mathbf{j} + (a_x b_y - b_x a_y) \hat \mathbf{k}</math>

The cross product is distributive:

<math>\mathbf{a} \times ( \mathbf{b} + \mathbf{c}) = (\mathbf{a} \times \mathbf{b}) + \mathbf{a} \times \mathbf{c})</math>

But it is anti-commutative, meaning, it flips sign if you commute.

<math>\mathbf{a} \times \mathbf{b} = - \mathbf{b} \times \mathbf{a}</math>

It is not associative. So don't abandon the parentheses.

Identities

There are some cute identities and properties of the cross and dot products.

A vector dot itself is simply its length squared. So we just write it this way:

<math>\mathbf{A} \dot \mathbf{A} = A^2</math>

The length of a vector is simply the square root of the above:

<math>| \mathbf{A} | = A = \sqrt{\mathbf{A} \dot \mathbf{A}}</math>

A unit vector dot itself is simply 1.

<math>\hat \mathbf{A} \dot \hat \mathbf{A} = 1</math>

Any two perpendicular vectors dot each other is zero:

<math>\mathbf{A} \dot \mathbf{B} = 0 \text{ when perpendicular}</math>

In Cartesian coordinates, the dot products of any two basis vectors is either 1 (if they are the same) or 0 (if they are not.)

The cross product of any vector with a parallel vector is always zero:

<math>\mathbf{A} \times \mathbf{A} = 0</math>

Multivariable Calculus

Geometric Interpretation

Although we'll be doing a lot of math, most of what we will do isn't too far divorced from reality. Almost every step of the calculations we perform uncover some interesting ideas that can be understood with geometry.

Rather than memorize just the formulas, you'll be much better off if you get a 3D geometrical feel of the formulas. I'll present the geometric interpretations first, and then the formulas, so you can "see" what the symbols really mean.

Simple Derivatives

You can take any function of one variable and find the slope between two points. This simple formula describes that.

<math>\text{slope between }a\text{ and }b = { f(a) - f(b) \over a - b} = { f(x + \Delta x) - f(x) \over \Delta x}</math>

This looks like calculus. But look at what it means geometrically: You take two values from the x axis, and draw lines up to f(x). Subtract the lengths of the lines, and divide by the distance along the x axis. Simple.

What about the slope, or tangent, AT a point? Well, we can't divide by zero, but you can see for yourself what happens as you approach it. Let's suppose we pick a very, very, VERY tiny delta x, dx. Then f(x) will change by a small amount, df. Then the slope AT that point is simply df/dx. Or, in other words,

<math>df = \left ( {df \over dx} \right ) dx</math>.

This quantity, df over dx, is the derivative. Simple, right?

You should already have a number of derivatives memorized. Let's review some important ones.

Partial Derivatives

What if you have a function that takes multiple parameters, and you want to find the "slope" of it? Well, this doesn't make a whole lot of sense at first glance. I mean, if you held all but one of the parameters constant, you could pretty easily find the slope along that parameter. We call this the partial derivative.

The Total Derivative is something else entirely. It says if you change ALL the parameters by a small amount, how much the function will change altogether.

3 Geometrical Derivatives

Just like there are 3 ways to multiple vectors, there are 3 ways to take a derivative of things related to vectors.

Gradient

First, suppose you have a function that varies over the 3 dimensions x, y, and z. A good example would be the temperature in a room. Not every point inside the room has the same temperature at any given moment.

The gradient tells us how the temperature is varying at each point. Where it's zero, the temperature immediately around it is constant. This represents a sort of saddle or peak or valley. At other places, the gradient is a vector that points to where the temperature increases most rapidly.

To find the gradient, simply take the partial derivative of each axis, and multiple it by the unit vector that axis. This should remind you of the dot product we talked about earlier, except we're "multiplying" by these partial derivative operators.

Del

By the way, this funny upside down triangle is called "del." It is an operator, just like d by dx, except it is really a vector operator. We're going to be using this a lot.

Divergence

The divergence of a vector field shows you whether vectors around it are pointing in or out or just flowing by. Imagine a tiny box surrounding a point, and the six vectors immediately around that point. Are they all pointing in or out or do you have some sort of flow?

The divergence is calculated by the dot product of this del operator with the vector field function. It's rather straight forward.

Curl

The curl is slightly less intuitive, but it is aptly named. Imagine a point, and the six vectors immediately around it. Do they twist around that point? If so, what is the plane of that twist? Follow the twist with your hand, and the thumb points in the direction of the twist. The magnitude says how tight the twist is.

To calculate, take the cross product of del and the vector field.

It's a lot more difficult to understand WHY this works. Imagine simple circular vector field. now pretend like we know the curl is going to be perpindicular to this. The question is, how strong will it be? We need to kind of twist the terms around to pull it out, which is exactly what the cross product does.

Conclusion

We've covered materials that would take mathematicians a whole lot longer to cover. Meaning, we're scratching the surface, and missing a whole lot of details that may be important. Don't worry, we'll go over those details later as we examine concrete roblebs.