Electrodynamics Tutorial Series/Math Background
Contents
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 cross product is weird. Just cope with it. We use this x symbol you learned about in 1st grade. The result is a vector that is perpendicular to both vectors.
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:
<math>\mathbf{a} \times \mathbf{b} = - \mathbf{b} \times \mathbf{a}</math>
Identities
There are some cute identities and properties of the cross and dot products.
