Lapis the Nevrean

This is going to be a short introduction to vector math and the operations required to discuss the equations themselves. It's just a crash course, so if you want to learn more about the subject, LibreTexts hosts an excellent OpenStax textbook on the subject of calculus which includes extensive material on vector mathematics.

A vector is a mathematical object that represents both a magnitude and a direction. In the simplest sense, you can think of a set of coordinates as a vector; for example, in the standard Cartesian coordinate plane, the point (3, 4) could represent a two-dimensional vector with length 5 (the magnitude) and angle (the direction) of ~0.928 radians.

The length of any vector is calculated, based on Pythagoras, as the square root of the sum of the squares of its component values. For a vector v, we call this the magnitude or the modulus of v, or sometimes the norm, and it is represented as |v|:

$$\begin{array}{cc}\mathrm{𝐯<!--\; -->}& =<!--\; -->⟨<!--\; -->{\mathrm{v<!--\; -->}}_{\mathrm{1<!--\; -->}},<!--\; -->{\mathrm{v<!--\; -->}}_{\mathrm{2<!--\; -->}},<!--\; -->{\mathrm{v<!--\; -->}}_{\mathrm{3<!--\; -->}}⟩<!--\; -->\\ |<!--\; -->\mathrm{𝐯<!--\; -->}|<!--\; -->& =<!--\; -->\sqrt{{\mathrm{v<!--\; -->}}_{\mathrm{1<!--\; -->}}^{\mathrm{2<!--\; -->}}+<!--\; -->{\mathrm{v<!--\; -->}}_{\mathrm{2<!--\; -->}}^{\mathrm{2<!--\; -->}}+<!--\; -->{\mathrm{v<!--\; -->}}_{\mathrm{3<!--\; -->}}^{\mathrm{2<!--\; -->}}}\end{array}$$

For example, for our vector (3, 4) above, we would calculate the length like:

$$\begin{array}{cc}\mathrm{𝐯<!--\; -->}& =<!--\; -->⟨<!--\; -->\mathrm{3<!--\; -->},<!--\; -->\mathrm{4<!--\; -->}⟩<!--\; -->\\ |<!--\; -->\mathrm{𝐯<!--\; -->}|<!--\; -->& =<!--\; -->\sqrt{{\mathrm{3<!--\; -->}}^{\mathrm{2<!--\; -->}}+<!--\; -->{\mathrm{4<!--\; -->}}^{\mathrm{2<!--\; -->}}}\\ & =<!--\; -->\mathrm{5<!--\; -->}\end{array}$$

The direction of a two-dimensional vector is expressible as an angle counterclockwise from the x axis, typically expressed in radians. This angle is the direction of the vector, and sometimes called the argument of the vector, especially when treating complex numbers as vectors. Given a vector with two components, x and y, the direction θ is given by the inverse tangent of y/x:

$$\begin{array}{cc}\mathrm{𝐯<!--\; -->}& =<!--\; -->⟨<!--\; -->{\mathrm{v<!--\; -->}}_{\mathrm{1<!--\; -->}},<!--\; -->{\mathrm{v<!--\; -->}}_{\mathrm{2<!--\; -->}}⟩<!--\; -->\\ \mathrm{\theta <!--\; -->}& =<!--\; -->\mathrm{arctan<!--\; -->}<!--\; -->\frac{{\mathrm{v<!--\; -->}}_{\mathrm{2<!--\; -->}}}{{\mathrm{v<!--\; -->}}_{\mathrm{1<!--\; -->}}}\end{array}$$

In three dimensions, the direction of a vector is a much more complicated question, and we don't usually try to treat it as an independent quantity; we only really use angles between vectors, and in three dimensions it's not necessarily easy to decide what the angle of a vector in its own right should be measured against. Instead, we deal with what are called unit vectors, a vector that points the same direction as the vector whose direction we are interested in but has a length of exactly 1. For a vector v, the unit vector is called $\stackrel{^<!--\; -->}{\mathbf{v<!--\; -->}}$, pronounced "v-hat", and no, I assure you, that's not a joke. What the fuck is going on with mathematicians.

We can obtain a unit vector by dividing a vector by its own length:

$$\begin{array}{cc}\mathrm{𝐯<!--\; -->}& =<!--\; -->⟨<!--\; -->{\mathrm{v<!--\; -->}}_{\mathrm{1<!--\; -->}},<!--\; -->{\mathrm{v<!--\; -->}}_{\mathrm{2<!--\; -->}},<!--\; -->{\mathrm{v<!--\; -->}}_{\mathrm{3<!--\; -->}}⟩<!--\; -->\\ \stackrel{^<!--\; -->}{\mathrm{𝐯<!--\; -->}}& =<!--\; -->⟨<!--\; -->\frac{{\mathrm{v<!--\; -->}}_{\mathrm{1<!--\; -->}}}{|<!--\; -->\mathrm{𝐯<!--\; -->}|<!--\; -->},<!--\; -->\frac{{\mathrm{v<!--\; -->}}_{\mathrm{2<!--\; -->}}}{|<!--\; -->\mathrm{𝐯<!--\; -->}|<!--\; -->},<!--\; -->\frac{{\mathrm{v<!--\; -->}}_{\mathrm{3<!--\; -->}}}{|<!--\; -->\mathrm{𝐯<!--\; -->}|<!--\; -->}⟩<!--\; -->\end{array}$$

Before we go any further, let's talk about notation a little bit. First, a common way to represent vector quantities is to write an arrow above the variable name. We saw this earlier with the function $\stackrel{\to <!--\; -->}{\mathbf{F<!--\; -->}}$. I'm only writing functions this way but it is conventional to write vector variables this way, especially in cases where boldface is not available or not easily distinguishable from regular text. Second, there are a number of equivalent ways to express a vector quantity. All of the following express precisely the same vector:

$$\begin{array}{cc}\mathrm{𝐯<!--\; -->}& =<!--\; -->⟨<!--\; -->{\mathrm{v<!--\; -->}}_{\mathrm{1<!--\; -->}},<!--\; -->{\mathrm{v<!--\; -->}}_{\mathrm{2<!--\; -->}},<!--\; -->{\mathrm{v<!--\; -->}}_{\mathrm{3<!--\; -->}}⟩<!--\; -->\\ \mathrm{𝐯<!--\; -->}& =<!--\; -->(<!--\; -->\begin{array}{c}{\mathrm{v<!--\; -->}}_{\mathrm{1<!--\; -->}}\\ {\mathrm{v<!--\; -->}}_{\mathrm{2<!--\; -->}}\\ {\mathrm{v<!--\; -->}}_{\mathrm{3<!--\; -->}}\end{array})<!--\; -->\\ \mathrm{𝐯<!--\; -->}& =<!--\; -->{\mathrm{v<!--\; -->}}_{\mathrm{1<!--\; -->}}<!--\; -->\stackrel{^<!--\; -->}{\mathrm{𝐢<!--\; -->}}+<!--\; -->{\mathrm{v<!--\; -->}}_{\mathrm{2<!--\; -->}}<!--\; -->\stackrel{^<!--\; -->}{\mathrm{𝐣<!--\; -->}}+<!--\; -->{\mathrm{v<!--\; -->}}_{\mathrm{3<!--\; -->}}<!--\; -->\stackrel{^<!--\; -->}{\mathrm{𝐤<!--\; -->}}\end{array}$$

In the third style, $\stackrel{^<!--\; -->}{\mathrm{𝐢<!--\; -->}}$, $\stackrel{}{\stackrel{^<!--\; -->}{\mathrm{𝐣<!--\; -->}}}$ and $\stackrel{^<!--\; -->}{\mathrm{𝐤<!--\; -->}}$ represent the standard basis vectors, unit vectors directly along the x, y and z axes, respectively:

$$\begin{array}{cc}\stackrel{^<!--\; -->}{\mathrm{𝐢<!--\; -->}}& =<!--\; -->⟨<!--\; -->\mathrm{1<!--\; -->},<!--\; -->\mathrm{0<!--\; -->},<!--\; -->\mathrm{0<!--\; -->}⟩<!--\; -->\\ \stackrel{^<!--\; -->}{\mathrm{𝐣<!--\; -->}}& =<!--\; -->⟨<!--\; -->\mathrm{0<!--\; -->},<!--\; -->\mathrm{1<!--\; -->},<!--\; -->\mathrm{0<!--\; -->}⟩<!--\; -->\\ \stackrel{^<!--\; -->}{\mathrm{𝐤<!--\; -->}}& =<!--\; -->⟨<!--\; -->\mathrm{0<!--\; -->},<!--\; -->\mathrm{0<!--\; -->},<!--\; -->\mathrm{1<!--\; -->}⟩<!--\; -->\end{array}$$

All of these are valid and have their uses; the columnar form is usually seen more in pure math contexts but is convenient when doing matrix math. We'll be seeing all three, so don't get confused.

Next, we're going to talk about operations you can do on vectors, starting with some basic arithmetic. First, multiplication is the main well-defined operation between a vector and a scalar. Given a vector v and a scalar x:

$$\begin{array}{cc}\mathrm{x<!--\; -->}<!--\; -->\mathrm{𝐯<!--\; -->}& =<!--\; -->⟨<!--\; -->\mathrm{x<!--\; -->}<!--\; -->{\mathrm{v<!--\; -->}}_{\mathrm{1<!--\; -->}},<!--\; -->\mathrm{x<!--\; -->}<!--\; -->{\mathrm{v<!--\; -->}}_{\mathrm{2<!--\; -->}},<!--\; -->\mathrm{x<!--\; -->}<!--\; -->{\mathrm{v<!--\; -->}}_{\mathrm{3<!--\; -->}}⟩<!--\; -->\\ \frac{\mathrm{𝐯<!--\; -->}}{\mathrm{x<!--\; -->}}& =<!--\; -->⟨<!--\; -->\frac{{\mathrm{v<!--\; -->}}_{\mathrm{1<!--\; -->}}}{\mathrm{x<!--\; -->}},<!--\; -->\frac{{\mathrm{v<!--\; -->}}_{\mathrm{2<!--\; -->}}}{\mathrm{x<!--\; -->}},<!--\; -->\frac{{\mathrm{v<!--\; -->}}_{\mathrm{3<!--\; -->}}}{\mathrm{x<!--\; -->}}⟩<!--\; -->\end{array}$$

Simple stuff, you just multiply each component by x, or divide through in the same way (which is really just multiplying by 1/x). Note that adding a vector and a scalar, subtracting one from the other or dividing a scalar by a vector are all undefined because they're meaningless operations; a vector has a direction and a scalar doesn't, so what could those operations even mean? Multiplication, on the other hand, can be pretty clearly interpreted as scaling the length of the vector while leaving its direction unaltered (which should also make it very clear how you get a unit vector by taking v/|v|).

Now, vector-vector operations! Now we're getting into the fun stuff. First, addition and subtraction. Given two vectors, u and v:

$$\begin{array}{cc}\mathrm{𝐮<!--\; -->}+<!--\; -->\mathrm{𝐯<!--\; -->}& =<!--\; -->⟨<!--\; -->{\mathrm{u<!--\; -->}}_{\mathrm{1<!--\; -->}}+<!--\; -->{\mathrm{v<!--\; -->}}_{\mathrm{1<!--\; -->}},<!--\; -->{\mathrm{u<!--\; -->}}_{\mathrm{2<!--\; -->}}+<!--\; -->{\mathrm{v<!--\; -->}}_{\mathrm{2<!--\; -->}},<!--\; -->{\mathrm{u<!--\; -->}}_{\mathrm{3<!--\; -->}}+<!--\; -->{\mathrm{v<!--\; -->}}_{\mathrm{3<!--\; -->}}⟩<!--\; -->\\ \mathrm{𝐮<!--\; -->}-<!--\; -->\mathrm{𝐯<!--\; -->}& =<!--\; -->⟨<!--\; -->{\mathrm{u<!--\; -->}}_{\mathrm{1<!--\; -->}}-<!--\; -->{\mathrm{v<!--\; -->}}_{\mathrm{1<!--\; -->}},<!--\; -->{\mathrm{u<!--\; -->}}_{\mathrm{2<!--\; -->}}-<!--\; -->{\mathrm{v<!--\; -->}}_{\mathrm{2<!--\; -->}},<!--\; -->{\mathrm{u<!--\; -->}}_{\mathrm{3<!--\; -->}}-<!--\; -->{\mathrm{v<!--\; -->}}_{\mathrm{3<!--\; -->}}⟩<!--\; -->\end{array}$$

Again, just add or subtract the components, easy. The geometric interpretation of this is pretty interesting though. Let's say we have two 2d vectors, (3,4) and (1, -2), which give (4,2) when added:

Adding or subtracting two vectors gives a third vector that points to the location that's the sum or difference of the two original vectors.

Now, multiplication. Here, uh... Leave your phone on the table and let's go out to the car for a minute.

...

Alright, I'll be straight: There's no "multiplication" operation between two vectors. In fact, there's two different operations you can do that are like multiplication, but neither of them is straight-up multiplication. For that, you want to be dealing in complex mathematics and you get that nice rotational behavior, but that's only for two-dimensional vectors. Here in general vector space land we have to make do with the dot product and the cross product.

First, the dot product. Given two vectors, u and v, the dot product can be determined as follows:

$$\begin{array}{cc}\mathrm{𝐮<!--\; -->}\cdot <!--\; -->\mathrm{𝐯<!--\; -->}& =<!--\; -->{\mathrm{u<!--\; -->}}_{\mathrm{1<!--\; -->}}<!--\; -->{\mathrm{v<!--\; -->}}_{\mathrm{1<!--\; -->}}+<!--\; -->{\mathrm{u<!--\; -->}}_{\mathrm{2<!--\; -->}}<!--\; -->{\mathrm{v<!--\; -->}}_{\mathrm{2<!--\; -->}}+<!--\; -->{\mathrm{u<!--\; -->}}_{\mathrm{3<!--\; -->}}<!--\; -->{\mathrm{v<!--\; -->}}_{\mathrm{3<!--\; -->}}\end{array}$$

It's the sum of the products of each term, and gives a scalar result. For instance, the dot product of (3,4) and (1,-2):

$$\begin{array}{cc}⟨<!--\; -->\mathrm{3<!--\; -->},<!--\; -->\mathrm{4<!--\; -->}⟩<!--\; -->\cdot <!--\; -->⟨<!--\; -->\mathrm{1<!--\; -->},<!--\; -->-<!--\; -->\mathrm{2<!--\; -->}⟩<!--\; -->& =<!--\; -->\mathrm{3<!--\; -->}\cdot <!--\; -->\mathrm{1<!--\; -->}+<!--\; -->\mathrm{4<!--\; -->}\cdot <!--\; -->-<!--\; -->\mathrm{2<!--\; -->}\\ & =<!--\; -->\mathrm{3<!--\; -->}-<!--\; -->\mathrm{8<!--\; -->}\\ & =<!--\; -->-<!--\; -->\mathrm{5<!--\; -->}\end{array}$$

The dot product of two vectors can be defined as follows:
$$\begin{array}{c}\mathrm{𝐮<!--\; -->}\cdot <!--\; -->\mathrm{𝐯<!--\; -->}=<!--\; -->|<!--\; -->\mathrm{𝐮<!--\; -->}|<!--\; --><!--\; -->|<!--\; -->\mathrm{𝐯<!--\; -->}|<!--\; --><!--\; -->\mathrm{cos<!--\; -->}<!--\; -->\mathrm{\theta <!--\; -->}\end{array}$$

In practice, what this means is that the dot product encodes the angle θ between two vectors, which can be obtained as such:

$$\begin{array}{cc}\mathrm{\theta <!--\; -->}& =<!--\; -->\mathrm{arccos<!--\; -->}<!--\; -->\frac{\mathrm{𝐮<!--\; -->}\cdot <!--\; -->\mathrm{𝐯<!--\; -->}}{|<!--\; -->\mathrm{𝐮<!--\; -->}|<!--\; --><!--\; -->|<!--\; -->\mathrm{𝐯<!--\; -->}|<!--\; -->}\end{array}$$

Now, we have the cross product, also called the vector product. The cross product of two vectors u and v, denoted by u×v, encodes the angle between the two vectors and their lengths, much like the dot product, but also a unit vector perpendicular to the two original vectors, called the normal vector. If you're familiar with 3D graphics, you'll know about the use of normal vectors for calculating incidence of light, texture mapping and myriad other applications, and here it has a number of similar applications.

There's a few different ways to calculate the cross product, depending if you're using formal linear algebra or not, but a simple method is:

$$\begin{array}{cc}\mathrm{𝐮<!--\; -->}\times <!--\; -->\mathrm{𝐯<!--\; -->}& =<!--\; -->(<!--\; -->\begin{array}{c}{\mathrm{u<!--\; -->}}_{\mathrm{2<!--\; -->}}<!--\; -->{\mathrm{v<!--\; -->}}_{\mathrm{3<!--\; -->}}-<!--\; -->{\mathrm{u<!--\; -->}}_{\mathrm{3<!--\; -->}}<!--\; -->{\mathrm{v<!--\; -->}}_{\mathrm{2<!--\; -->}}\\ {\mathrm{u<!--\; -->}}_{\mathrm{3<!--\; -->}}<!--\; -->{\mathrm{v<!--\; -->}}_{\mathrm{1<!--\; -->}}-<!--\; -->{\mathrm{u<!--\; -->}}_{\mathrm{1<!--\; -->}}<!--\; -->{\mathrm{v<!--\; -->}}_{\mathrm{3<!--\; -->}}\\ {\mathrm{u<!--\; -->}}_{\mathrm{1<!--\; -->}}<!--\; -->{\mathrm{v<!--\; -->}}_{\mathrm{2<!--\; -->}}-<!--\; -->{\mathrm{u<!--\; -->}}_{\mathrm{2<!--\; -->}}<!--\; -->{\mathrm{v<!--\; -->}}_{\mathrm{1<!--\; -->}}\end{array})<!--\; -->\end{array}$$

For instance, the cross product between the vectors (3,4,5) and (1,-2,0) (and please note that the cross product is specific to three-dimensional vectors; there is a related concept called the exterior product for other types of vectors, and there's also a seven dimensional cross product for octonions, but that's well beyond what we're interested in here):

$$\begin{array}{cc}(<!--\; -->\begin{array}{c}\mathrm{3<!--\; -->}\\ \mathrm{4<!--\; -->}\\ \mathrm{5<!--\; -->}\end{array})<!--\; -->\times <!--\; -->(<!--\; -->\begin{array}{c}\mathrm{1<!--\; -->}\\ -<!--\; -->\mathrm{2<!--\; -->}\\ \mathrm{0<!--\; -->}\end{array})<!--\; -->& =<!--\; -->(<!--\; -->\begin{array}{c}\mathrm{4<!--\; -->}\cdot <!--\; -->\mathrm{0<!--\; -->}-<!--\; -->\mathrm{5<!--\; -->}\cdot <!--\; -->-<!--\; -->\mathrm{2<!--\; -->}\\ \mathrm{5<!--\; -->}\cdot <!--\; -->\mathrm{1<!--\; -->}-<!--\; -->\mathrm{3<!--\; -->}\cdot <!--\; -->\mathrm{0<!--\; -->}\\ \mathrm{3<!--\; -->}\cdot <!--\; -->-<!--\; -->\mathrm{2<!--\; -->}-<!--\; -->\mathrm{4<!--\; -->}\cdot <!--\; -->\mathrm{1<!--\; -->}\end{array})<!--\; -->\\ & =<!--\; -->(<!--\; -->\begin{array}{c}\mathrm{10<!--\; -->}\\ \mathrm{5<!--\; -->}\\ -<!--\; -->\mathrm{10<!--\; -->}\end{array})<!--\; -->\end{array}$$

Similarly to the dot product, the cross product has a trigonometric definition:

$$\begin{array}{c}\mathrm{𝐮<!--\; -->}\times <!--\; -->\mathrm{𝐯<!--\; -->}=<!--\; -->|<!--\; -->\mathrm{𝐮<!--\; -->}|<!--\; --><!--\; -->|<!--\; -->\mathrm{𝐯<!--\; -->}|<!--\; --><!--\; -->\mathrm{sin<!--\; -->}<!--\; -->(<!--\; -->\mathrm{\theta <!--\; -->})<!--\; --><!--\; -->\stackrel{^<!--\; -->}{\mathrm{𝐧<!--\; -->}}\end{array}$$

Where θ is the angle between u and v, just as in the dot product, and $\stackrel{^<!--\; -->}{\mathbf{n<!--\; -->}}$ is the normal vector described above. Note that the direction of the normal vector follows what is called the right-hand rule; if you point your index finger along the direction of the first vector, u, then turn your hand so that you can point your middle finger along the direction of the vector v, your thumb will point in the direction of $\stackrel{^<!--\; -->}{\mathbf{n<!--\; -->}}$. Seen here (image taken from Wikipedia's article on the right-hand rule, which explains it in more depth):

And that's our basic vector operations. All of our other vector operations, in particular the vector calculus we'll be talking about later, builds off these operations.

Again, this is just going to be a quick crash course. LibreTexts hosts an excellent OpenStax textbook on calculus. MIT also has excellent OpenCourseware courses on single-variable calculus and (multivariate and vector calculus)[https://ocw.mit.edu/courses/18-02-multivariable-calculus-fall-2007/].

Okay: Calculus is the study of change. How the behavior of functions change across their domain, how functions change as they approach their limits, how series of sums or products change as they approach infinite numbers of terms, and the like. Physics uses calculus heavily because almost all physics is interested, in some way or another, in change over time.

At the bottom of calculus we have this idea of a limit. Instead of necessarily being concerned with the result of a function at a particular point, f(x) where x is some value, the limit of a function is how it behaves as it approaches some value. As a basic example, let's think about the classic problem of division by zero. If you have a function $\mathrm{f<!--\; -->}(<!--\; -->\mathrm{x<!--\; -->})<!--\; -->=<!--\; -->\frac{\mathrm{1<!--\; -->}}{\mathrm{x<!--\; -->}}$, it is pretty well-known that you can't just plug a zero in for x and get anything back, since division by zero is undefined and, in fact, is a logical dead end since it introduces a logical contradiction; if you've ever seen bogus "proofs" that purport to show something like 2=1, the usual way that's accomplished is by doing a hidden division by zero somewhere, introducing a contradiction which allows any statement to be "proved". However! At any non-zero value of x, even those very close to zero, this function has a defined value! The way we use a limit, then, is by using this idea of approaching a particular value infinitely closely to see what value the function approaches as x gets infinitely close to that value.

So let's use that to look at our function there. First, let's take a graph of f(x):

We can see from this that when approached from the left (where x is < 0), the function goes down, seemingly to -∞, while when approached from the right (x > 0), it goes up:

$$\begin{array}{cc}\mathrm{f<!--\; -->}<!--\; -->(<!--\; -->\mathrm{x<!--\; -->})<!--\; -->& =<!--\; -->\frac{\mathrm{1<!--\; -->}}{\mathrm{x<!--\; -->}}\\ \underset{\mathrm{x<!--\; -->}\to <!--\; -->{\mathrm{0<!--\; -->}}^{-<!--\; -->}}{lim<!--\; -->}\mathrm{f<!--\; -->}<!--\; -->(<!--\; -->\mathrm{x<!--\; -->})<!--\; -->& =<!--\; -->-<!--\; -->\mathrm{\infty <!--\; -->}& & \text{Left-handed limit<!-- -->}\\ \underset{\mathrm{x<!--\; -->}\to <!--\; -->{\mathrm{0<!--\; -->}}^{+<!--\; -->}}{lim<!--\; -->}\mathrm{f<!--\; -->}<!--\; -->(<!--\; -->\mathrm{x<!--\; -->})<!--\; -->& =<!--\; -->\mathrm{\infty <!--\; -->}& & \text{Right-handed limit<!-- -->}\end{array}$$

What we have here is a function that has a vertical asymptote at x=0, meaning the function diverges (goes to infinity). It has a left-handed limit (where we "approach" the limit from the left so our variable is always strictly less than the limit we want to find) and a right-handed limit (approaching from the right so that the variable is always strictly greater than the limit we're investigating), but they're different so the two-sided limit (where we would be approaching the value from both sides) doesn't exist.

There's a number of different techniques for determining limits, but we're not going to go too much into it here; it's a big subject and finding really difficult limits can employ some fairly advanced math. In most of the cases we'll be caring about, we'll be employing fairly simple algebra or some basic calculus that can help us; the main thing to know is that if a function is defined at the limit value (ie, you can plug in the limit and get a result out of the function) then the limit is identical to the value of the function at that point.

Next, we're going to talk about the derivative. Let's say we have a big truck that we know goes from 0 to 100 kph in seven seconds. What if we wanted to know what its acceleration was (assuming it acccelerates linearly, ie it doesn't change acceleration)?

So the acceleration at any given point is equal to the slope of the graph at that point in the function. For this case, the answer is obvious because we already have that, it's the slope of the line which is 100/7, meaning the constant acceleration is 100 k/h/7 s ≈ 3.97 m/s².

But what if we wanted the slope of a function at some point x for a non-linear function? For instance, a basic parabola, $\mathrm{f<!--\; -->}(<!--\; -->\mathrm{x<!--\; -->})<!--\; -->=<!--\; -->{\mathrm{x<!--\; -->}}^{\mathrm{2<!--\; -->}}$. What's the slope of the graph at, say, x=3?

Well, let's think about our linear acceleration again. Because the function $\mathrm{f<!--\; -->}(<!--\; -->\mathrm{x<!--\; -->})<!--\; -->=<!--\; -->\frac{\mathrm{100<!--\; -->}}{\mathrm{7<!--\; -->}}\mathrm{x<!--\; -->}$ is linear, the acceleration, the rate of change of f(x), is just constant. $\mathrm{f<!--\; -->}(<!--\; -->\mathrm{x<!--\; -->})<!--\; -->=<!--\; -->{\mathrm{x<!--\; -->}}^{\mathrm{2<!--\; -->}}$, however, is not linear, it changes exponentially, faster and faster, meaning that the rate of change is not constant but is going to be a function in its own right.

That rate of change function is the derivative. For a single-variable function $\mathrm{f<!--\; -->}(<!--\; -->\mathrm{x<!--\; -->})<!--\; -->$, the derivative of that function is a function that expresses the slope of a tangent line at any given point on the curve. There's a few different kinds of notation for the derivative - notation for the operation of differentiation - but mostly we're going to stick with Leibniz's notation, $\frac{\mathrm{d<!--\; -->}}{\mathrm{d<!--\; -->}\mathrm{x<!--\; -->}}\mathrm{f<!--\; -->}(<!--\; -->\mathrm{x<!--\; -->})<!--\; -->$, or Lagrangian notation, ${\mathrm{f<!--\; -->}}^{\prime <!--\; -->}(<!--\; -->\mathrm{x<!--\; -->})<!--\; -->$.

$$\begin{array}{cc}\mathrm{f<!--\; -->}<!--\; -->(<!--\; -->\mathrm{x<!--\; -->})<!--\; -->& =<!--\; -->{\mathrm{x<!--\; -->}}^{\mathrm{2<!--\; -->}}\\ {\mathrm{f<!--\; -->}}^{\prime <!--\; -->}<!--\; -->(<!--\; -->\mathrm{x<!--\; -->})<!--\; -->& =<!--\; -->\mathrm{2<!--\; -->}<!--\; -->\mathrm{x<!--\; -->}\end{array}$$

Now, explaining how to do differentiation is a massive topic in its own right, but at its base it's defined as a pretty simple limit formula:

$$\begin{array}{cc}{\mathrm{f<!--\; -->}}^{\prime <!--\; -->}<!--\; -->(<!--\; -->\mathrm{x<!--\; -->})<!--\; -->& =<!--\; -->\underset{\mathrm{h<!--\; -->}\to <!--\; -->\mathrm{0<!--\; -->}}{lim<!--\; -->}\frac{\mathrm{f<!--\; -->}<!--\; -->(<!--\; -->\mathrm{x<!--\; -->}+<!--\; -->\mathrm{h<!--\; -->})<!--\; -->-<!--\; -->\mathrm{f<!--\; -->}<!--\; -->(<!--\; -->\mathrm{x<!--\; -->})<!--\; -->}{\mathrm{h<!--\; -->}}\end{array}$$

What this is saying is that we start with the difference between two values of the function separated by some distance between them h, $\mathrm{f<!--\; -->}<!--\; -->(<!--\; -->\mathrm{x<!--\; -->}+<!--\; -->\mathrm{h<!--\; -->})<!--\; -->-<!--\; -->\mathrm{f<!--\; -->}<!--\; -->(<!--\; -->\mathrm{x<!--\; -->})<!--\; -->$, and divide that by the distance on the x-axis between those two points, creating a function that returns the slope of a line that runs between those two points on the curve. We then take the limit as h, the distance between the two points, approaches zero to get a function that returns the slope of a tangent line to f(x) at any infinitesimal point x. For instance, here's how we would differentiate our function $\mathrm{f<!--\; -->}(<!--\; -->\mathrm{x<!--\; -->})<!--\; -->=<!--\; -->{\mathrm{x<!--\; -->}}^{\mathrm{2<!--\; -->}}$ using the limit definition:

$$\begin{array}{cc}\mathrm{f<!--\; -->}<!--\; -->(<!--\; -->\mathrm{x<!--\; -->})<!--\; -->& =<!--\; -->{\mathrm{x<!--\; -->}}^{\mathrm{2<!--\; -->}}\\ {\mathrm{f<!--\; -->}}^{\prime <!--\; -->}<!--\; -->(<!--\; -->\mathrm{x<!--\; -->})<!--\; -->& =<!--\; -->\underset{\mathrm{h<!--\; -->}\to <!--\; -->\mathrm{0<!--\; -->}}{lim<!--\; -->}\frac{{(<!--\; -->\mathrm{x<!--\; -->}+<!--\; -->\mathrm{h<!--\; -->})<!--\; -->}^{\mathrm{2<!--\; -->}}-<!--\; -->{\mathrm{x<!--\; -->}}^{\mathrm{2<!--\; -->}}}{\mathrm{h<!--\; -->}}\\ & =<!--\; -->\underset{\mathrm{h<!--\; -->}\to <!--\; -->\mathrm{0<!--\; -->}}{lim<!--\; -->}\frac{{\mathrm{x<!--\; -->}}^{\mathrm{2<!--\; -->}}+<!--\; -->\mathrm{2<!--\; -->}<!--\; -->\mathrm{h<!--\; -->}<!--\; -->\mathrm{x<!--\; -->}+<!--\; -->{\mathrm{h<!--\; -->}}^{\mathrm{2<!--\; -->}}-<!--\; -->{\mathrm{x<!--\; -->}}^{\mathrm{2<!--\; -->}}}{\mathrm{h<!--\; -->}}& & \text{Expanding f(x)<!-- -->}\\ & =<!--\; -->\underset{\mathrm{h<!--\; -->}\to <!--\; -->\mathrm{0<!--\; -->}}{lim<!--\; -->}\frac{\mathrm{h<!--\; -->}<!--\; -->(<!--\; -->\mathrm{2<!--\; -->}<!--\; -->\mathrm{x<!--\; -->}+<!--\; -->\mathrm{h<!--\; -->})<!--\; -->}{\mathrm{h<!--\; -->}}& & \text{Factor an h out of the numerator<!-- -->}\\ & =<!--\; -->\underset{\mathrm{h<!--\; -->}\to <!--\; -->\mathrm{0<!--\; -->}}{lim<!--\; -->}\mathrm{2<!--\; -->}<!--\; -->\mathrm{x<!--\; -->}+<!--\; -->\mathrm{h<!--\; -->}& & \text{This is continuous at h=0 so we can just plug it in!<!-- -->}\\ & =<!--\; -->\mathrm{2<!--\; -->}<!--\; -->\mathrm{x<!--\; -->}\end{array}$$

Calculating derivatives gets tricky fast and using the limit definition obviously gets incredibly unwieldy, so lots of rules and definitions have been discovered over the last few centuries to make getting derivatives easier. It's a topic worth entire textbooks in its own right so I'm going to refer you to Paul's Online Notes at Lamar University on derivatives. We'll be using derivatives throughout but as long as you understand the concepts you shouldn't need to know much of the specifics of differential calculus to follow along. That being said, here's a quick table of some derivative rules that will help you follow along with my blithering:

Now, we've been talking about functions of a single variable so far, f(x). When you take the derivative of a single-variable function, you're always differentiating with respect to the only variable. What about functions with multiple variables? For instance, since we're dealing in physical phenomena here, magnetism and and electricity, we might have functions that deal with points in space, so instead of a single variable, or one variable whose value is strictly a function of another one, we might have two or three variables that we're dealing with in a single function. How do we handle differentiation in that case?

Well, there's a few different approaches depending on what you're trying to accomplish. In many cases you'll only care about the derivative in terms of a single variable in the function, in which case you're taking the partial derivative, which uses the symbol ∂. The partial derivative in terms of a single variable in a multivariate function allows you to isolate the rate of change of that variable alone. In turn, if you want to get the derivative of the entire function in all variables, you're getting what's called the total derivative.

The way a partial derivative works is by treating all variables except the one we're differentiating in terms of as constants. Intuitively, this makes a certain amount of sense; if we isolate one variable and only allow it to vary, we can see the behavior of that variable alone.

As an example, let's say we have a function in two variables, $\begin{array}{c}\mathrm{f<!--\; -->}<!--\; -->(<!--\; -->\mathrm{x<!--\; -->},<!--\; -->\mathrm{y<!--\; -->})<!--\; -->=<!--\; -->{\mathrm{y<!--\; -->}}^{\mathrm{3<!--\; -->}}+<!--\; -->{\mathrm{x<!--\; -->}}^{\mathrm{2<!--\; -->}}<!--\; -->\mathrm{y<!--\; -->}-<!--\; -->\mathrm{x<!--\; -->}<!--\; -->\mathrm{y<!--\; -->}\end{array}$. First, let's graph it to see what we're up against:

(I put in the color map just for looks; the colors just equal the z value, so don't break your brain trying to decipher it. There's no ARG here... OR IS THERE?!)

It's a nice-looking curved surface. Now, let's take the partial derivatives in terms of x and y and see what we get:

$$\begin{array}{cc}\mathrm{f<!--\; -->}<!--\; -->(<!--\; -->\mathrm{x<!--\; -->},<!--\; -->\mathrm{y<!--\; -->})<!--\; -->& =<!--\; -->{\mathrm{y<!--\; -->}}^{\mathrm{3<!--\; -->}}+<!--\; -->{\mathrm{x<!--\; -->}}^{\mathrm{2<!--\; -->}}<!--\; -->\mathrm{y<!--\; -->}-<!--\; -->\mathrm{x<!--\; -->}<!--\; -->\mathrm{y<!--\; -->}\\ \frac{\partial <!--\; -->}{\partial <!--\; -->\mathrm{x<!--\; -->}}<!--\; -->\mathrm{f<!--\; -->}<!--\; -->(<!--\; -->\mathrm{x<!--\; -->},<!--\; -->\mathrm{y<!--\; -->})<!--\; -->& =<!--\; -->\mathrm{y<!--\; -->}<!--\; -->(<!--\; -->\mathrm{2<!--\; -->}<!--\; -->\mathrm{x<!--\; -->}-<!--\; -->\mathrm{1<!--\; -->})<!--\; -->\\ \frac{\partial <!--\; -->}{\partial <!--\; -->\mathrm{y<!--\; -->}}<!--\; -->\mathrm{f<!--\; -->}<!--\; -->(<!--\; -->\mathrm{x<!--\; -->},<!--\; -->\mathrm{y<!--\; -->})<!--\; -->& =<!--\; -->\mathrm{3<!--\; -->}<!--\; -->{\mathrm{y<!--\; -->}}^{\mathrm{2<!--\; -->}}+<!--\; -->{\mathrm{x<!--\; -->}}^{\mathrm{2<!--\; -->}}-<!--\; -->\mathrm{x<!--\; -->}\end{array}$$

On the left we have the graph of the partial derivative in terms of x, and on the right in terms of y. You can see how the surface has been partially deformed in different ways in both cases but still is recognizable as a transformation of the original surface.

Taking the total derivative is pretty easy in most cases that we will care about. Essentially, if certain conditions are met (conditions we will not go into because it gets real complicated real quickly and it's nothing we're going to need to deal with here), the total derivative is simply taking each partial derivative in turn, one after another. Really interestingly, it doesn't matter what order you take them in either:

$$\begin{array}{cc}\mathrm{f<!--\; -->}<!--\; -->(<!--\; -->\mathrm{x<!--\; -->},<!--\; -->\mathrm{y<!--\; -->})<!--\; -->& =<!--\; -->{\mathrm{y<!--\; -->}}^{\mathrm{3<!--\; -->}}+<!--\; -->{\mathrm{x<!--\; -->}}^{\mathrm{2<!--\; -->}}<!--\; -->\mathrm{y<!--\; -->}-<!--\; -->\mathrm{x<!--\; -->}<!--\; -->\mathrm{y<!--\; -->}\\ \frac{\partial <!--\; -->}{\partial <!--\; -->\mathrm{x<!--\; -->}}<!--\; -->\mathrm{f<!--\; -->}<!--\; -->(<!--\; -->\mathrm{x<!--\; -->},<!--\; -->\mathrm{y<!--\; -->})<!--\; -->& =<!--\; -->\mathrm{y<!--\; -->}<!--\; -->(<!--\; -->\mathrm{2<!--\; -->}<!--\; -->\mathrm{x<!--\; -->}-<!--\; -->\mathrm{1<!--\; -->})<!--\; -->\\ \frac{\partial <!--\; -->}{\partial <!--\; -->\mathrm{y<!--\; -->}}<!--\; -->\frac{\partial <!--\; -->}{\partial <!--\; -->\mathrm{x<!--\; -->}}<!--\; -->\mathrm{f<!--\; -->}<!--\; -->(<!--\; -->\mathrm{x<!--\; -->},<!--\; -->\mathrm{y<!--\; -->})<!--\; -->& =<!--\; -->\mathrm{2<!--\; -->}<!--\; -->\mathrm{x<!--\; -->}-<!--\; -->\mathrm{1<!--\; -->}\\ \frac{\partial <!--\; -->}{\partial <!--\; -->\mathrm{y<!--\; -->}}<!--\; -->\mathrm{f<!--\; -->}<!--\; -->(<!--\; -->\mathrm{x<!--\; -->},<!--\; -->\mathrm{y<!--\; -->})<!--\; -->& =<!--\; -->\mathrm{3<!--\; -->}<!--\; -->{\mathrm{y<!--\; -->}}^{\mathrm{2<!--\; -->}}+<!--\; -->{\mathrm{x<!--\; -->}}^{\mathrm{2<!--\; -->}}-<!--\; -->\mathrm{x<!--\; -->}\\ \frac{\partial <!--\; -->}{\partial <!--\; -->\mathrm{x<!--\; -->}}<!--\; -->\frac{\partial <!--\; -->}{\partial <!--\; -->\mathrm{y<!--\; -->}}<!--\; -->\mathrm{f<!--\; -->}<!--\; -->(<!--\; -->\mathrm{x<!--\; -->},<!--\; -->\mathrm{y<!--\; -->})<!--\; -->& =<!--\; -->\mathrm{2<!--\; -->}<!--\; -->\mathrm{x<!--\; -->}-<!--\; -->\mathrm{1<!--\; -->}\\ \mathrm{d<!--\; -->}<!--\; -->\mathrm{f<!--\; -->}& =<!--\; -->\mathrm{2<!--\; -->}<!--\; -->\mathrm{x<!--\; -->}-<!--\; -->\mathrm{1<!--\; -->}\end{array}$$

And a graph of the total derivative, df f(x,y) = 2x - 1:

If you compare this with the original graph, you can see that it does give a tangent plane to the original function at any (x,y)!

Since this is all groundwork for the Maxwell-Heaviside Equations, we'll mainly be dealing with vector functions, which are effectively multivariate functions, so we'll be dealing with partial derivatives a lot; they have major applications for differential operators. Total derivatives of this type, we won't be dealing with so much.

Now, integration, aka "the hard part". There's a lot of ways to talk about integration, but let's start with the little fact we already talked about right up top: Integration is, in some senses, an inverse operation of differentiation. But what does that mean exactly? We have this interpretation of the derivative of a function as being a function describing the rate of change of the original function. By a similar token, the integral of a function is the area under the function, either over the entire domain of the function (an indefinite integral) or over a specific region (the definite integral).

The last paragraph is a really blockheaded version of the fundamental theorem of calculus, which relates derivation and the antiderivative (what we call the indefinite integral) to definite integration. You don't need to worry about the details of that; everything important about it, I just described and now you have it too.

Anyway, an example! We already know that for $\mathrm{f<!--\; -->}(<!--\; -->\mathrm{x<!--\; -->})<!--\; -->=<!--\; -->{\mathrm{x<!--\; -->}}^{\mathrm{2<!--\; -->}}$, the derivative is $\mathrm{f<!--\; -->}\text{'}<!--\; -->(<!--\; -->\mathrm{x<!--\; -->})<!--\; -->=<!--\; -->\mathrm{2<!--\; -->}\mathrm{x<!--\; -->}$. if you look at it in reverse, $\mathrm{f<!--\; -->}(<!--\; -->\mathrm{x<!--\; -->})<!--\; -->=<!--\; -->\mathrm{2<!--\; -->}\mathrm{x<!--\; -->}$, then the antiderivative would be:

$$\begin{array}{cc}\mathrm{f<!--\; -->}<!--\; -->(<!--\; -->\mathrm{x<!--\; -->})<!--\; -->& =<!--\; -->\mathrm{2<!--\; -->}<!--\; -->\mathrm{x<!--\; -->}\\ \int <!--\; -->\mathrm{f<!--\; -->}<!--\; -->(<!--\; -->\mathrm{x<!--\; -->})<!--\; --><!--\; -->𝑑<!--\; -->\mathrm{x<!--\; -->}& =<!--\; -->{\mathrm{x<!--\; -->}}^{\mathrm{2<!--\; -->}}+<!--\; -->\mathrm{C<!--\; -->}\end{array}$$

Quick explanation of the notation: The big stretched out S is a big stretched out S and it's the integral sign; it's a Declaration of Independence-ass S because conceptually integration is kind of a fucked-up sum of an infinite number of terms. The "dx" tells us that we're integrating in terms of x, and in particular it's an infinitesimal delta which I will explain in a minute. Finally, the + C is called the constant of integration, and it's there because a math teacher will stab you to death in front of the whole class if you forget it; it's actually important because the derivative of any constant term is zero, meaning that 2x is the derivative of x² + anything, and so we express that "anything" in the indefinite integral as the arbitrary constant C.

As mentioned, integration can be thought of as a fucked-up infinite sum. Think of it this way: The area under any curve could be approximated by a bunch of rectangles of equal width, whose height is the value of the function at the left (or right, or middle or whatever) side of the rectangle. Something like this:

With narrow rectangles like in the graph, we have a pretty good approximation already, but the more rectangles we get, the better approximation we get. As the number of rectangles goes to infinity, the rectangles become infinitesimally thin and the sum of the areas of the infinite number of boxes is exactly equal to the area under the graph. If we say we have k rectangles, and we want the area under a function f(x) from x values a to b, we can describe our approximation as a sum of the areas of those rectangles:

$$\begin{array}{cc}\mathrm{\Delta <!--\; -->}<!--\; -->\mathrm{x<!--\; -->}& =<!--\; -->\frac{\mathrm{b<!--\; -->}-<!--\; -->\mathrm{a<!--\; -->}}{\mathrm{k<!--\; -->}}\\ \mathrm{A<!--\; -->}& =<!--\; -->\underset{\mathrm{n<!--\; -->}=<!--\; -->\mathrm{0<!--\; -->}}{\overset{\mathrm{k<!--\; -->}-<!--\; -->\mathrm{1<!--\; -->}}{\sum <!--\; -->}}\mathrm{f<!--\; -->}<!--\; -->(<!--\; -->\mathrm{a<!--\; -->}+<!--\; -->\mathrm{n<!--\; -->}<!--\; -->\mathrm{\Delta <!--\; -->}<!--\; -->\mathrm{x<!--\; -->})<!--\; --><!--\; -->\mathrm{\Delta <!--\; -->}<!--\; -->\mathrm{x<!--\; -->}\end{array}$$

This is usually expressed in a slightly different form (a bit more general and a bit less of a sloppy fucking for loop) but broadly speaking this is called a Riemann sum. If we take the limit of A as k goes to infinity, we get a rough limit definition of the Riemann integral. That Δx becomes the little infinitesimal dx in our integral.

Now, much like differentiation, yes, we have a limit definition of integrals (although, in keeping with integration being a much bigger pain in the ass than differentiation, it's quite a bit more complex than the difference quotient and Riemann sums aren't the only way of defining integrals either), but we don't want to try to use it all the time to evaluate integrals. In particular, taking antiderivatives with the limit definition is ugly as hell. So! We have, instead, a whole course worth of integration rules, the most basic ones I'm going to address here, just as I did with the differentiation rules.

As noted, integrals come in two main flavors, definite integrals and indefinite integrals. The definite integral gives the exact area under a curve for a particular set of endpoints (formally a closed interval). To get the definite integral, there's a few different approaches but the simplest way (assuming the integral is amenable to it and you don't have to use horrible things like contour integration that we're not going to talk about here) is to take the indefinite integral and evaluate it at the endpoints (all that means is you plug in the upper and lower endpoints and subtract the term with the lower endpoint from the term with the upper endpoint). For instance, for our graph of x² from 0 to 5 up above:

$$\begin{array}{cc}{\int <!--\; -->}_{\mathrm{0<!--\; -->}}^{\mathrm{5<!--\; -->}}{\mathrm{x<!--\; -->}}^{\mathrm{2<!--\; -->}}<!--\; -->𝑑<!--\; -->\mathrm{x<!--\; -->}& =<!--\; -->{\frac{{\mathrm{x<!--\; -->}}^{\mathrm{3<!--\; -->}}}{\mathrm{3<!--\; -->}}|<!--\; -->}_{\mathrm{0<!--\; -->}}^{\mathrm{5<!--\; -->}}\\ & =<!--\; -->\frac{{\mathrm{5<!--\; -->}}^{\mathrm{3<!--\; -->}}}{\mathrm{3<!--\; -->}}-<!--\; -->\frac{{\mathrm{0<!--\; -->}}^{\mathrm{3<!--\; -->}}}{\mathrm{3<!--\; -->}}\\ & =<!--\; -->\frac{\mathrm{125<!--\; -->}}{\mathrm{3<!--\; -->}}\end{array}$$

Note that we didn't bother with the constant of integration because it always disappears in a definite integral; you can see when we evaluate it, we'd just end up with C - C so there's no point.

Now, we've been talking about single-variable integration, and now it's time to talk about integration in multiple variables. This lets us think about integrating higher-dimensional volumes instead of just 2-d areas.

The first thing we need to know is that there's no such thing as an antiderivative in multiple variables, so we generally don't think about indefinite integrals for multiple integrals. Instead, we extend the definite integral to integrating over a domain, and we notate that with multiple integral signs. For instance, for a domain D:

$$\begin{array}{cccc}\mathrm{𝐃<!--\; -->}& \subseteq <!--\; -->{\mathrm{ℝ<!--\; -->}}^{\mathrm{2<!--\; -->}}& & \text{Domain is a subregion of<!-- -->}<!--\; -->{\mathrm{ℝ<!--\; -->}}^{\mathrm{2<!--\; -->}}\\ \mathrm{I<!--\; -->}& =<!--\; -->{\iint <!--\; -->}_{\mathrm{D<!--\; -->}}\mathrm{f<!--\; -->}<!--\; -->(<!--\; -->\mathrm{x<!--\; -->},<!--\; -->\mathrm{y<!--\; -->})<!--\; --><!--\; -->𝑑<!--\; -->\mathrm{x<!--\; -->}<!--\; -->𝑑<!--\; -->\mathrm{y<!--\; -->}& & \text{The double integral of f on D<!-- -->}\\ \mathrm{𝐃<!--\; -->}& \subseteq <!--\; -->{\mathrm{ℝ<!--\; -->}}^{\mathrm{3<!--\; -->}}& & \text{Domain is a subregion of<!-- -->}<!--\; -->{\mathrm{ℝ<!--\; -->}}^{\mathrm{3<!--\; -->}}\\ \mathrm{I<!--\; -->}& =<!--\; -->{\iiint <!--\; -->}_{\mathrm{D<!--\; -->}}\mathrm{f<!--\; -->}<!--\; -->(<!--\; -->\mathrm{x<!--\; -->},<!--\; -->\mathrm{y<!--\; -->},<!--\; -->\mathrm{z<!--\; -->})<!--\; --><!--\; -->𝑑<!--\; -->\mathrm{x<!--\; -->}<!--\; -->𝑑<!--\; -->\mathrm{y<!--\; -->}<!--\; -->𝑑<!--\; -->\mathrm{z<!--\; -->}& & \text{The triple integral of f on D<!-- -->}\end{array}$$

Evaluating multiple integrals is a large topic but we're going to restrict ourselves to the simple case; for most continuous functions over a closed region and other math words we can evaluate these types of multiple integrals as iterated integrals, much like what we did for the derivative. For instance:

$$\begin{array}{cc}\mathrm{f<!--\; -->}<!--\; -->(<!--\; -->\mathrm{x<!--\; -->},<!--\; -->\mathrm{y<!--\; -->})<!--\; -->& =<!--\; -->{\mathrm{x<!--\; -->}}^{\mathrm{2<!--\; -->}}+<!--\; -->{\mathrm{y<!--\; -->}}^{\mathrm{2<!--\; -->}}\\ \mathrm{𝐃<!--\; -->}& =<!--\; -->\{<!--\; -->(<!--\; -->\mathrm{x<!--\; -->},<!--\; -->\mathrm{y<!--\; -->})<!--\; -->\in <!--\; -->{\mathrm{ℝ<!--\; -->}}^{\mathrm{2<!--\; -->}}:<!--\; -->-<!--\; -->\mathrm{1<!--\; -->}\le <!--\; -->\mathrm{x<!--\; -->}\le <!--\; -->\mathrm{1<!--\; -->};<!--\; -->-<!--\; -->\mathrm{1<!--\; -->}\le <!--\; -->\mathrm{y<!--\; -->}\le <!--\; -->\mathrm{1<!--\; -->}\}<!--\; -->& & \text{x is over -1 to 1, y is over the same<!-- -->}\\ \mathrm{I<!--\; -->}& =<!--\; -->{\iint <!--\; -->}_{\mathrm{D<!--\; -->}}\mathrm{f<!--\; -->}<!--\; -->(<!--\; -->\mathrm{x<!--\; -->},<!--\; -->\mathrm{y<!--\; -->})<!--\; --><!--\; -->𝑑<!--\; -->\mathrm{x<!--\; -->}<!--\; -->𝑑<!--\; -->\mathrm{y<!--\; -->}\\ & =<!--\; -->{\int <!--\; -->}_{-<!--\; -->\mathrm{1<!--\; -->}}^{\mathrm{1<!--\; -->}}{\int <!--\; -->}_{-<!--\; -->\mathrm{1<!--\; -->}}^{\mathrm{1<!--\; -->}}{\mathrm{x<!--\; -->}}^{\mathrm{2<!--\; -->}}+<!--\; -->{\mathrm{y<!--\; -->}}^{\mathrm{2<!--\; -->}}<!--\; -->\mathrm{d<!--\; -->}<!--\; -->\mathrm{x<!--\; -->}<!--\; -->\mathrm{d<!--\; -->}<!--\; -->\mathrm{y<!--\; -->}\\ & =<!--\; -->{\int <!--\; -->}_{-<!--\; -->\mathrm{1<!--\; -->}}^{\mathrm{1<!--\; -->}}\frac{{\mathrm{x<!--\; -->}}^{\mathrm{3<!--\; -->}}}{\mathrm{3<!--\; -->}}+<!--\; -->{\mathrm{x<!--\; -->}<!--\; -->{\mathrm{y<!--\; -->}}^{\mathrm{2<!--\; -->}}|<!--\; -->}_{\mathrm{x<!--\; -->}=<!--\; -->-<!--\; -->\mathrm{1<!--\; -->}}^{\mathrm{1<!--\; -->}}<!--\; -->\mathrm{d<!--\; -->}<!--\; -->\mathrm{y<!--\; -->}& & \text{Integrate and evaluate over x…<!-- -->}\\ & =<!--\; -->{\int <!--\; -->}_{-<!--\; -->\mathrm{1<!--\; -->}}^{\mathrm{1<!--\; -->}}\frac{\mathrm{2<!--\; -->}}{\mathrm{3<!--\; -->}}+<!--\; -->\mathrm{2<!--\; -->}<!--\; -->{\mathrm{y<!--\; -->}}^{\mathrm{2<!--\; -->}}<!--\; -->\mathrm{d<!--\; -->}<!--\; -->\mathrm{y<!--\; -->}\\ & =<!--\; -->\frac{\mathrm{2<!--\; -->}<!--\; -->\mathrm{y<!--\; -->}}{\mathrm{3<!--\; -->}}+<!--\; -->{\frac{\mathrm{2<!--\; -->}<!--\; -->{\mathrm{y<!--\; -->}}^{\mathrm{3<!--\; -->}}}{\mathrm{3<!--\; -->}}|<!--\; -->}_{\mathrm{y<!--\; -->}=<!--\; -->-<!--\; -->\mathrm{1<!--\; -->}}^{\mathrm{1<!--\; -->}}& & \text{Then over y!<!-- -->}\\ & =<!--\; -->\frac{\mathrm{8<!--\; -->}}{\mathrm{3<!--\; -->}}\end{array}$$

Triple integrals work the same way. Additionally, according to Fubini's theorem, for multiple integrals that are continuous in their domains (and so can be evaluated at all), the order of evaluation of the variables doesn't matter, and you'll always get the same result.

Now, there's a lot of complexity we haven't addressed here; for instance, if the domain of the integral is more complicated than a simple rectangle/rectangular prism so that it's defined as a function in its own right, we have more work to do to figure out the endpoints - the bounds of integration - of each iterated integral. We will also run into (you can see it up above) notation like this:

$${\displaystyle {∯<!--\; -->}_{\partial <!--\; -->\mathrm{\Omega <!--\; -->}}}\mathrm{𝐁<!--\; -->}\cdot <!--\; -->d<!--\; -->\mathrm{𝐒<!--\; -->}=<!--\; -->\mathrm{0<!--\; -->}$$

That circle describes this as a surface integral with some special rules on determining the bounds of integration (or, equivalently, transforming the integrand to be able to use linear bounds). We also saw single integral symbols with a circle; that describes an analogous line integral. You don't need to worry too much about these; I'll describe what's going on when we actually deal with them, since there's a limit to how much abstract-seeming bullshit I want to frontload here. Just know that at the end of the day, after we do some minor fuckery these will reduce to exactly the kinds of multiple integrals we've been looking at already and will make sense in context.