#3-space

I. Volume, Volumes, and the Volume Integral

We model our three-dimensional physical environment with a mathematical equivalent known as real 3-space (or simply 3-space). It's essentially three copies of the real line $\mathbb{R}$ strung together. We express this symbolically by writing $\mathbb{R}^3=\mathbb{R}\times \mathbb{R}\times \mathbb{R}$. It’s standard practice in physics to call the first copy of $\mathbb{R}$ the $x$-axis, the second copy the $y$-axis, and the third copy the $z$-axis.

Consequently, any point $\vec{a}$ in 3-space is uniquely characterized by three real numbers, collectively called rectangular coordinates: an $x$-coordinate $a_x$, a $y$-coordinate $a_y$, and a $z$-coordinate $a_z$. Each coordinate selects a real number from its corresponding axis (so that, for example, the $x$-coordinate is a specific value on the $x$-axis). We often write $3$-space points with an overhead arrow, thus why I've been doing that. We also sometimes write points as a list of their coordinates, such as $\vec{a}=(a_x,a_y,a_z)$.

One of the simplest ways to make a three-dimensional shape in $3$-space is by restricting each of our coordinates to be in a certain interval, i.e. we look at the shape composed of points $(x,y,z)$ such that $x\in [x_L,x_R]$, $y\in [y_L,y_R]$, and $z\in [z_L,z_R]$. The resulting geometric object is called a right rectangular prism (abbreviated as RRP) or a $3$-cell.

Just as line segments have length and rectangles have area, $3$-cells possess a natural measure of size called volume, created by multiplying the lengths of each interval together:

In this context, any of these widths is called a side-length of the $3$-cell. We like $3$-cells because we can approximate any three-dimensional shape as a union of $3$-cells. This enables us to extend the definition of volume to any three-dimensional shape.

Although generic $3$-cells will get the job done, for convenience we typically specialize to $3$-cells which have a uniform side-length, say $L = |x_R-x_L| = |y_R-y_L| = |z_R-z_L|$. We call such an object a cube. By definition,

Having discussed the building blocks, let's move on to three-dimensional shapes proper. A generic three-dimensional shape is (somewhat confusingly) also called a volume. As already mentioned, we can approximate generic volumes using cubes. In the same way that smaller squares make for better approximations of surfaces, smaller cubes make for better approximations of volumes.

Cutting to the chase, if we utilize an infinitely-fine partition of infinitely-small cubes, we obtain the volume integral over a volume $V$:

The quantity $\Delta \tau_i$ in this definition is the volume of the $i$th cube $\text{CUBE}_i$ contained in our approximation of $V$. Meanwhile, the index $n$ labels the stages of our partition refinement, so that as $n\rightarrow \infty$, the number of cubes used to approximate $V$ diverges as well: $N^{(n)}\rightarrow \infty$. Per usual, we can reweight the contribution of each cube in our partition according to a function defined on $V$, resulting in what we call the volume integral of a (real-valued) function $f$ over a volume $V$:

Like we saw last week in the real plane, there are ways to weight cubes such that the weights can depend on the specific partition we use. These possibly partition-dependent weight prescriptions are called distributions. (Reminder: functions are a subset of distributions.) I bring up distributions because we’re just as interested in Dirac delta distributions in $\mathbb{R}^3$ as we were in $\mathbb{R}^2$.

Now that we know how to construct generic three-dimensional objects in a controlled manner, let’s discuss all of the other objects that live in $\mathbb{R}^3$. In doing so, we’ll derive a handful of Dimensional Bridges and obtain corresponding Dirac delta distributions. We’ll also learn about zero-dimensional objects and zero-dimensional integrals.

II. Bridging the Many Shapes of 3-Space

There exist four kinds of dimensional shapes in $\mathbb{R}^3$: volumes, surfaces, curves, and points.

Despite their different dimensionalities, any of these shapes can be reconstructed by a sequence of volumes. We proceed like we did last week: we create the appropriately thickened version of the desired object and then take some of its widths to zero. This is how we build Dimensional Bridges.

For example, suppose we would like to have a certain surface $S$ in $\mathbb{R}^3$, but are only allowed to build it using volumes. To do so, we could craft a volume $\mathbf{S}_{W_1}$ that resembles the desired surface but has a thickness $W_1$ to it:

While it’s technically three-dimensional, we can imagine making $\mathbf{S}_{W_1}$ thinner and thinner until--once its thickness $W_1$ goes to zero--we recover $S$. This procedure constructs a Dimensional Bridge:

In this case, we’ve eliminate precisely one of the volume’s three dimensions.

Alternatively, we could create a curve $C$ from a curve-like volume $\mathbf{C}_{W_1,W_2}$ by eliminating two dimensions:

This leads to another Dimensional Bridge equation:

We can even create a point $P$ from a point-like volume $\mathbf{P}_{W_1,W_2,W_3}$ by eliminating all three of the volume’s dimensions.

The Dimensional Bridge of this case is tricky. By following the pattern of the other cases, we can guess its right-hand side:

But what should the left-hand side of the equation be? If it follows the pattern of the previous cases, it should be a measure of zero-dimensional objects, which leads us to ask: what is the natural measure of a point?

III. Making Sum-Thing Out of One-Thing

Every point in $\mathbb{R}^3$ is a zero-dimensional object. This expresses the fact that points lack any spatial extent: they simply exist “at a point.”

To phrase it another way, imagine we scale-up all of our shapes by a factor of two. As a result, our curves become twice as long ($2=2^1$), our surfaces possess four times as much area ($4=2^2$), and our volumes encompass eight times as much volume ($8=2^3$)... but our points look exactly the same as before. Points are scale-invariant.

This gets to the heart of the problem with finding a natural measure of a point: there’s no such thing as a larger or smaller point. What does it mean to "measure" a point when they’re all identical?

Let’s entirely ignore the fact that we don’t know what’s happening on the left-hand side of our unfinished Dimensional Bridge and try to evaluate its right-hand side instead. In doing so, we should approximate the point $\vec{P}$ with a point-like volume $\text{P}_{W_1,W_2,W_3}$. Let’s use a (small) $3$-cell for this purpose. Specifically, let’s use a $3$-cell that contains the point $\vec{P}$ and has side-lengths $W_1=|x_R-x_L|$, $W_2= |y_R-y_L|$, and $W_3=|z_R-z_R|$.

Because our point-like $3$-cell has volume $W_1\cdot W_2\cdot W_3$, we can simplify the right-hand side of our unfinished equation:

Whoa! According to this, whatever the natural measure of a point might be, a point always contributes an amount $1$.

Now we’re getting somewhere! To clear things up, let’s generalize a little bit. What if we instead had $N$ points? 

Let’s label the points $\vec{P}_1$, $\vec{P}_2$, and so-on, and label their entire collection as $A = \{\vec{P}_1,\cdots,\vec{P}_N\}$. We’ll approximate each point with a $3$-cell with side-lengths $W_1$, $W_2$, and $W_3$. Let’s call the collection of these point-like volumes $\mathbf{A}_{W_1,W_2,W_3}$.

Because each point-like object has volume $W_1\cdot W_2\cdot W_3$, our conglomerate $\mathbf{A}_{W_1,W_2,W_3}$ has $N$-times as much volume, yielding a total volume of $N\cdot (W_1\cdot W_2\cdot W_3)$. That means the right-hand side of our unfinished Dimensional Bridge equals, in this case,

Huh, look at that: we’ve recovered the number of points in $A$! We therefore surmise that the natural way to measure points is to count how many there are; we call this zero-dimensional measurement the set size of a finite set, aka size.

Note how size respects the scale-invariance of points because making everything twice as big doesn’t change how many points there are.

Following this thread to completion allows us to write the final Dimensional Bridge. Given a finite set $A$ of points, the Dimensional Bridge reads,

While this is great, we’ve introduced size via a three-dimensional integral, which is the oppposite of how we treated other natural measures. In all of the previous cases, we first expressed each measure as an integral over an object of the same relevant dimensionality (e.g., expressing area as a surface integral, length as a line integral, and so-on). Let’s rectify this now by writing size as a zero-dimensional integral.

The zero-dimensional integral is unique because it’s the only integral for which partitions are irrelevant. See, points are the basic zero-dimensional building block, and every zero-dimensional object is just a finite set of points. As a result, there’s no need to make any approximations: we can exactly reconstruct zero-dimensional objects.

Like our previous integration routines, if we have $N$ points in a finite set $A=\{\vec{P}_1,\cdots,\vec{P}_N\}$, then we’ll find the size of $A$ by breaking $A$ into individual zero-dimensional pieces (points) and adding up the size of each piece (always $1$). This yields the zero-dimensional integral we desired, aka the point integral over a finite set $A$:

But we can go further. We recovered the natural measure of our zero-dimensional object by associating each point with a value of $1$... which looks like a weight!  Following the usual integration script, we can reweight each point in $A$ according to a function $f$ on $\mathbb{R}^3$, and in doing so we obtain the point integral of a function $f$ over a finite set $A$:

where we’ve written the coordinates of $\vec{P}_i$ as $(x_i,y_i,z_i)$. We thereby arrive at a peculiar result: zero-dimensional integrals are finite sums. It’s for this reason I will probably never call zero-dimensional integrals “point integrals” again and instead opt to call them “sums” from here on out.

(Hey, this means you’re technically evaluating an integral every time you count or add two numbers together! If you’ve learned anything today, I hope it’s that--if you're being super technical--schools teach integral calculus to kindergartners.)

IV. Turning Bridges into Dirac Deltas

Last week, we saw how a Dimensional Bridge equation can yield a Dirac delta distribution. In particular, we saw that we could use a Dimensional Bridge to write line integrals as surface integrals instead:

Notice I tweaked my notation slightly by writing the Dirac delta on $C$ as $\delta^2_C$ instead of $\delta_C$. This establishes that it’s meant to be integrated over a 2-dimensional surface as opposed to--for example--a 3-dimensional volume. It also reminds us that distributions are, by definition, objects intended for integration. 

Aside: I should mention that the above equation holds true for surfaces and curves in $3$-space too, even though $3$-space admits stranger-looking surfaces than the real plane.

We built several additional Dimensional Bridges in $3$-space today. From every bridge we derive an analogous Dirac delta distribution:

The superscript $3$ on these Dirac deltas indicates they should be volume-integrated (as they are). The notation also allows us to quickly determine the physical units of any given Dirac delta. Let’s focus on $\delta_C^2$ and $\delta_C^3$ first. If you’ll allow me to slip into inexact physics notation for a moment, the weights generated by these Dirac deltas go like this:

where each $\chi$ is a characteristic function. Because characteristic functions only output $0$ or $1$, they’re unitless. Therefore, $\delta_C^2$ has units of width$^{-1}$, while $\delta_C^3$ has units of width$^{-2}$. Generally, if the Dimensional Bridge that generates a Dirac delta requires removing $n$ dimensions, then the Dirac delta will have units of width$^{-n}$.

V. Some Final Generic Properties of Integrals

I wouldn’t be able to forgive myself if I concluded this integral calculus series without mentioning certain important integral properties. In particular, because integrals are a special kind of weighted sum, they inherit all of the nice properties that sums have, including linearity. 

By linearity, I mean that sums respect addition and scalar multiplication. Given two finite collections of real numbers labelled $\{a_i\}$ and $\{b_i\}$ and any real number $\kappa$, the following statements are true:

Both of these properties carry through to integrals in the form of weights, and therefore functions. A technical way to express this is to say that integrals are linear operators with respect to functions. Given two real functions $f$ and $g$ defined on a subset $\alpha$ of $\mathbb{R}^3$ and any real number $\kappa$, integrals satisfy

The fact that integrals behave nicely with addition will be extremely useful for us next week when we discuss the superposition principle of classical electromagnetism. That’s right: after a month of calculus, we’re returning to physics!

No need to worry, calculus fans. It’s only a matter of time before the derivative calculus demands our attention again...