Calculus III: the Dirac Delta, a Bridge Between Dimensions
Today, I'm going to teach you a magic trick. This trick connects two very different tools from two very different worlds: the area integral of two-dimensional regions and the line integral of one-dimensional curves. The time has come to join these disparate items under a common framework.
Last week, we described how to find the length of a curve using a line integral. Today’s trick tells us how to find the length of a curve using a surface integral instead.
To accomplish this, we're going to need something unlike anything we've seen before: an ethereal mathematical object called the Dirac delta distribution, which possesses the ability to bridge dimensions.
Come one, come all! The show is about to begin...
I. The Dimensional Bridge Equation: Measuring Length Using Area
In order to discuss the length of a curve, we're going to need a curve to work with. Let's call it $C$:
Like any other curve, $C$ is infinitesimally thin. However, suppose we thicken $C$ so that we obtain a similarly-shaped region with a finite (but small) width $W$:
Let's call this region $\mathbf{C}_W$. Despite their dimensional differences, $C$ and $\mathbf{C}_W$ are aesthetically similar. We expect that we should somehow "recover" $C$ in the limit that $\mathbf{C}_W$ becomes infinitesimally thin, aka $W\rightarrow 0$. To maintain this aesthetic similarity, let's partition $C$ and $\mathbf{C}_W$ in analogous fashions:
The finer we make our partition along the length of $C$, the more its partition pieces look like line segments. Simultaneously, if we keep the width of $\mathbf{C}_W$ small, the partition pieces of $\mathbf{C}_W$ look increasingly like rectangles. Let’s pursue this by focusing on one piece of $C$ and the corresponding piece of $\mathbf{C}_W$:
If $C$'s piece has a length approximately equal to the line segment length $\Delta \ell_i$, then $\mathbf{C}_W$'s piece will have an area approximately equal to $\Delta A_i \approx W\cdot \Delta \ell_i$. By combining the contributions of every partition piece in this way, we discover an approximate relationship between lengths and areas:
This relationship becomes exact when our partition is infinitely fine ($N\rightarrow\infty$) and $\mathbf{C}_W$ is infinitely thin ($W\rightarrow 0$), in which case we may write,
Now, while this is a true result, it's not very useful to us. The right-hand side always vanishes, so this equation simply says "the area of an infinitely-thin region equals zero." We'll find a more useful expression if we move all of the $W$-dependent information to one side before taking the limit. That is, let's take the limit as $N\rightarrow \infty$ and $W\rightarrow 0$ of the following expression instead:
Because the length of $C$ is entirely independent of the width of $\mathbf{C}_W$, it's unaffected as $W$ shrinks to zero. Therefore, taking the limits of both sides gives us what I'll call the Dimensional Bridge equation:
This is an awesome result! The Dimensional Bridge equation is an exact relationship between one-dimensional and two-dimensional objects.
Although we used a very specific partition to derive the Dimensional Bridge, all of the quantities appearing in the equation are partition-independent. As we continue to work with the Dimensional Bridge, we'd find our current partition cumbersome; therefore, let's develop a better partition that also simplifies the double-limit ($W\rightarrow 0$ and $N\rightarrow \infty$) buried in the right-hand side of the equation.
II. Switching to a Lattice Partition
Integration relies on a sequence of increasingly-accurate approximations of some desired quantity. We typically increase accuracy by gradually refining the partition of a relevant shape. Let's refer to each partition in this refinement sequence as a stage and let an index $n$ label each stage. We'll let $n=0$ correspond to the first partition, then let $n=1$ correspond to its subsequent refinement, and so-on.
Next, label the number of partition pieces in the $n$th stage as $N^{(n)}$. For example, that means our initial partition will divide the relevant shape into $N^{(0)}$ pieces, then the subsequent partition will utilize $N^{(1)}$ pieces, and so-on. Because it's a refinement, we'd make sure $N^{(n)}$ increases an $n$ increases. When $n\rightarrow \infty$, we obtain $N^{(n)}\rightarrow \infty$, just like before.
Now instead of dividing $\mathbf{C}_W$ into approximate rectangles along its length, let's surround $\mathbf{C}_W$ in a region $R$ and let's divide $R$ into increasingly-refined square lattices. (Note: this is the partition we used when we first defined the surface integral.)
We've seen that the Dimensional Bridge equation involves two limits: $W\rightarrow 0$ and $N\rightarrow \infty$. The $W\rightarrow 0$ limit leads to a series of progressively-thinner curves:
Meanwhile, the $N\rightarrow \infty$ limit yields a series of progressively-finer lattices:
We can combine these two procedures in a single grid, where moving downward decreases $W$ and moving rightward increases $N$:
When we calculate the right-hand side of the Dimensional Bridge equation, we ultimately want to travel down-and-right through the grid. We could do this by first heading forever rightward (in the $N \rightarrow \infty$ direction), and then heading forever downward (in the $W\rightarrow 0$ direction); this is the order implied by the equation.
Alternatively, if we're careful, we can combine the two limits into a single limit by traveling diagonally through the grid. This would mean that as $n$ increases we're simultaneous refining the partition AND thinning the shape whose area we're approximating. I've indicated such a sequence of stages in orange:
There are ways to mess this up, so we have to be very careful. As an extreme example, suppose we decide to take $W$ to zero first (so that we're travelling down the leftmost column of the grid). In this case, we'd be using the coarsest partition for every approximation. As $\mathbf{C}_W$ grows narrower, the number of squares we can fit in $\mathbf{C}_W$ will decrease until $\mathbf{C}_W$ contains no squares at all. In that case, even if we take $N\rightarrow \infty$ afterwards, the Dimensional Bridge equation would seem to imply our curve has no length! In other words, we'll break our math if the width $W$ decreases too quickly.
Therefore, we want to refine the partition quicker than we shrink $\mathbf{C}_W$. We might ensure this by choosing our widths at each stage so that $\mathbf{C}_W$ contains an increasing number of squares as $n$ increases. However we manage to do it, let's label the width we choose for the $n$th stage as $W^{(n)}$ and label the corresponding thickened curve as $C_{W}^{(n)}$.
From here on, we assume we've successfully collapsed the two limits $W\rightarrow 0$ and $N\rightarrow \infty$ into a single limit $n\rightarrow \infty$. Combining the limits like this allows us perform a powerful reorganization of the Dimensional Bridge equation. In particular, we'll use it to augment how we write the area of $\mathbf{C}_W$. To streamline this manipulation, let's introduce the characteristic function of a set in the plane.
III. Characteristic Weight: Measuring the Area of Subsets
Given a set $A$ of points in the real plane, we define its characteristic function $\chi_A$ as a function on the real plane that equals $1$ for points in $A$ and $0$ for points outside of $A$:
When $A$ is a two-dimensional region, $\chi_A$ sort of mimics a plateau sitting directly above $A$.
By construction, the surface integral of $\chi_A$ over $A$ gives us the area of $A$:
Now, suppose our two-dimensional region $A$ is contained within a larger two-dimensional region $B$, and consider integrating $\chi_A$ over $B$. Because $\chi_A$ gives zero weight to points outside of $A$, the pieces of $B$ that aren't also in $A$ contribute nothing to the integral, such that we end up with the area of $A$ again:
But what if instead $B$ contains only some of $A$? In that case, we'll only add up those pieces of $A$ that are contained in $B$, aka those pieces in the intersection of $A$ and $B$, denoted $A\cap B$:
We introduce the characteristic function because it let's us rewrite the area of $\mathbf{C}_W$. Particularly, at the $n$th stage of our partition procedure we're interested in the area of $\mathbf{C}_W^{(n)}$, which has width $W^{(n)}$ and will be approximated via a partition of $N^{(n)}$ squares. Like any other region, $\mathbf{C}_W^{(n)}$ has an associated characteristic function, which we’ll label $\chi^{(n)}_{\mathbf{C}_W}$. We may approximate the area of $\mathbf{C}_W^{(n)}$ via its characteristic function:
where $R$ can be any region that contains $\mathbf{C}^{(0)}_W$ (containing $\mathbf{C}^{(0)}_W$ ensures $R$ contains all of the thickened curves $\mathbf{C}^{(n)}_W$ because $W$ only gets smaller). Let's then apply this expression to the Dimensional Bridge equation, which allows us to write the length of $C$ as,
And just like that, we did it! We've managed to express a curve's length as a surface integral! This is the trick I alluded to in the intro. This trick (as well as its generalizations) is extremely powerful, and is used extensively throughout physics. Of course, like any other magic trick, there's more going on here than meets the eye.
IV. The Dirac Delta Distribution
To see what's up, recall how we defined the surface integral of a function $f$ over a region $R$, and compare it to our latest result:
This definition suggests that whenever we want the length of a curve $C$, we should just weight a surface integral with the following function:
Except this weight is not a function at all!
First of all, this weight depends on $n$: in other words, it changes as we refine our partition. This makes sense (it results from thinning $\mathbf{C}_W$ as we increase $n$), but it also immediately hints that something weird is afoot. In our earlier examples of integrating weight functions, the weight functions remained the same from partition to partition.
Second, this weight outputs nonsense when our partition becomes infinitely fine. For points not on the curve $C$, things aren't so bad. Consider a specific point $(x,y)\not\in C$. At some stage of refinement, $\mathbf{C}_W^{(n)}$ will be too narrow to contain $(x,y)$, but its width $W^{(n)}$ will still be finite. That means that at this stage (and every stage beyond), $\chi^{(n)}_{\mathbf{C}_W}(x,y)=0$, such that the ratio $\chi^{(n)}_{\mathbf{C}_W}(x,y)/W^{(n)}$ vanishes, and
Functions are allowed to output zero, so this isn't really a problem. Unfortunately, the circumstances are worse for points on $C$. The characteristic function $\chi_{\mathbf{C}_W}(x,y)$ always equals $1$ for points on $C$, even as the width $W^{(n)}$ becomes infinitesimally small. As a result, their ratio grows larger and larger, and the weight diverges!
Real-valued functions are not allowed to output infinities!
Therefore, this object we're integrating is definitely not a function. This is the secret at the heart of the trick: instead of a function, we've utilized a mathematical object called a distribution.
Distributions tell us how to weight partition pieces during our integration procedure. While all functions are distributions, many distributions are not functions. This is because distributions need only make sense when integrated. One consequence of this freedom is that weights from a distribution can vary between different partitions (as we saw earlier).
Particularly, the distribution we derived today is the Dirac delta distribution $\delta_C$ corresponding to the curve $C$. It's defined by the expression we found:
However, the utility of the Dirac delta comes from the Dimensional Bridge equation, which (when applied to our definition) yields a more popular expression of the Dirac delta distribution:
Note that I’ve written the line integral over $C\cap R$ instead of simply $C$. This reflects the fact that (just as we saw with the characteristic functions) the only parts of a curve $C$ that can contribute to an integral are those parts that are included in the integration region $R$.
It is through that last equation that a Dirac delta connects objects of different dimensions. It is important to recognize that by definition the Dirac delta distribution only makes sense within integrals. It's literally meaningless otherwise. This fact unfortunately doesn't stop physicists from trying to manipulate Dirac deltas outside of integrals. It's a dangerous practice that can lead to nonsensical results if the physicist isn't careful. Here at SineOfPsi, we'll be more cautious than the average physicist.
Finally, I want to point out that all of our arguments above still work if we multiply the weight $\chi^{(n)}_{\mathbf{C}_W}$ by a function $f(x,y)$ (so long as it's defined on all of $R$), in which case we derive a relationship between generic line integrals and surface integrals:
We'll be using this equation (and generalizations thereof) A LOT on SineOfPsi. Dirac deltas show up all over the place in physics. For example, in classical electromagnetism we often model point charges using Dirac deltas. But before we do that, we have to tie up a few loose ends regarding the integral calculus; that will be the topic of next week's post. Until then, best wishes!
Thanks for reading today’s post! Follow sineofpsi.tumblr.com for new physics content every Friday. Have questions about anything we’ve talking about? Send me an ask. I’m wishing you the best!
