James Camacho's Shortform

programjames

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There is a lot of confusion around the determinant, and that's because it isn't taught properly. To begin talking about volume, you first need to really understand what space is. The key is that points in space like aren't the thing you actually care about—it's the values you assign to those points. Suppose you have some generic function, fiber, you-name-it, that takes in points and spits out something else. The function may vary continuously along some dimensions, or even vary among multiple dimensions at the same time. To keep track of this, we can attach tensors to every point:

d x_{1} \otimes d x_{2} \otimes \dots \otimes d x_{n}

Or maybe a sum of tensors:

d x_{1} \otimes d x_{2} \otimes \dots \otimes d x_{n} + d x_{2} \otimes d x_{1} + \dots

The order in that second tensor is a little strange. Why didn't I just write it like

d x_{1} \otimes d x_{2} ?

It's because sometimes the order matters! However, what we can do is break up the tensors into a sum of symmetric pieces. For example,

d x_{2} \otimes d x_{1} = \frac{1}{2} [d x_{1} \otimes d x_{2} + d x_{2} \otimes d x_{1}]      symmetric - \frac{1}{2} [d x_{1} \otimes d x_{2} - d x_{2} \otimes d x_{1}]      antisymmetric .

To find all the symmetric pieces in higher dimensions, you take the symmetric group $S_{n}$ and compute its irreducible representations (re: the Schur-Weyl duality). Irreducible representations are orthogonal, so each of these symmetric pieces don't really interact with each other. If you only care about one of the pieces (say the antisymmetric one) you only need to keep track of the coefficient in front of it. So,

d x_{2} \otimes d x_{1} = - \frac{1}{2} d x_{1} \land d x_{2}      antisymmetric + other pieces .

We could also write the antisymmetric piece as

+ \frac{1}{2} d x_{2} \land d x_{1}      reverse order!

and this is where the determinant comes from! It turns out that our physical world seems to only come from the antisymmetric piece, so when we talk about volumes, we're talking about summing stuff like

d x_{1} \land d x_{2} \land \dots \land d x_{n} .

If we have vectors

\begin{matrix} a_{1} & = a_{11} d x_{1} + a_{12} d x_{2} + \dots + a_{1 n} d x_{n} a_{2} & = a_{21} d x_{1} + a_{22} d x_{2} + \dots + a_{2 n} d x_{n} ⋮ a_{n} & = a_{n 1} d x_{1} + a_{n 2} d x_{2} + \dots + a_{n n} d x_{n} \end{matrix}

then the volume between them is

a_{1} \land a_{2} \land \dots \land a_{n} .

Note that

d x_{i} \land d x_{i} = - d x_{i} \land d x_{i} ⟹ d x_{i} \land d x_{i} = 0

so we're only looking for terms where no $d x_{i}$ overlap, or equivalently terms with every $d x_{i}$ . These are

\sum σ \in S_{n} (a_{1 σ_{1}} d x_{σ_{1}}) \land (a_{2 σ_{2}} d x_{σ_{2}}) \land \dots \land (a_{n σ_{n}} d x_{σ_{n}})

or rearranging so they show up in the same order,

\sum σ \in S_{n} (- 1)^{s g n (σ)} n \prod i = 1 a_{i σ_{i}} n ⋀ i = 1 d x_{i} .

To begin talking about volume, you first need to really understand what space is.

No, stop it, this is a terrible approach to math education. "Ok kids, today we're learning about the area of a circle. First, recall the definition of a manifold." No!!

2Algon1y

I always liked this way of looking at the determinant.

4speck14471y

I think this is the other way around. We could decide to only care about the alternating tensors, sure, but your explanation for why we care about this component of the tensor algebra in particular is just "it turns out" that the physical world works like this. I'm trying to explain that we could have known this in advance, it's natural to expect the alternating component to be particularly nice. I think my answers to your other two questions are basically the same so I'll write a longer paragraph here: We can think of derivatives and integrals as coming from a boundary operator, sure, but this isn't "the" boundary operator, because it's only defined up to simplicial structure and we might choose many simplicial structures! In practice we don't depend on a simplicial structure at all. We care a lot about doing calculus on (smooth) manifolds - we want to integrate over membranes or regions of spacetime, etc. With manifolds we get smooth coordinate charts, local coordinates with some compatibility conditions to make calculus work, and this makes everything nice! To integrate something on a manifold, you integrate it on the coordinate charts via an isomorphism to Rn (we could work over complex manifolds instead, everything is fine), you lift this back to the manifold, everything works out because of the compatibility. Except that we made a lot of choices in this procedure: we chose a particular smooth atlas, we chose particular coordinates on the charts. Our experience from Rn tells us that these choices shouldn't matter too much, and we can formalize how little they should matter. There's a particularly nice smooth atlas, where every point gets local coordinates that are essentially the projection from its tangent space. Now what sorts of changes of coordinates can we do? The integral maps nice functions on our manifold to our base field (really we want to think of this as a map of covectors). That should give us a linear map on tangent spaces, and this map should