In a different business that I'm more familiar with, it's called the Geometric Product. The case of divergence and curl is just when one of the multiplicands is the 4D spacetime gradient.

In N dimensions, there are N(N-1)/2 components to the curl function. each value of the curl simply corresponds to each possible plane. Notice: 2d -> xy 3d -> xy, yx, xz 4d will have 6 components, and so on. due to the linearity of everything, the case of N+1 dimensions can be generated from N dimensions: every vector can be broken up as an element parallel to the N+1st axis, and an element perpendicular to it. The perpendicular components have a curl of the same kind as in N dimensions, and notably any curling occurring in N dimensions will have no effect on the N+1st axis. Finally, for each of the N dimensions, the N+1st axis forms a plane, which can then experience curl.

@@MrRyanroberson1 N(N-1)/2 I suppose is one way of formulating it, but I usually see it as N choose 2.

@@angeldude101 in the most literal sense, there are (n choose k) many unique K dimensional orthonormal subspaces in N dimensions, yes.

"All I can figure out is that it somehow corresponds to how many times you've switched a pair." That's pretty much exactly what it is. I personally don't like thinking about it in terms of signs at all, especially since they won't work the moment there is more than one plane the field can be rotating in. The two directions of rotation in 2D are x->y and y->x. It's not too hard to see that rotating in one direction is equivalent to rotating backwards in the other direction, so x->y = -(y->x). In practice, most representations of curl just pretend that the value is a scalar in 2D and instead represent the direction with a sign. Which direction is positive and which is negative is completely arbitrary, but convention usually sets x->y as positive and y->x as negative. df/dy is x->y and dg/dx is y->x. In order to add them into a single rotation, you need to flip one so that they have the same unit: -dg/dx in x->y or -df/dy in y->x. In 3D, x->y and x->z are along completely different planes and therefor can't be converted into each other with just a sign flip. x->y->z on the other hand is equivalent to z->y->x, while also opposite to x->z->y. In the 3D volume case, I tend to think of a helix that you can rotate all you like, but can't be mirrored without changing the sign, though that shouldn't be relevant for curl, which only cares about the 2D plane.

Ooh, nice point about the helix. Also, maybe the curl-ing itself only cares about the 2D plane, but you still need an axis. So I think your imaginary helix is still justified. I suppose the sign-switching = rotation-switching trick might not be all that helpful for explaining 3D curl. Although maybe the idea can be pushed a little further. If you travel through a helix that obeys the right-hand rule, with your thumb pointing from the origin towards [1,1,1], then the closest axis to you will follow a pattern of being ...x, y, z, x, y, z... If your thumb points towards [-1,1,1] instead, keeping with the right-hand rule, the pattern becomes ...z, y, x, z, y, x... It looks to me that the other 6 octants (I learnt a new word) all work the same: reverse a sign, and you reverse the cycle (which is equivalent to flipping a pair when you have a 3 element cycle, of course). Admittedly, that feels like a slightly contrived pattern, and I'm not sure it helps much for intuiting why 3d curl works. Ultimately, I think it's more interesting to think about how pair-switching = sign-switching when calculating each term of the determinant in any dimension. [I haven't thoroughly checked, but] I think the rule can work as a replacement of the cheeky, recursive definition of the n-dimensional determinant. Although I'm not sure there's an easy intuition for why it's true above 3D.

​@@plasticpeepsperformances "but you still need an axis." Says who? The whole point in 2D is that there is no axis. To try to find such an axis would be to invent an entire extra dimension. In 4D it gets even worse since there are infinitely many perpendicular vectors. How would you ever choose just one? The helix that I visualize is just for an oriented volume to show that it can be rotated into equivalent volumes, but can't be mirrored without changing it. The volume actually acts more like a scalar rather than a vector in some ways. In 3D, the curl is still usually just an oriented plane. In fact, the oriented volume is so much like a scalar that it's usually generated through the scalar triple product, whose use of a dot product means that it's usually addressed as the divergence rather than the curl.

@@angeldude101 Fair enough, I'll tweak what I said. With rotation in 3D, you end up with an axis (or multiple I suppose). And with 2D rotation, since it's maths, you are allowed to assert the existence of a 3rd dimension as long as it doesn't interfere with anything else (I see no inherent reason why it should). But ok, you got me, maybe rotation is better thought of fundamentally as a trade between 2 (or more?) dimensions. How would you explain it? I'd be curious to know. (You are free to point me to some resources if you don't want to explain) And I can see your helix point better now that you mention the scalar triple product. I can't say much more, unfortunately, since I'm getting to the limits of my knowledge. (Thanks for responding, btw)

@@plasticpeepsperformances I recommend looking up Geometric/Clifford Algebra. It includes an operation called the wedge/outer product that acts like the cross product in 3D, but scales to any dimension and gives an object called a bivector which acts as an oriented plane segment. x ^ y = the unit bivector in the xy plane. Since 2D only has one plane available, there's only one unit bivector. 3D has 3 orthogonal unit bivectors and each one has a correspondence to a unique orthogonal unit vector. In 4D, the number of orthogonal unit bivectors reaches 6, which is more than the number of possible vectors since there are 6 ways to combine each of the 4 basis vectors. The sign behavior also becomes rather intuitive since the bivector v ^ u is the one that rotates v to u. u ^ v instead rotates from u to v in the opposite direction. In 3D, there's actually no consensus for which basis bivectors to us, but it's doesn't actually cause a problem since the units are explicit in specifying whether you're working the the right-handed zx plane or the lexicographic xz plane. Since the ordering is explicit in the basis, it only takes a glance to tell if you're going to have to flip the sign. Regarding curl and divergence specifically, Geometric Algebra included the geometric product, which acts as a combination of the inner and outer products. As such, you can do both operations in a single step. Since the magnetic field is formed through the cross product, it can be substituted with its corresponding bivector form (which actually transforms better under reflection than "pseudovectors") and then added to the electric field without the two interfering. Similarly the time derivative can be grouped with the spatial gradient and charge can be grouped with the current (along with some unit conversions) yielding the Geometric Algebra formulation of Maxwell's Equations: ∇F = J. I'm sorry, did I say Maxwell's Equations? I meant Maxwell's _Equation._ This geometric product has an inverse by the way, so you can totally divide by the values shown. And it works in arbitrary dimensions too, though the magnetic field doesn't exist in only 1 dimension (since it requires a plane), so electrostatics are kind of boring there.

Oh hi fellow cross product hater! It's really hard to find a vector orthogonal to a plane within 2D isn't it, and it's even harder to find a unique one in 4D or higher. Having the direction made explicit with xy or yx instead of just a sign is also nice.

Another cross product hater here. Either use geometric products or wedge products