What’s an intuitive way to understand determinants, div’s, and curl’s?

Answer by Alex Kritchevsky:

This will be a big answer if I get through it.. you asked about six different questions.

All of these concepts are extremely geometric, and it's a damn shame they aren't taught that way.

Any (square) matrix is a transformation on vectors: [math]\mathbf{A}x=y[/math]. In n dimensions, the vector x is a linear combination of n basis vectors, and since matrix multiplication is linear you can figure out what A does to x by the sum of what it does to each basis vector.

Imagine that you start with basis vectors x, y, and z, each at right angles to each other. Applying the matrix takes those three lines and puts them somewhere in space. It might stretch them any direction, even unevenly. It might twist them around each other or invert one. It might flatten the three down into a plane. It might delete two and just leave the last one. It might delete all three. It might rotate them without changing their lengths at all. It might stretch one by 2x and leave the other two the same. It might rotate all three.

The determinant is a scalar that wraps up all of these things, and it can be extremely intuitive. If you start with x,y,z, the determinant of A is the signed volume of the parallelohedron <Ax, Ay, Az> after the transformation. "Signed' volumes are like regular volumes but if you mirror the world they become negative.

Let's go through the list from above..

  • It might stretch them any direction, even unevenly. (det A = anything)
  • It might invert one direction. (det A = -1 – the parallelogram has the opposite orientation) 
  • It might flatten the three down into a plane. (det A = 0. It's a 'projection' onto the plane. A is singular, aka non-invertible. You took away one dimension – how could you invert it? If you took a vector in the plane you have no idea what to map it back to; you lost a dimension of information).
  • It might delete two and just leave the last one. (det A = 0, you projected onto a line)
  • It might delete all three. (A = 0)
  • It might rotate them without changing their lengths at all. (det A = 1, it's  a rotation and doesn't change the volume at all)
  • It might stretch one by 2x and leave the other two the same. (det A = 2, the volume doubled)
  • It might rotate all three. (det A = 1, rotations don't change volume).

And, by the way:
Two transformations can be composed and they change volumes by multiplying the amounts (det AB = det A det B).
And of course, inverting a transformation undoes the volume transformation. ([math] det A^{-1} = (det A)^{-1} [/math]).

So that's neat.
… now, let's get more advanced. I'm sort of glossing here because this stuff is pretty complicated to  with any sort of rigor.

0. Suppose you have a scalar.
1. And you have three vectors.
2. Then you take their "wedge product", [math]a\wedge b[/math], to get three bivectors. What the hell is a bivector? Why, it's an "area vector". When you take the cross product of the sides of a parallogram and the result is an 'area vector pointing up', that's really a bivector. It's not quite the same thing. Why not?

Suppose you turn around your axes so every vector becomes the negative version of itself. You've performed a 'spatial inversion', a kind of discrete transformation. All your vectors get negative: [math]x \rightarrow -x [/math]. But all your bivectors stay the same: [math] b \rightarrow b [/math]. So trust me – there's something different about bivectors. If you don't believe me, draw it out.

Bivectors are called 'pseudovectors' in physics cause they act different. These are your angular momentum and things like that – anything that came from a cross product.

"Technically speaking", the cross product is:
[math] a \times b = *(a \wedge b)[/math]. You take the wedge product to get a bivector, and then you map it back to a vector using another funky operation called the Hodge Star. But don't worry about that…

3. You can do one more level of wedge producting: take the wedge product of a vector and a bivector. If they're not in the same plane, you have three sides and you get an oriented volume, as I mentioned above. This is your parallohedron-volume. This is your triple product, [math] a \cdot b \times c [/math]. This is the determinant of the matrix with a, b, and c as columns.

In physics we call that a pseudoscalar. It's a lot like a scalar, except that if you invert space it flips its sign.

Oof, this is long. I'll try to come back and hit gradient, divergence, and curl later. (hint, they're all the same thing. As are the divergence, Green's, and Stokes' theorems..).

What's an intuitive way to understand determinants, div's, and curl's?

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