Gobbo Is Bad At Math

seriously i'm like a decade out of practice and goign to be slamming my head against the wall. idfk why you're here.

[grandgobbobarb]

I'll be on cohost

[grandgobbobarb]

You should be too. Especially cis folk.

[mathgobbobarb]

"For constant area and material properties" seems like it's doing a lot of work

But then i think about when it's not. I have trouble thinking of a real situation where it's not locally true. So we could chop it up and stitch it together under appropriate emergent boundary conditions.

[mathgobbobarb]

Hodge Dual baybeeeeeee

I'm back in business and it's even more convenient

Muwahahaha I'm absolutely fucking everything up because I don't remember my hodge star

[mathgobbobarb]

The heat equation has the same teardrop symmetry that I am annoyed by with navier stokes, but it's simpler.

[mathgobbobarb]

If you think about my insanity regarding not treating time special, and let v = (v,1), we just get

v • □u = 0

I used □ cuz I ain't got a del on my phone

[mathgobbobarb]

Lukaszewicz & Kalita Navier-Stokes C2S1

Let's not give time a special status and instead define (x1,x2,x3,x4) and the transformations x(X). Then we can consider a set of paths pi for allowing any individual component of x to vary while the others remain fixed.

Just because our perception of time seems superficially special doesn't mean I gotta treat it so.

This is the source of my derangement.

Here is my first headache. I wanna just go X = X(x). Does that work?

for i=/= 4 it is the same. So I'm basically just requiring that I can do for time what I do for the others. Which is EXACTLY what I'm wanting to assume from the start so DUH BARBIE.

The determinant form remains unchanged too we just have an extra layer to the matrix

the determinant considered in the text is the determinant of space, the coefficient of the dt/dT portion. the mixed determinants have the dx/dT, dy/dT, etc. if Those are 0 then it collapses to the same that the authors consider.

here's the first annoying deviation. By giving special consideration to time, we have F(X_space, t) instead of F(X_space, T) = F(X).

need to be considerate of the material Time as well as the material Space.

I need a little bit extra to account for t vs T. I generally suspect that in most cases L&K cosider here dT/dt = dt/dT = 1, but in the case of generality I want to keep it in my head that they're using this.

It's probably not *super* important until we consider the material derivative

because we're not giving special consideration to time, pulling df/dt out is meaningless. especially since we have a u_t = 1 sitting in front. so

our material derivative becomes a scaled gradient. Ain't that nice and comfy?

This is probably something else to hold on to

so we can bring in our Material Time derivative of f by scaling our scaling factor and our spacial coordinates.

This probably needs derivation, but it seems like a good idea to make an equivalent material derivative for material space and time. I suspect V and v are the same, but I don't wanna assume that without doing a derivation that I don't have time for right now since I wanna get this chapter re-read done before work.

I'm gonna go out on a limb and say that our revision is more elegant.

Now there is a bit of ??? about if our new divergence works, but if u_t = 1 then of course it does so it's fine.

There are Exercises to do after work. In addition I want to poke at my new divergence and at the v vs V question.