Make Derivative axis-free

[dlfivefifty]

I think Derivative should behave like UniformScaling and not require axes.

[jagot]

👍, but how does that scale to multiple dimensions? I.e. how do you distinguish between partial derivatives, along e.g.x only, and the gradient operator?

[dlfivefifty]

Good question. I suppose we could have:

D = Derivative()
D2 = D^2 # stored lazily as `applied(^,D,2)`
Dx = D ⊗ I # stored lazily as `applied(⊗,D, I)`
Dy = I ⊗ D  # stored lazily as `applied(⊗,I,D)`
∇ = [Dx; Dy] # stored lazily as `applied(vcat,Dx,Dy)`

Alternatively, we could try to do things "properly" with differential forms...

[jagot]

I guess the ⊗ approach works for curvilinear coordinates (polar/spherical coordinates, etc), but if we ever want to do differential geometry, i.e. with a spatially varying metric, we need differential forms (?)

[dlfivefifty]

It's interesting to think what differentiable forms would look like in quasi-array language: I'm not aware of finite-dimensional analogues of differential forms. Though we'll still need gradients and partial derivatives anyways so maybe best not to overthink it: just have Gradient and Laplacian types?

[jagot]

Yes, I agree that Gradient and Laplacian will be enough for multiple dimensions, for now. I guess that PartialDerivative will still need to be attached to an axis though?

[dlfivefifty]

Yes, since partial derivatives are inherently coordinate dependent they probably don’t need their own type, and the construction above is fine