Make Derivative axis-free about continuumarrays.jl HOT 6 OPEN

👍, 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?

from continuumarrays.jl.

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...

from continuumarrays.jl.

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 (?)

from continuumarrays.jl.

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?

from continuumarrays.jl.

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?

from continuumarrays.jl.

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

from continuumarrays.jl.