I am using a time-dependent invertible map to define a moving geometry: ξ=f(x,t) where ξ,x have the same dimension. For example, this might be a time-varying shift and rotation ξ=R(θ(t))*(x-x₀(t)), or it might be something a little more strange, like a traveling wave deformation.

The velocity of the geometry is then ẋ=ḟ⁻¹(ξ,t)=ḟ⁻¹(f(x,t),t). Is it possible automatically compute this function for a given invertible f? I could ask the user to specify f⁻¹ and then use autodiff, but I feel like the whole thing should be easy enough to do using some form of reverse-diff… But I’m not sure.