# Transformation of derivatives of coordinates

I am quite new to this topic. Please bear with me.

Suppose we are given a transformation of both time and space coordinate's derivatives as $$\partial_t\to D_t=\partial_t-f(t,x)\partial_t\\ \nabla\to \mathbf{D}=\nabla+\mathbf{F}(t,x)\partial_t \tag{1}$$ $$f$$ is a scalar field and $$\mathbf{F}$$ is a vector field. Is there anyway to know what kind of coordinate transformation made the derivative transformation look like $$(1)$$?

Personally, the time derivative transformation given in $$(1)$$ looks okay to me but the space derivative transformation looks "weird" to me. I mean the space derivative is changing into time derivative... what does it mean? Is it a normal thing? Could someone please explain what is the physical meaning of this transformation.

I can’t explain the physical meaning for all $$f$$ but I can say that Lorentz transformations also mix time and space derivatives so in a sense this is normal. This is because the derivative $$\partial^\mu$$ 4-vector is well… a 4-vector so it transforms under Lorentz transformations like $$x^\mu$$.
With that in mind, if we take the typical 1D $$x$$-boost transformation (which now we are considering needs to mix space and time coordinates), $$x' = x'(x,t)$$ Consider some function $$f$$ of $$x'$$. By the chain rule $$\frac{\partial f(x')}{\partial t'} = \frac{\partial f(x')}{\partial x}\frac{\partial x}{\partial t'} +\frac{\partial f(x')}{\partial t}\frac{\partial t}{\partial t'}$$ and $$\frac{\partial f(x')}{\partial x'} = \frac{\partial f(x')}{\partial x}\frac{\partial x}{\partial x'} +\frac{\partial f(x')}{\partial t}\frac{\partial t}{\partial x'}$$ so it is natural these derivatives mix when the coordinates mix (so the physical intuition follows from the quote above).