I am doing a math project on volume preserving transformations. It seemed like a good topic at the time but all I really know is that to preserve volume the jacobian must be 1 and that seems to be the end of the story.

My project was to investigate the "solution" to the jacobian differential equation = 1. But to me its just the set of all vector fields (if thats the right word) that preserve volume.

I found some stuff on representation theorems for volume preserving transformations but I don't know what to do with that either. Its just another equation.

Does anyone know any applications of representation theorems for volume preserving transformations or anything about the subject at all?