Conceptual question. I'm thinking about this because I'm studying directional derivatives in differential geometry.
Are vector functions uniquely determined by their partial derivatives up to some constant?
Even kind of more specifically, are vector functions uniquely determined by their Jacobian (up to a constant)?
I would think that they are because of the fundamental theorem of calculus, but I just haven't seen anything that answers my question and I'm pretty new to the material.
I do understand that directional derivatives are linear transformations and that they consider a very geometric relationship. Specifically, they consider the relationship between a collection of curves in the domain associated with some point and the corresponding initial velocities of the transformations of those vectors.
Thank you in advance for any help!