I'm very interested to know how the concept of vector field was created. What problems or phenomena led mathematicians to introduce the concept of vector field?
I would point to Euler's Principes généraux du mouvement des fluides, Mém. Acad. Sci. Berlin 11 (1755) (1757) 274-315 for a pretty clear notion of a fluid's velocity field (whose integral curves are the lines of flow, and satisfying what we now call Euler's equations and the continuity equation).
O. Darrigol in Worlds of flow: A history of hydrodynamics from the Bernoullis to Prandtl has some of this anticipated in earlier memoirs of Euler and D'Alembert, but I find the modern concept quite a bit harder to identify there.