Help connecting vector fields in ODE and Vector Calc

[musik132]

The vector field F=<y,x> looks exactly like the the direction field for the system
dY/dt = {dx/dt = y}
{dy/dt = x}
A few questions on this:
Are the direction field of a system of ODE's the same as a vector field of calculus?
In vector calc we take the line integral of a vector field over some trajectory to find something like work. So if the direction field and vector field are the same does it mean that if given initial values and the work I can find the trajectory?

One more question: Suppose the potential of F=<y,x> is f=xy. The level curves of this function f are hyperbolas. When plotted they look like the isoclines of the ODE. If they are then do all isoclines of an ODE which is in the form of a conservative field, have isoclines that are formed by level curves of the potential? Does this extend to non conservative fields in anyway?I had trouble phrasing this question in google to find some relevant sources, does anyone have a good source to read up on this connection, if there is one?Thanks ahead of time

 

[HallsofIvy]

Yes, the "vectors" in ODEs are the same as "vectors" in Calculus. Yes, the level curves are curves of constant potential and so are "isoclines". You can also think of them as being the curves of constant altitude in a geodetic map.