An extremely important way of defining curves is through a vector field. In the plane, this means attaching an arrow to every point and interpreting it as a velocity. Here, velocity is meant in its physical sense: it has a direction, an orientation, and a magnitude (the last of these being proportional to the length of the arrow). We can therefore picture a vector field as specifying, at each point, the velocity of the water in a river whose flow does not vary over time.
Approximating the path
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Vector field modeling the flow of a river.
A cork placed on the surface of this river will follow the path dictated by the vector field. A natural question is this: given the field and the cork's initial position B0, what will that path be?
