I am searching for an expression with integrals or derivatives which let me take any vector field, and calculate a vector that is the integral along the field in some sense.
The result should be a vector pointing out the sink we end up in from the current coordinate.
Simplest example: One point sink and no sources: All vectors of our integral point to the universal sink.
Second example: Planetary bodies and gravity force. Each vector points to the mass center of the body which an object at rest in that point would end up hitting.
Third example: Ideal plate capacitor, two really close electrically opposite charged plates. Here I imagine the integral result should be the vector pointing to the point on the charged plate which a charge would hit if dropped from rest at point of origin.
EDIT
Not explicitly stated before: length of travel in some sense being equal to norm of vector.
Fourth example: circular movement: A rotation $\cases{{\bf v}_x = -k\pi y\\{\bf v}_y=k\pi x}$ should follow the field for "length" $k\pi$ (or less if field hits a sink). Any point on a circle would under this "integral" rotate $k\pi$ radians along the circumference of a circle of radius $\sqrt{x^2+y^2}$.