Thread: Can someone give me a good definition of a conservative vector field?

1. Can someone give me a good definition of a conservative vector field?

So basically I don't really understand what a conservative vector field means. I understand how to determine if a vector field is conservative, but I do not really know why Py = Qy etc makes a vector field conservative.

And I get that a conservative vector field is path independent. But a nice dumbed down definition would set the record straight for me.

2. If you have two points on a graph, we could connect them with a straight line, concave up line, and concave down line. The line integral of the path really doesn't matter if it is conservative. That means you can take any path and arrive at the same answer. Also, conservative vector fields are irrotational. It follows Green's Theorem that the double integral of conservative vector fields are equal to zero since the partial derivatives are equal.

3. I personally dislike using physics terms for mathematics concepts!

In physics, a force field is "conservative" if and only if the work done in moving from point A to point B is independent of the path used.

In mathematics, a differential field (which can be expressed as a vector field) is "exact" if and only if the integral from point a to point B is independent of the path used.

4. Thank you. I didn't realize at first was that the main point of a conservative vector field is path independence.