Line Integral of Vector Field: Is 0 a Meaningful Value?

[madachi]

Can line integral of a vector field ever be zero? If can, what is the interpretation of this value (0) ?

Thanks.

 

[HallsofIvy]

That's a very peculiar question. One of the first things students learn about path integrals is that if a f(x,y)dx+ g(x,y)dy is an "exact differential" (There exist a function F(x,y) such that $\partial F/\partial x= f(x)$ and $\partial F/\partial y= g(x,y)$ then the integral around any closed path is 0. And, since you refer to "vector fields", we can think of that as the integral of the vector field $f(x,y)\vec{i}+ g(x,y)\vec{j}$ over the path.

As for the interpretation, that would depend upon the interpretation of the vector field wouldn't it? If the vector field is a force field, in a physics problem, then the integral along a path is the work done in moving along that path against that force. In particular, if the integral is 0, there is no work done and if moving around any closed the integral is 0, then the force field is a "conservative" force field.