How can I check whether the given function or vector field is path-dependent or path-independent? I understand that the fundamental theorem on line integrals can be used only to those vector fields that are path-independent.
It would be so troublesome if I go on checking different paths and then stating it as path-independent. Is there any methods/techniques using which I can clearly differentiate a path-independent vector field?
 A: If you're working in 3-D Euclidean space, then a vector field $\vec{A}: \mathbb{R}^3 \to \mathbb{R}^3$ can be written as the gradient of a scalar field if and only if its curl is zero:
$$
\vec{\nabla} \times \vec{A} = 0 \Leftrightarrow \vec{A} = \vec{\nabla} f \text{ for some $f: \mathbb{R}^n \to \mathbb{R}$.}
$$
Proving the arrow going to the left above is relatively easy (it follows from assuming that the mixed partials of $f$ are equal).  Proving the arrow going to the right is noticeably harder, but can still be done.  
This can easily be extended to 2-D Euclidean space by briefly "pretending" that you have a vector field that doesn't depend on a third Euclidean coordinate, taking the curl, and seeing if it vanishes.  In higher-dimensional Euclidean spaces, the idea of the "curl" must be replaced by something called the exterior derivative, but a similar statement still holds.  
And in non-Euclidean spaces, the failure of the above statement to be true tells you something deep and beautiful about the topology of the space;  the study of this phenomenon is called de Rham cohomology, which I never miss an opportunity to mention if I'm given an opening because I think it's so darn cool.
