The text I am reading says that a line integral,
$$\int_{C}{\mathbf{F}\cdot\textrm{ d}\mathbf{r}}$$
is path-independent whenever $\mathbf{F}$ is a gradient field (or in the realm of physics, a conservative field).
My question is, when is a line integral path-dependent? Under the characterization of path-independence I described above, it seems like every line integral must be path-independent because every vector field is the gradient field of some potential function. Or is this false?
Best Answer
Consider the vector field $\mathbf{F}=(y,-x)$. Is it the gradient field of some potential function? Note that if $C$ is a circle centered at the origin of radius $R>0$ counterclockwise oriented, then $$\int_{C}{\mathbf{F}\cdot\textrm{ d}\mathbf{r}}=\int_0^{2\pi} R^2(\sin^2(t)+\cos^2(t))\,dt=2\pi R^2\not=0.$$ So $\mathbf{F}$ is not a conservative field.