TITLE: Determining from its graph whether a vector field is conservative
QUESTION [7 upvotes]: Given the graph of a vector field, how can I tell whether it is conservative or non-conservative?

REPLY [4 votes]: One cannot always make this determination visually, but one can apply some ad hoc tests that guarantee one or the other. For example (here we assume the given vector field $\bf F$ is $C^1$):

If one can find, for example, an smooth, oriented loop $\gamma$ such that at every point of $\gamma$ the unit tangent vector makes a zero, acute, or right angle, and at least at one point where the vector field is nonzero makes a zero or acute angle with the vector field, then the vector field cannot be conservative: This condition guarantees that $\int_{\gamma} {\bf F} \cdot d{\bf s} > 0$, but for a conservative vector field that integral is zero for all loops $\gamma$.
If the vector field is invariant under rotation about some point, then it is conservative: By translating we may take the distinguished point to be the origin, and by construction $\bf F$ has potential $f\left(\sqrt{x^2 + y^2}\right)$, where $f(r) := \int_a^r {\bf F}(x, 0) \cdot d{\bf x}$, where $d{\bf x}$ is the infinitesimal vector pointing in the positive $x$-direction and $(a, 0)$ is some point in the domain of $\bf F$ on the positive $x$--half-axis. (Strictly speaking, this construction assumes that the domain of $\bf F$ is connected.)
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