Rules for curl test applicability

[mathwizeguy]

trying to remember rules for curl test applicability.

is it just simple closed curve?

is F=-ysin(x)i+cos(x)j able to use the curl test?

 

[HallsofIvy]

It would help if you would cite the "curl test" you are talking about. I think you are referring to the fact that a two variable differential, f(x,y)dx+ g(x,y)dy, is an "exact differential", that is, that there exist F(x,y) such that dF= f(x,y)dx+ g(x,y)dy, if and only if
$$\frac{\partial f}{\partial y}= \frac{\partial g}{\partial x}$$
which the same as saying
$$curl \vec{f}= \nabla\times\vec{f}= \vec{0}$$ where
$$\vec{f}= f(x,y)\vec{i}+ g(x,y)\vec{j}$$.
It follows from that that the integral of f(x,y)dx+ g(x,y)dy around any closed path is 0 and that the integral from one point to another in the xy-plane is independent of the path. Which of those are you referring to?

 

[mathwizeguy]

i was referring to the df/dy-dg/dx.

 

[HallsofIvy]

Actually, that doesn't answer my question- were you referring to the fact that if df/dy- dg/dx= 0 implies that the integral around a closed path is 0 or to the fact that it implies the integral, from one point to another, is independent of the path.

The first obviously requires a closed path- it says so! The second does not.