how can you find if f is conservative vector field

Question

anonymous · Answer

Determine whether or not F is a conservative vector field. If it is, find a function f such that F = f. If it is not, enter NONE. 
F(x, y) = (3x2 - 3y2)i + (6xy + 3)j

anonymous · Answer

A field F is conservative if there is a function such that when you take its partials you get the field back.

in this case, F is conservation if there exists a function such that
$$\int\limits 3x^2 - 3y^2 dx$$is the same as
$$\int\limits_{}^{}6xy + 3 dy$$

anonymous · Answer

$$x^3 + 3xy^2$$
the same as
$$3xy^2 + 3y$$

?

since they are not, then this vector field is not conservative

(its been a while since calc 3, but i tried to dig up some stuff )

dan815 · Answer

a conservative vector field is a field where no matter what path you take in that field you will get the same value between those 2 points

dan815 · Answer

and any close loop line integral = 0 this is just another consequence of a conservative vector field

dan815 · Answer

now from greens theorem or stokes theorem, you know that a line integral can be broken down into 
integral of del x f da

dan815 · Answer

so this integral of del x f must = 0 then you know that for any closed loop integral of this function you get 0 so it must be a conservative vector field

dan815 · Answer

another way is if you can write it as a gradient of a function

anonymous · Answer

Curl(f) = 0  => conservative vector field