Is F = ∇f if DF1/Dy = DF2/Dx for F(x,y) = (ycos(x), xsin(y))?

[ak123456]

Homework Statement

consider a function F : R^2 $$\rightarrow$$R^2 given as F(x,y)=(F1(x,y),F2(x,y)).Show that if F=$$\nabla$$f for some function f : R^2$$\rightarrow$$R,then
(for partial derivative )
DF1/Dy=DF2/Dx
show that F(x,y)=(ycos(X),xsin(y))is not the gradient of a function

Homework Equations

The Attempt at a Solution

i don't know how to set about this question
any clue ?

 

[yyat]

I am assuming F has continuous partial derivatives (otherwise the result is wrong). The second partial derivatives of f are symmetric (http://en.wikipedia.org/wiki/Symmetry_of_second_derivatives#Clairaut.27s_theorem"). What does this tell you about the first partial derivatives of F?