Suppose $\vec{V}=p(x,y)\hat{i}+q(x,y) \hat{j}$ is a continuously differentiable vector field defined in a donmain D in $\mathbb{R}^2$. Which of the following statements is NOT equivalent to the remaining ones?
(a) There exists a function $\phi (x,y)$ such that $\frac{\partial \phi}{\partial x}=p(x,y)$ and $\frac{\partial \phi}{\partial y}= q(x,y)$ for all $(x,y)\in D$
(b)$\frac{\partial p}{\partial y}=\frac{\partial q}{\partial x}$
(c)$\oint_{C} \vec V.d\vec r=0$ for every piecewise smooth closed curve C in D
(d) The differential $pdx+qdy $ is exact in D
My try
Let (a) is holds i.e.
$\frac{\partial \phi}{\partial x}=p(x,y)$ and $\frac{\partial \phi}{\partial y}= q(x,y)$
$\Rightarrow \frac{\partial \phi}{\partial y \partial x}=\frac{\partial p}{\partial y} \ and\ \frac{\partial \phi}{\partial x \partial y}=\frac{\partial q}{\partial x} $
Thus $(a)\Rightarrow(b)$
$(b)\Rightarrow (d)$ Therefore $(a)\Rightarrow (b)\Rightarrow (d)\Rightarrow (a)$
Now let statement (c) holds then there $\vec V=\bigtriangledown \phi$
Which implies all the above statements .
Anyone can tell me which is not equivalent to remaining ones?
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