calculus - Integral of a gradient over a plane area

Let $A$ be a plane area bounded by a curve $\partial A$. Then, is

$$ \iint_A \nabla f\, \textrm{d}x \textrm{d}y = \oint_{\partial A} f\ \hat{\mathbf{n}}\ dl $$

where $f=f(x,y)$ and $\mathbf{\hat n}$ is the outward unit normal?