If the integral over an area is zero is the integral of the gradient also zero?

Question

Say I know that $\int\int v_z dx dy = 0$ over some area with $dA = dx dy$. $v_z$ is a function of $x$ that "points" in $z$. Is this enough to say that $\int\int \frac{\partial v_z}{\partial x} dx dy = 0$ ?

Thanks

Answer 1

No. Let $v_z = (0,0,x)$ and the region of integration be the unit circle. Clearly, by symmetry, $\int \int v_z \mathrm{d}A = 0$. However, $\frac{\partial v_z}{\partial x} = 1$ is positive everywhere, so its integral is not zero. (In fact, it's $\pi$.)