Conservative field

(i) Let $f \in \mathcal{C}^2(\mathbb{R})$ such that ${\rm div(grad \;f)}=0$ . Let $\mathbb{D} \subseteq \mathbb{R}^2$ be a $\mathcal{C}^1$ normal area. Prove that

$\oint \limits_{\partial \mathbb{D}} \left ( \frac{\partial f}{\partial y} , -\frac{\partial f}{\partial x} \right ) \cdot {\rm d}(x, y) =0$

(ii) Examine if $\bar{f}(x, y)=(2x \cos y , -x^2 \sin y)$ is a conservative field. If so, find a force of it.

Solution

(i) Using Green’s theorem we have that

$\begin{align*} \oint \limits_{\partial \mathbb{D}} \left ( \frac{\partial f}{\partial y} , -\frac{\partial f}{\partial x} \right ) \cdot {\rm d}(x, y) &= \iint \limits_{\mathbb{D}} \left ( -\frac{\partial^2 f}{\partial x^2} - \frac{\partial^2 f}{\partial y^2} \right )\, {\rm d}(x, y) \\ &= \iint \limits_{\mathbb{D}} \left ( -\nabla^2 f \right ) \, {\rm d}(x, y)\\ &= 0 \end{align*}$

(ii) Yes, it is. We note that

$\nabla \left ( x^2 \cos y \right ) = 2x\cos y - x^2 \sin y$

and of course a force is $g(x, y)=x^2\cos y$ .

Conservative field

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