Is it a conservative field?

(i) Let $\mathbb{D} \subset \mathbb{R}^2$ be the unit disk and $\partial \mathbb{D}$ be its positive oriented boundary. Evaluate the line integral

$\displaystyle \mathcal{J} = \ointctrclockwise \limits_{\partial \mathbb{D}} (x-y^3, x^3-y^2)\, {\rm d}(x, y)$

(ii) Can you deduce if the function

$f(x, y) =(x-y^3, x^3-y^2)$

is a conservative field using the above question?

Solution

(i) We are invoking Green’s theorem. Thus:

$\begin{aligned} \ointctrclockwise \limits_{\partial \mathbb{D}} \left ( x-y^3, x^3-y^2 \right ) {\rm d}\left ( x, y \right ) &=\iint \limits_{\mathbb{D}} \left ( \frac{\partial }{\partial x}Q(x, y) - \frac{\partial }{\partial y} P(x, y)\right )\, {\rm d}(x, y) \\ &= \iint \limits_{\mathbb{D}} \left ( 3x^2 + 3y^2 \right )\, {\rm d}(x, y)\\ &= 3\int_{0}^{1}\int_{0}^{2\pi} \, {\rm d} \theta \, {\rm d} \rho\\ &= 6\pi \neq 0 \end{aligned}$

(ii) If $f$ was a conservative field then then the line integral over all closed curves would have to be $0$ . But in the previous question we found one closed curve whose line integral is not $0$ . Thus $f$ is not a conservative field.

The exercise can also be found in the Jom Forum here.

Is it a conservative field?

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