How do I apply Green's theorem to evaluate $\iint_{\Omega} xy\ dxdy$ where $\Omega$ is a triangle in the $z=0$ plane?

Question

I specifically want to know how I can apply Green's Theorem to solve the above when $\Omega$ is an arbitrary triangle in the $z=0$ plane with vertices $\{\mathbf{v}_0,\mathbf{v}_1,\mathbf{v}_2\}$ where $\mathbf{v}_i = [v_{i,x}, v_{i,y}]$.

Answer 1

Green's Theorem in the Plane:

Let $\Omega$ be a closed, bounded domain in the $(x,y)$ plane with piecewise smooth boundary $\partial \Omega$. Then $$\iint_\Omega \bigg(\frac{\partial p}{\partial x} + \frac{\partial q}{\partial y}\bigg) \, dx dy = \oint _{\partial \Omega} (p,q) \cdot \mathbf n \, ds$$ where $\partial \Omega$ is positively-oriented, and $\mathbf n$ is the outward unit normal to $\partial \Omega$.

In your case, $\Omega$ is the triangular domain, and $\partial \Omega$ the edges of the triangle.

Let $\partial \Omega = C_1 \cup C_2 \cup C_3$ where $C_1, C_2,C_3$ are the three edges (oriented counter-clockwise) with corresponding outward unit normals $\mathbf n_1, \mathbf n_2, \mathbf n_3$. We then have

$$\iint_\Omega xy \, dxdy = \iint_\Omega \bigg[\frac{\partial}{\partial x}\bigg(\frac 12 x^2y\bigg) + \frac {\partial}{\partial y}(0)\bigg] \,dxdy = \sum_{i=1}^3 \int_{C_i}\bigg(\frac12 x^2y \, , \, 0\bigg) \cdot \mathbf n_i \, ds$$

and it is then a matter of finding $\mathbf n_i$ for each edge and parameterising each edge, then computing the integrals.