Question

Verify Green's Theorem in the plane for integral_c [(x^2-xy^3)dx +(y^2-2xy)dy] where C is a square with vertices at (0,0), (2,0), (2,2), (0,2).

Answer 1

Conditions

Verify Green's Theorem in the plane for integral_c [(x^2 - xy^3)dx + (y^2 - 2xy)dy] where C is a square with vertices at (0,0), (2,0), (2,2), (0,2).

Solution

As we know, the Green's Theorem claims:

Let $C$ be a positively oriented, piecewise smooth, simple closed curve in a plane, and let $D$ be the region bounded by $C$. If $L$ and $M$ are functions of $(x, y)$ defined on an open region containing $D$ and have continuous partial derivatives there, then:

$$\oint_{C} (L \, \mathrm{d}x + M \, \mathrm{d}y) = \iint_{D} \left( \frac{\partial M}{\partial x} - \frac{\partial L}{\partial y} \right) \, \mathrm{d}x \, \mathrm{d}y.$$

Let's check the conditions of this theorem to check whether it works for our case.

As the C is a square with vertices at (0,0), (2,0), (2,2), (0,2), then this is positively oriented, piecewise smooth (because each side of this square could be represented as a linear function on a plane), simple closed curve. So, the conditions for C is completed.

Let's check the partial derivatives for functions:

$$L(x, y) = x^2 - x y^3$$ $$M(x, y) = y^2 - 2x y$$

As these functions are polynomials, then each their derivative is a continuous function. And these functions are defined in all $R^2$, so they are defined on each open region containing $D$.

That's why the Green's Theorem is verified for our example.