Section 5.6 Green’s Theorem
The following theorem provides a simplification technique for some vector line integrals.
Theorem 5.6.1. Green’s Theorem.
For a simple, closed region \(D\) inside a curve \(C\) parameterized so that \(D\) is on the left of \(C\) and a vector field \(\vec{F}=(M(x,y),N(x,y))\) whose derivatives are continuous in the region
\begin{align*}
\oint_C \vec{F} \cdot d\vec{r} = & \int_a^b \vec{F}(x(t),y(t)) \cdot \left( \frac{dx}{dt}, \frac{dy}{dt} \right) \; dt\\
= & \int_a^b M(x(t),y(t)) \; dx + N(x(t),y(t)) \; dy\\
= & \dint_D \left( \frac{\partial N}{\partial x}-\frac{\partial M}{\partial y}\right) \; dx \: dy.
\end{align*}
Activity 77.
Confirm the theorem by completing the following steps. \(\vec{F}(x,y)=(xy,y^2) \) enclosed within \(y=x, y=x^2. \)
(a)
Parameterize the curves.
(b)
Integrate the line integral \(\int_a^b \vec{F}(x(t),y(t)) \cdot \left( \frac{dx}{dt}, \frac{dy}{dt} \right) \; dt \)
(c)
Integrate \(\dint_D \left( \frac{\partial N}{\partial x}-\frac{\partial M}{\partial y}\right) \; dx \: dy \)
(d)
Do they match?
Exercises Exercises
1.
Integrate \(\vec{F}(x,y)=(2xy+2,x^2+3x+1)\) along the triangle with vertices \((-3,1),\) \((3,1),\) and \((1,3)\) traversed counterclockwise. Do not use Green’s Theorem.
2.
Repeat the previous problem using Green’s Theorem.
