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# 3.8: Extensions and Applications of Green’s Theorem

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## Simply Connected Regions

##### Definition: Simply Connected Regions

A region $$D$$ in the plane is simply connected if it has “no holes”. Said differently, it is simply connected for every simple closed curve $$C$$ in $$D$$, the interior of $$C$$ is fully contained in $$D$$.

##### Examples Figure $$\PageIndex{1}$$: Examples

$$D_1$$ - $$D_5$$ are simply connected. For any simple closed curve $$C$$ inside any of these regions the interior of $$C$$ is entirely inside the region.

Note: Sometimes we say any curve can be shrunk to a point without leaving the region.

The regions below are not simply connected. For each, the interior of the curve $$C$$ is not entirely in the region. Figure $$\PageIndex{2}$$: Not simply connected regions. (CC BY-NC; Ümit Kaya)

## Potential Theorem

Here is an application of Green’s theorem which tells us how to spot a conservative field on a simply connected region. The theorem does not have a standard name, so we choose to call it the Potential Theorem.

##### Theorem $$\PageIndex{1}$$: Potential Theorem

Take $$F = (M, N)$$ defined and differentiable on a region $$D$$.

1. If $$F = \nabla f$$ then $$\text{curl}F = N_x - M_y = 0$$.
2. If $$D$$ is simply connected and $$\text{curl} F = 0$$ on $$D$$, then $$F = \nabla f$$ for some $$f$$.

We know that on a connected region, being a gradient field is equivalent to being conservative. So we can restate the Potential Theorem as: on a simply connected region, $$F$$ is conservative is equivalent to $$\text{curl} F = 0$$.

Proof

Proof of (a): $$F = (f_x, f_y)$$, so $$\text{curl} F = f_{yx} - f_{xy} = 0$$.

Proof of (b): Suppose $$C$$ is a simple closed curve in $$D$$. Since $$D$$ is simply connected the interior of $$C$$ is also in $$D$$. Therefore, using Green’s theorem we have,

$\oint_{C} F \cdot dr = \int \int_{R} \text{curl} F\ dA = 0.$ Figure $$\PageIndex{3}$$: Potential Theorem

This shows that $$F$$ is conservative in $$D$$. Therefore, this is a gradient field.

Summary: Suppose the vector field $$F = (M, N)$$ is defined on a simply connected region $$D$$. Then, the following statements are equivalent.

1. $$\int_P^Q F \cdot dr$$ is path independent.
2. $$\oint_{C} F \cdot dr = 0$$ for any closed path $$C$$.
3. $$F = \nabla f$$ for some $$f$$ in $$D$$.
4. $$F$$ is conservative in $$D$$.
If $$F$$ is continuously differentiably then 1, 2, 3, 4 all imply 5:
5. $$\text{curl} F = N_x - M_y = 0$$ in $$D$$

## Why we need simply connected in the Potential Theorem

If there is a hole then $$F$$ might not be defined on the interior of $$C$$. (Figure $$\PageIndex{4}$$) Figure $$\PageIndex{4}$$: A hole in the region. (CC BY-NC; Ümit Kaya)

## Extended Green’s Theorem

We can extend Green’s theorem to a region $$R$$ which has multiple boundary curves.

Suppose $$R$$ is the region between the two simple closed curves $$C_1$$ and $$C_2$$. Figure $$\PageIndex{5}$$: Multiple boundary curves as an example of the extended Green's theorem. (CC BY-NC; Ümit Kaya)

(Note $$R$$ is always to the left as you traverse either curve in the direction indicated.)

Then we can extend Green’s theorem to this setting by

$\oint_{C_1} F \cdot dr + \oint_{C_2} F \cdot dr = \int \int_R \text{curl} F \ dA.$

Likewise for more than two curves:

$\oint_{C_1} F \cdot dr + \oint_{C_2} F \cdot dr + \oint_{C_3} F \cdot dr + \oint_{C_4} F \cdot dr = \int \int_R \text{curl} F \ dA.$

$$Proof$$. The proof is based on the following figure. We ‘cut’ both $$C_1$$ and $$C_2$$ and connect them by two copies of $$C_3$$, one in each direction. (In the figure we have drawn the two copies of $$C_3$$ as separate curves, in reality they are the same curve traversed in opposite directions.)

Now the curve $$C = C_1+ C_3 + C_2 - C_3$$ is a simple closed curve and Green’s theorem holds on it. But the region inside $$C$$ is exactly $$R$$ and the contributions of the two copies of $$C_3$$ cancel. That is, we have shown that

$\int \int_R \text{curl} F\ dA = \int_{C_1 + C_3 + C_2 - C_3} F \cdot dr = \int_{C_1 + C_2} F \cdot dr.$

This is exactly Green's theorem, which we wanted to prove. Figure $$\PageIndex{6}$$: The punctured plane. (CC BY-NC; Ümit Kaya)
##### Example $$\PageIndex{1}$$

Let $$F = \dfrac{(-y, x)}{r^2}$$ ("tangential field")

$$F$$ is defined on $$D$$ = plane - (0, 0) = the punctured plane (Figure $$\PageIndex{7}$$). Figure $$\PageIndex{7}$$: A punctured plane. (CC BY-NC; Ümit Kaya)

It’s easy to compute (we’ve done it before) that $$\text{curl}F = 0$$ in $$D$$.

Question: For the tangential field $$F$$ what values can $$\oint_{C} F \cdot dr$$ take for $$C$$ a simple closed curve (positively oriented)?

Solution

We have two cases (i) $$C_1$$ not around 0 (ii) $$C_2$$ around 0 Figure $$\PageIndex{8}$$: Two cases. (CC BY-NC; Ümit Kaya)

In case (i) Green’s theorem applies because the interior does not contain the problem point at the origin. Thus,

$\oint_{C_1} F \cdot dr = \int\int_R \text{curl} F\ dA = 0.$

For case (ii) we will show that

let $$C_3$$ be a small circle of radius $$a$$, entirely inside $$C_2$$. By the extended Green’s theorem we have

$\oint_{C_2} F \cdot dr - \oint_{C_3} F \cdot dr = \int\int_R \text{curl} F\ dA = 0.$

Thus, $$\oint_{C_2} F \cdot dr = \oint_{C_3} F \cdot dr$$.

Using the usual parametrization of a circle we can easily compute that the line integral is

$\int_{C_3} F \cdot dr = \int_{0}^{2\pi} 1 \ dt = 2\pi. \ \ \ \ QED.$ Figure $$\PageIndex{1}$$: A punctured region. (CC BY-NC; Ümit Kaya)

Answer to the question: The only possible values are 0 and $$2\pi$$.

We can extend this answer in the following way:

If $$C$$ is not simple, then the possible values of $$\oint_C F \cdot dr$$ are $$2\pi n$$, where $$n$$ is the number of times $$C$$ goes (counterclockwise) around (0,0).

Not for class: $$n$$ is called the winding number of $$C$$ around 0. $$n$$ also equals the number of times $$C$$ crosses the positive $$x$$-axis, counting +1 from below and -1 from above. Figure $$\PageIndex{1}$$: An example of a non-simple region. (CC BY-NC; Ümit Kaya)