# 3.8: Extensions and Applications of Green’s Theorem

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## 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 \(D\).

\(D_1\) - \(D_5\) 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.

## 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.

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

- If \(F = \nabla f\) then \(\text{curl}F = N_x - M_y = 0\).
- 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,\[\oint_{C} F \cdot dr = \int \int_{R} \text{curl} F\ dA = 0.\]

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.

- \(\int_P^Q F \cdot dr\) is path independent.
- \(\oint_{C} F \cdot dr = 0\) for any closed path \(C\).
- \(F = \nabla f\) for some \(f\) in \(D\).
- \(F\) is conservative in \(D\).

If \(F\) is continuously differentiably then 1, 2, 3, 4 all imply 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 \(\PageIndex{4}\))

## 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 \(C_2\).

(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.

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

\(F\) is defined on \(D\) = plane - (0, 0) = the punctured plane (Figure \(\PageIndex{7}\)).

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

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

\[\int_{C_3} F \cdot dr = \int_{0}^{2\pi} 1 \ dt = 2\pi. \ \ \ \ QED.\]

**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.