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Green's Theorem  

Relationships Between Green's Theorem, Stokes' Theorem and the Divergence TheoremGreen's Theorem has two forms, the circulation form and the divergence form. Green's Theorem is in two dimensions, While Stokes' Theorem is the threedimensional form of the circulation form of Green's Theorem. Similarly, the Divergence Theorem is the threedimensional form of the divergence form of Green's Theorem. The following table summarizes these relationships.

Green's Theorem 

IF WE HAVE 
 a simplyconnected region \(\mathcal{R}\) which is bounded by a smooth curve \(\mathcal{C}\), oriented anticlockwise (counterclockwise) and traversed once, and 
THEN THIS EQUATION HOLDS 
\(\displaystyle{ \oint_{\mathcal{C}}{M~dx+N~dy} = \iint\limits_{\mathcal{R}}{\frac{\partial N}{\partial x}  \frac{\partial M}{\partial y} dA } }\) 
Notation   The circle on the integral sign clarifies that the curve \(\mathcal{C}\) is a closed curve. It is not necessary and not all books and instructors use it. 
If you want a complete lecture on this topic, here is a video.
Prof Leonard  Green's Theorem [1hr45min53secs]  
Green's Theorem relates the calculation of line integrals to double integrals over area in the special case when the line integral completely encloses a single region in the plane. This enclosed region must be simplyconnected, i.e. the enclosing line must not cross itself and it must consist of one simple closed curve. This theorem is very powerful and allows us to evaluate line integrals using double integrals.
Let's get started with a video. This video is excellent in explaining Green's Theorem, including several examples. It covers the idea of simplyconnected regions as well.
Dr Chris Tisdell  What is Green's Theorem?  
Alternative Forms of Green's Theorem 
Okay, so the theorem at the top of the page is the basic form of Green's Theorem. There are two other alternative forms that are equivalent to this basic form. These forms were touched on in the previous video. The next video, below the table, explains them in more detail.
1. circulation or curl ( in 3dim called Stokes' Theorem )  

\(\displaystyle{ \oint_{\mathcal{C}}{ \vec{F} \cdot \vec{T} ~ds} = }\) \(\displaystyle{ \iint_{\mathcal{R}}{(curl ~\vec{F}) \cdot \hat{k} ~ dA} }\) 
\(\displaystyle{ \oint_{\mathcal{C}}{\vec{F} \cdot d\vec{r} } = }\) \(\displaystyle{ \iint_{\mathcal{R}}{ \left( \frac{\partial N}{\partial x}  \frac{\partial M}{\partial y}\right) dA} }\) 
 
2. flux or divergence ( in 3dim called the Divergence Theorem )  
\(\displaystyle{ \oint_{\mathcal{C}}{ \vec{F} \cdot \vec{n} ~ds} = }\) \(\displaystyle{ \iint_{\mathcal{R}}{div~\vec{F}~dA} }\) 
\(\displaystyle{ \iint_{\mathcal{R}}{ \left( \frac{\partial N}{\partial y} + \frac{\partial M}{\partial x}\right) dA} }\) 
In the table above, \(\vec{T}\) is the unit tangent vector and \(\vec{n}\) is the outward unit normal vector.
This next video explains these alternative forms in more detail and has some examples.
Dr Chris Tisdell  Green's Theorem  
Another byproduct of Green's Theorem is that we can now use a line integral to calculate area. Here is the equation.
\(\displaystyle{ A = \frac{1}{2} \oint_{\mathcal{C}}{x~dy  y~dx} }\)
The previous video shows the derivation of this equation. If you skipped it, now is the time to go back and watch it.
Meaning of Green's Theorem 
So, now that you know how to use Green's Theorem, what is really going on here? This next video gives a quick review of the Theorem and then explains the interpretation and meaning of it.
Michael Hutchings  Multivariable calculus 4.4.1: Review and interpretation of Green's theorem  
Proofs of Green's Theorem 
Here are several video proofs of Green's Theorem. Each instructor proves Green's Theorem differently. So it will help you to understand the theorem if you watch all of these videos.
Khan Academy  Green's Theorem Proof (2 videos)  
Michael Hutchings  Multivariable calculus 4.3.4: Proof of Green's theorem  
David Metzler  Greens Theorem Proof  
Okay, you are now ready for some practice problems. Next, you need to understand how to parameterize surfaces before you go on to surface integrals and the three dimensional versions of Green's Theorem, Stokes' Theorem and the Divergence Theorem. 
parametric surfaces → 
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Practice Problems 

Instructions   Unless otherwise instructed, use Green's Theorem to evaluate these integrals.
Level A  Basic 
Practice A01  

Evaluate \(\displaystyle{\oint_{\mathcal{C}}{(y^2~dx+3xy~dy)}}\) where \(\mathcal{C}\) is the boundary of the semiannular region D in the upper half plane between the circles \(x^2+y^2=1\) and \(x^2+y^2=4\).  
answer 
solution 
Practice A02  

Calculate the outward flux of \(\vec{F}=x\hat{i}+2y\hat{j}\) over the square with corners \((\pm 1, \pm 1)\) where the unit normal is outwardpointing.  
solution 
Practice A03  

Evaluate \(\displaystyle{\oint_{\mathcal{C}}{y^2~dx+x^2~dy}}\) where \(\mathcal{C}\) is the triangle bounded by the lines \(x=0, x+y=1, y=0 \)  
solution 
Practice A04  

Let \(\mathcal{C}\) be the circle \(x^2+y^2=4\).  
solution 
Practice A05  

Evaluate \(\displaystyle{ \int_{\mathcal{C}}{x~dxx^2y^2~dy}}\) where \(\mathcal{C}\) is the triangle with vertices \((0,0)\), \((0,1)\) and \((1,1)\).  
solution 
Practice A06  

Evaluate \(\displaystyle{\oint_{\mathcal{C}}{P~dx+Q~dy}}\) where \(\mathcal{C}\) is the square with corners \((\pm 1, \pm 1)\); \(P(x,y)=x+y^2\) and \(Q(x,y)=y+x^2\).  
solution 
Practice A07  

Evaluate \(\displaystyle{\oint_{\mathcal{C}}{P~dx+Q~dy}}\) where \(\mathcal{C}\) is the triangle with corners \((0,0)\), \((1,1)\), \((2,0)\), \(P(x,y)=y+e^x\) and \(Q(x,y)=2x^2+\cos y\).  
solution 
Practice A08  

Evaluate \(\displaystyle{\int_{\mathcal{C}}{\vec{F} \cdot d\vec{r}}}\) where \(\mathcal{C}\) is the circle \(x^2+y^2=4\) and \(\vec{F}=\langle y, x \rangle\).  
solution 
Practice A09  

Evaluate \(\displaystyle{\int_{\mathcal{C}}{\vec{F} \cdot d\vec{r}}}\) where \(\mathcal{C}\) is along \(y=x^2/2\) and \(y=x\) and \(\vec{F}=\langle y^2, x^2 \rangle\).  
solution 
Practice A10  

Evaluate \(\displaystyle{\int_{\mathcal{C}}{\vec{F} \cdot d\vec{r}}}\) where \( \vec{F}=\langle y^3, 6xy^2 \rangle\) and \(\mathcal{C}\) is the curve with the 3 segments:  
solution 
Practice A11  

Evaluate \(\displaystyle{\int_{\mathcal{C}}{\vec{F} \cdot d\vec{r}}}\) where \(\mathcal{C}\) is boundary between the two circles \(x^2+y^2=1\) and \(x^2+y^2=3\) that lies in the first quadrant and \(\vec{F}=\langle e^x+2xy, 4x^2+\sin y \rangle\).  
solution 
Practice A12  

Determine the flux of the field \(\vec{F}=\langle x, y \rangle\) across \(x^2+y^2=16\).  
solution 
Practice A13  

Evaluate \(\displaystyle{\oint_{\mathcal{C}}{(x^2y^2)~dx + 2xy~dy}}\) on the curve which bounds the region \(\mathcal{R}: \{(x,y): 0 \leq x \leq 1, 2x^2 \leq y \leq 2x \}\)  
solution 
Practice A14  

Evaluate \(\displaystyle{\oint_{\mathcal{C}}{2y~dx3x~dy}}\) on the unit circle.  
solution 
Practice A15  

Calculate the outward flux of \(\vec{F}=x\hat{i}+2y\hat{j}\) over the square with corners \((\pm 1, \pm 1)\) where the unit normal is outwardpointing.  
answer 
solution 
Level B  Intermediate 
Practice B01  

Verify Green's theorem for the line integral \(\displaystyle{\oint_{\mathcal{C}}{xy~dx+x~dy}}\) where \(\mathcal{C}\) is the unit circle.  
solution 