# Green's Theorem

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Green's Theorem is a vector identity that is equivalent to the curl theorem in two dimensions. It relates the line integral around a simple closed curve  with the double integral over the plane region .

The theorem is named after the British mathematician George Green. It can be applied to various fields in physics, among others flow integrals.

## Mathematical Statement in two dimensions

Let  be a region in  with a positively oriented, piecewise smooth, simple closed boundary .  and  are functions defined on a open region containing  and have continuous partial derivatives in that region. Then Green's Theorem states that



The theorem is equivalent to the curl theorem in the plane and can be written in a more compact form as



### Application: Area Calculation

Green's theorem is very useful when it comes to calculating the area of a region. If we take  and , the area of the region , with boundary  can be calculated by



This formula gives a relationship between the area of a region and the line integral around its boundary.

If the curve is parametrized as , the area formula becomes



## Statement in three dimensions

Different ways of formulating Green's theorem in three dimensions may be found. One of the more useful formulations is



### Proof



where  is defined by  and  is the outward-pointing unit normal vector field.

Insert



and use



so that we obtain the result to be proved,