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Gauss’ Divergence Theorem

In vector calculus, the divergence theorem, also known as Gauss’ theorem (Carl Friedrich Gauss), Ostrogradsky’s theorem (Mikhail Vasilyevich Ostrogradsky), or Gauss–Ostrogradsky theorem is a result that relates the flow of a vector field through a surface to the behaviour of the vector field inside the surface (i.e., volume).

This theorem states that the divergence of a vector field over a volume is equal to the surface integral of the normal component of the vector through the closed surface bounding the volume.
 Description: 75480.png
 where V is the volume enclosed by the closed surface S.

Green’s Identities

Green’s identities are corollaries of the divergence theorem and can be derived as follows.
We consider, Description: 75204.png
where S1 and S2 are scalar functions, continuous together with their partial derivatives of first and second orders.
{By vector identity}
Applying the divergence theorem,
Description: 75190.png
Substituting the value of Description: 75182.png and Description: 75173.png, we get,
Description: 75163.png------------(i)
This is the first form of Green’s identity.
Now, interchanging the functions S1 and S2, we get,
Description: 75154.png----------(ii)

Subtracting Eq. (ii) from Eq. (i), we get,
Description: 75144.png
This is the second form of Green’s identity.
Using Green’s identities, it can be proved that the specifications of both divergence and curl of a vector with boundary conditions are sufficient to make the function unique (known as uniqueness theorem).

Green’s Theorem (in a Plane)

We consider an oriented closed path C. The integral Description: 75136.png represents the circulation of Description: 75129.png around C.

This circulation can be computed by performing the line integral directly. But, if the line integral happens to be in two dimensions (i.e., if Description: 75122.png is a two-dimensional vector field and C is a closed path that lies in the plane) then Green’s theorem applies and we can use Green’s theorem as an alternative way to calculate the line integral.
Green’s theorem transforms the line integral around C into a surface integral over the region bounded by C. However, we have to find out the function that we should integrate over the region inside C so that we get the same answer as the line integral. This is done by considering the notion of circulation.

We consider the integral Description: 100691.png as the ‘macroscopic circulation’ of the vector field Description: 100683.png around the path C. This microscopic circulation at a point (x, y) indicates how much Description: 100676.png would circulate around a tiny closed curve centred around the point (x, y). These microscopic circulations are shown as a bunch of small closed curves [Fig. (b)], where each curve represents the tendency for the vector field to circulate at that location.
Description: 75058.png
Fig. (a) Region R bounded by the closed path C

Description: 75089.png
Fig. (b) Microscopic circulations inside the region
Green’s theorem provides a relationship between the macroscopic circulation around the curve C and the sum of all the microscopic circulations inside C. If C is a closed curve in the plane (two dimensional) then it surrounds some region R [shown in Fig. (a)] in the plane.

Green’s theorem says that the addition of all the microscopic circulation inside C (i.e., the microscopic circulation in the region R), is exactly the same as the macroscopic circulation around C. The addition of the microscopic circulation in R means taking the double integral of the microscopic circulation over R. Therefore, we can write Green’s theorem as,
Description: 75021.png 
The microscopic circulation of Green’s theorem has the same meaning with the curl of a three-dimensional vector field. The only difference is that Green’s theorem applies only with two-dimensional vector fields, e.g., for vector fields in the xy-plane. The microscopic circulation is the circulation in the xy-plane. Therefore, the microscopic circulation parallel to the xy plane turns out to be the z-component of the curl.
Description: 75010.png


Hence, Green’s theorem can be represented as,
Description: 75001.png 
Now, since Description: 74993.png here is a two-dimensional vector field, we can write,
Thus, the final form of Green’s theorem is written as,
 Description: 74979.png

This must be remembered here that, Green’s theorem is valid only for curves oriented counterclockwise. In this case, we say that C is a positively oriented boundary of the region R.

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