# Gaussâ€™ Divergence Theorem

In vector calculus, the divergence theorem, also known as Gaussâ€™ theorem (Carl Friedrich Gauss), Ostrogradskyâ€™s theorem (Mikhail Vasilyevich Ostrogradsky), or Gaussâ€“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).

Statement
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.

Mathematically,

where V is the volume enclosed by the closed surface S.

# Greenâ€™s Identities

Greenâ€™s identities are corollaries of the divergence theorem and can be derived as follows.
 We consider,

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,

Substituting the value of  and , we get,

------------(i)

This is the first form of Greenâ€™s identity.

Now, interchanging the functions S1 and S2, we get,
----------(ii)

Subtracting Eq. (ii) from Eq. (i), we get,

This is the second form of Greenâ€™s identity.

Using Greenâ€™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).

# Greenâ€™s Theorem (in a Plane)

We consider an oriented closed path C. The integral  represents the circulation of  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  is a two-dimensional vector field and C is a closed path that lies in the plane) then Greenâ€™s theorem applies and we can use Greenâ€™s theorem as an alternative way to calculate the line integral.

Greenâ€™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  as the â€˜macroscopic circulationâ€™ of the vector field  around the path C. This microscopic circulation at a point (x, y) indicates how much  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.

Fig. (a) Region R bounded by the closed path C

Fig. (b) Microscopic circulations inside the region

Greenâ€™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.

Greenâ€™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 Greenâ€™s theorem as,

â€‹
The microscopic circulation of Greenâ€™s theorem has the same meaning with the curl of a three-dimensional vector field. The only difference is that Greenâ€™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.

Hence, Greenâ€™s theorem can be represented as,

Now, since  here is a two-dimensional vector field, we can write,

Thus, the final form of Greenâ€™s theorem is written as,

This must be remembered here that, Greenâ€™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.