# 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

*S*_{1}and*S*_{2}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,

This is the ------------(i)

*first form of Greenâ€™s identity*.

*S*

_{1}and

*S*

_{2}, we get,

----------(ii)

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

This is the
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

*second form of Greenâ€™s identity*.*uniqueness theorem*).

# Greenâ€™s Theorem (in a Plane)

We consider an oriented closed path

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

Greenâ€™s theorem transforms the line integral around

We consider the integral as the â€˜macroscopic circulationâ€™ of the vector field around the 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.*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

Greenâ€™s theorem says that the addition of all the microscopic circulation inside

*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*.