## Stokes Theorem Statement

According to this theorem, the line integral of a vector field **A vector **around any closed curve is equal to the surface integral of the curl of **A vector **taken over any surface **S** of which the curve is a bounding edge.

**Stokes Theorem Proof**

Let **A vector **be the vector field acting on the surface enclosed by closed curve **C**. Then the line integral of vector **A vector **along a closed curve is given by

where dl vector is the length of a small element of the path as shown in fig.

Now let us divide the area enclosed by the closed curve C into two equal parts by drawing a line ab as shown in fig. We have now two closed curves C1 and C2.

Therefore, the line integral of vector A vector along a closed curve C can be written as

If the area enclosed by the curve C is divided into a large number of small areas such as dS_{1}, dS_{2}, dS_{3}……………… dS_{n} bounded by the curves C_{1}, C_{2}……………C_{n} as shown in fig.

According to the definition of curl

Put this value in (2), we get

where, dS_{n} is the surface area in the case under consideration.

Hence eqn. (4) can be written as

which is the **Stokes Theorem**.