Stokes theorem is a fundamental theorem in vector calculus that relates the surface integral of a vector field over a closed surface to the line integral of the same vector field over the boundary of the surface.
Stokes theorem is important because it allows us to relate two different types of integrals (surface and line integrals) and can simplify calculations in many physical and engineering problems involving vector fields.
Stokes theorem can be derived from the fundamental theorem of calculus and the divergence theorem. It is also a special case of the more general theorem known as the generalized Stokes theorem.
Stokes theorem assumes that the vector field is continuously differentiable and that the surface and its boundary have certain smoothness properties. It also only applies to closed surfaces and their boundaries.
Stokes theorem is commonly used in physics and engineering to calculate flux through a surface, work done by a conservative force, and circulation of a fluid, among other applications. It is also used in more abstract areas of mathematics such as differential geometry and topology.