Stokes' theorem connects a line integral and a surface integral.

Introduction:
Stokes' theorem is a fundamental theorem in vector calculus that relates a line integral around a closed curve to a surface integral over the region bounded by that curve. It provides a powerful tool for calculating circulations and fluxes in vector fields.

Statement of Stokes' theorem:
Stokes' theorem states that the line integral of a vector field F around a closed curve C is equal to the surface integral of the curl of F over the surface S bounded by C. Mathematically, it can be written as:

∮_C F · dr = ∬_S (curl F) · dS

where ∮_C represents the line integral around the closed curve C, F is the vector field, dr is an infinitesimal vector along the curve C, ∬_S represents the surface integral over the surface S, curl F is the curl of the vector field F, and dS is an infinitesimal vector normal to the surface S.

Explanation:
Stokes' theorem connects a line integral and a surface integral by relating the circulation of a vector field around a closed curve to the flux of its curl through the surface bounded by that curve. Here's a step-by-step explanation of how it works:

1. Consider a closed curve C in a vector field F.
2. Divide the surface bounded by the curve C into infinitesimal surface elements.
3. Calculate the curl of the vector field F at each point on the surface.
4. Take the dot product of the curl of F with the infinitesimal vector normal to the surface at each point.
5. Sum up all the dot products over the entire surface to obtain the surface integral of the curl of F.
6. Calculate the line integral of the vector field F along the closed curve C.
7. According to Stokes' theorem, the line integral is equal to the surface integral.

Significance:
Stokes' theorem provides a powerful tool for calculating the circulation of a vector field around a closed curve or the flux of its curl through a surface. It allows us to relate these two seemingly different concepts and provides a bridge between line integrals and surface integrals. This theorem has applications in various fields, including fluid dynamics, electromagnetism, and differential geometry.