Stokes' Theorem Visualization

Understanding the relationship between surface integrals and line integrals
∂S F · dr = ∬S (∇ × F) · dS
F
Vector Field
∂S
Boundary of Surface
∇ × F
Curl of F
S
Surface

What is Stokes' Theorem?

Stokes' theorem is a fundamental result in vector calculus that generalizes several theorems from calculus. It establishes a relationship between the surface integral of the curl of a vector field over a surface S and the line integral of the vector field around the boundary curve ∂S.

Physical Interpretation

The theorem states that the total "circulation" around the boundary of a surface is equal to the total "vorticity" (curl) passing through the surface.

Applications

Stokes' theorem is used in electromagnetism, fluid dynamics, and general relativity. It helps convert difficult surface integrals into simpler line integrals.

Special Cases

When applied to a 2D plane, Stokes' theorem reduces to Green's theorem. In 3D space, it relates line and surface integrals.