Stokes' Theorem Calculator

Enter each component using variables x, y, z. Example: x^2 + y*z
Enter the bounds for surface parameters u and v

Results:

Line Integral: -

Surface Integral: -

Understanding Stokes' Theorem

What is Stokes' Theorem?

Stokes' Theorem relates the surface integral of the curl of a vector field over a surface S to the line integral of the vector field around the boundary curve C of that surface:

∮C F · dr = ∬S (∇ × F) · dS

Key Components

  • Surface S must be oriented and have a smooth boundary curve C
  • Vector field F must be smooth and defined on S and C
  • Curve C must be traversed in a direction consistent with the surface orientation
  • The theorem connects line integrals with surface integrals
  • Generalizes Green's theorem to three dimensions

Applications

Electromagnetic Theory

Used in Maxwell's equations to relate magnetic fields to electric currents

Fluid Dynamics

Analysis of fluid circulation and vorticity

Aerodynamics

Study of lift and drag forces on aircraft wings

Thermodynamics

Analysis of heat flow and energy transfer

Important Considerations

  • Surface orientation determines the positive direction of the boundary curve
  • Right-hand rule determines the relationship between surface normal and curve direction
  • Theorem fails if the surface is not simply connected
  • Vector field must be differentiable over the entire surface
  • Multiple surfaces can share the same boundary curve

Related Theorems

  • Green's Theorem (2D version of Stokes' Theorem)
  • Divergence Theorem (Gauss's Theorem)
  • Kelvin-Stokes Theorem (generalization to manifolds)
  • Conservative Field Theorem