Stokes' Theorem Calculator
Verify Stokes' theorem by computing both sides: line integral and surface integral of curl.
Vector Field F(x, y, z)
Surface
Stokes' Theorem
oint_C F dot dr = iint_S (curl F) dot dS
Theorem Verification
Verified
Difference: 0.004136
ointLine Integral
-6.283185
iintSurface Integral
-6.287321
ASurface Area
6.284219
DeltaDifference
0.004136
Calculation Details
Left side (circulation):
oint_C F dot dr = -6.283185
Right side (flux of curl):
iint_S (curl F) dot dS = -6.287321
Key Points
- Generalizes Green's theorem to 3D
- Any surface with same boundary gives same result
- Boundary orientation follows right-hand rule
- Related to electromagnetic induction
About Stokes' Theorem
Statement
Stokes' theorem relates the line integral of a vector field around a closed curve to the surface integral of the curl over any surface bounded by that curve.
Applications
- Faraday's law of induction
- Ampere's law in electromagnetism
- Fluid circulation analysis
- Simplifying integral computations