Stokes Theorem Calculator
Calculate using Stokes theorem
Frequently Asked Questions
What is Stokes' theorem?
Stokes' theorem relates a surface integral of the curl of a vector field to a line integral around the boundary: ∮_C F·dr = ∬_S (∇×F)·dS. It generalizes Green's theorem to three dimensions.
How is Stokes' theorem used in physics?
It is fundamental in electromagnetism (Maxwell's equations), fluid dynamics (circulation and vorticity), and thermodynamics. Faraday's law of induction is a direct application: the EMF around a loop equals the rate of change of magnetic flux through it.
What is the curl of a vector field?
Curl measures the rotation of a vector field at a point. For F = (P, Q, R): ∇×F = (∂R/∂y - ∂Q/∂z, ∂P/∂z - ∂R/∂x, ∂Q/∂x - ∂P/∂y). The calculator computes curl components from your input field.
What is the relationship between Stokes' and Green's theorem?
Green's theorem is a special case of Stokes' theorem in 2D. When the surface lies in the xy-plane, Stokes' theorem reduces to: ∮_C (P dx + Q dy) = ∬_D (∂Q/∂x - ∂P/∂y) dA, which is Green's theorem.
How do I verify Stokes' theorem for a given problem?
Compute both sides independently: (1) evaluate the line integral ∮F·dr around the boundary curve, and (2) evaluate the surface integral ∬(∇×F)·dS. If both give the same result, the theorem is verified. The calculator performs both computations.