1. What is Stokes's Theorem?

Stokes's Theorem relates a surface integral of the curl of a vector field over a surface S to a line integral of the vector field around the boundary curve C of S. It generalizes the fundamental theorem of calculus to higher dimensions.

2. Understanding the Theorem

The mathematical statement of Stokes's Theorem:

Where:

• \( \mathbf{F} \) — A vector field defined on S
• \( C \) — The boundary curve of S, oriented positively
• \( S \) — An oriented surface with boundary C
• \( \nabla \times \mathbf{F} \) — The curl of the vector field F

Explanation: The circulation of F around the boundary C equals the flux of the curl of F through the surface S.

3. Applications of Stokes's Theorem

Details: Stokes's Theorem is fundamental in physics and engineering, particularly in electromagnetism and fluid dynamics. It connects microscopic rotation (curl) to macroscopic circulation.

4. Using the Calculator

Tips: Enter the vector field components, describe the curve and surface, and specify orientation. The calculator will compute the line integral and surface integral to verify Stokes's Theorem.

5. Frequently Asked Questions (FAQ)

Q1: What's the difference between Stokes's and Green's Theorem?
A: Green's Theorem is a special case of Stokes's Theorem for 2D regions in the plane.

Q2: When can Stokes's Theorem be applied?
A: When S is an oriented, piecewise smooth surface bounded by a simple, closed, piecewise smooth curve C.

Q3: How is orientation determined?
A: The surface normal and curve direction follow the right-hand rule - fingers curl in curve direction, thumb points in normal direction.

Q4: What are common vector fields used with Stokes's Theorem?
A: Any differentiable vector field, but particularly useful for conservative fields and fields with rotational properties.

Q5: How is this related to the Divergence Theorem?
A: Both are part of the generalized Stokes's Theorem, relating integrals of derivatives to boundary integrals.