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's a fundamental theorem in vector calculus.

2. Understanding the Theorem

The theorem states:

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 F

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

3. Applications

Details: Stokes's Theorem is used in electromagnetism, fluid dynamics, and differential geometry. It simplifies calculations by converting surface integrals to line integrals or vice versa.

4. Using the Calculator

Tips: Enter the vector field components (e.g., [y, -x, z]), the surface equation (e.g., z = x² + y²), and the boundary curve (e.g., x² + y² = 1).

5. Frequently Asked Questions (FAQ)

Q1: How is Stokes's Theorem related to Green's Theorem?
A: Green's Theorem is a special case of Stokes's Theorem for 2D regions in the xy-plane.

Q2: What's the difference between Stokes's and Divergence Theorem?
A: Divergence Theorem relates volume integrals to surface integrals, while Stokes's relates surface integrals to line integrals.

Q3: When can Stokes's Theorem be applied?
A: When the surface is piecewise smooth and oriented, and the vector field is continuously differentiable.

Q4: How to determine the orientation of the boundary curve?
A: Use the right-hand rule - when fingers curl in the direction of the surface orientation, thumb points in boundary direction.

Q5: Can Stokes's Theorem be used for any surface?
A: The surface must be orientable (like a sphere or torus, but not a Möbius strip).