1. What is Stokes' Theorem?

Stokes' Theorem relates a surface integral of the curl of a vector field to a line integral of the vector field around the boundary of the surface. It generalizes the Fundamental Theorem of Calculus to higher dimensions.

2. How Does the Calculator Work?

The calculator uses Stokes' Theorem:

Where:

• \( \mathbf{F} \) — Vector field
• \( C \) — Boundary curve of surface S
• \( S \) — Oriented surface
• \( \nabla \times \mathbf{F} \) — Curl of F

Explanation: The theorem converts a line integral around a closed curve into a surface integral over the area bounded by the curve.

3. Importance of Stokes' Theorem

Details: Stokes' Theorem is fundamental in vector calculus, with applications in fluid dynamics, electromagnetism, and differential geometry.

4. Using the Calculator

Tips: Enter the vector field in [P, Q, R] format, define the surface equation, and specify the boundary curve. The calculator will compute both sides of Stokes' Theorem.

5. Frequently Asked Questions (FAQ)

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

Q2: How is orientation determined in Stokes' Theorem?
A: The surface normal and curve direction follow the right-hand rule.

Q3: Can Stokes' Theorem be applied to any surface?
A: The surface must be piecewise smooth and oriented, with a piecewise smooth boundary.

Q4: What are practical applications of Stokes' Theorem?
A: It's used in calculating work done by force fields, electromagnetic induction, and fluid circulation.

Q5: How does this relate to the Divergence Theorem?
A: Both relate different types of integrals, but the Divergence Theorem connects volume and surface integrals.