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 the vector field

Explanation: The theorem states that the circulation of a vector field around a closed curve is equal to the flux of its curl through any surface bounded by that curve.

3. Importance of Stokes' Theorem

Details: Stokes' Theorem is fundamental in vector calculus and has applications in physics, particularly in electromagnetism and fluid dynamics.

4. Using the Calculator

Tips: Enter the vector field components (i,j,k), the surface equation, and the boundary curve parametric equations. The calculator will evaluate either the line integral or surface integral using 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 that applies to 2D regions in the plane.

Q2: What are typical applications of Stokes' Theorem?
A: It's used in calculating work done by force fields, electromagnetic induction, and fluid flow calculations.

Q3: What orientation should the surface have?
A: The surface must be oriented consistently with the boundary curve (right-hand rule).

Q4: Can any surface be used with Stokes' Theorem?
A: Yes, as long as it's piecewise smooth and bounded by the curve C.

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