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 curve. 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 any surface bounded by that 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 components, select the surface type, and describe the boundary curve. The calculator will compute either the line integral or surface integral.

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: What orientation should the surface have?
A: The surface normal should follow the right-hand rule relative to the curve's orientation.

Q3: Can I use any surface bounded by C?
A: Yes, the theorem holds for any surface with boundary C, as long as the field is differentiable.

Q4: What are common applications?
A: Calculating work done by a force field, fluid flow circulation, and electromagnetic induction.

Q5: How is this related to divergence theorem?
A: Both relate integrals of different dimensions, but divergence theorem relates volume and surface integrals.