1. What is the Divergence Theorem?

The Divergence Theorem, also known as Gauss's Theorem, relates the flux of a vector field through a closed surface to the divergence of the field in the volume enclosed. It connects surface integrals to volume integrals.

2. How Does the Calculator Work?

The calculator uses the Divergence Theorem:

Where:

• \( \mathbf{F} \) — Vector field (F = [P, Q, R])
• \( S \) — Closed surface
• \( V \) — Volume enclosed by S
• \( \nabla \cdot \mathbf{F} \) — Divergence of F

Explanation: The theorem states that the outward flux of a vector field through a closed surface equals the volume integral of the divergence over the region inside.

3. Importance of the Divergence Theorem

Details: The theorem is fundamental in physics (electromagnetism, fluid dynamics) and engineering, converting difficult surface integrals into easier volume integrals.

4. Using the Calculator

Tips: Enter the vector field components as comma-separated values (e.g., "x,y,z"), select the region type, and specify dimensions. The calculator will compute both sides of the theorem.

5. Frequently Asked Questions (FAQ)

Q1: What types of surfaces can this handle?
A: The calculator currently supports standard surfaces like spheres, cubes, and cylinders.

Q2: How accurate are the results?
A: Results are mathematically exact for the given inputs and standard shapes.

Q3: Can I use any vector field?
A: The field must be continuously differentiable within the volume.

Q4: What's the difference between this and Stokes' Theorem?
A: Stokes' relates line integrals to surface integrals, while Divergence Theorem relates surface to volume integrals.

Q5: Why is this theorem important in physics?
A: It's used to derive fundamental laws like Gauss's Law for electricity and continuity equations in fluid dynamics.