Divergence Theorem (Gauss's Theorem) Calculator
Relates the outward flux of a vector field through a closed surface to the volume integral of the divergence of the field.
Formula first
Overview
This fundamental theorem provides a bridge between surface integrals and volume integrals, effectively showing that the total flow of a vector field out of a region is equal to the sum of all sources and sinks within that region. It is a three-dimensional generalization of the Fundamental Theorem of Calculus. In physical terms, it describes how the local density of a field's source (divergence) accumulates into a net transport across a boundary.
Symbols
Variables
V = Enclosed Volume, F = Vector Field, n = Normal Vector
Apply it well
When To Use
When to use: Use this theorem when evaluating a complex surface integral over a closed boundary is more difficult than computing a volume integral of the divergence.
Why it matters: It is essential in fluid dynamics, heat transfer, and electromagnetism to track how fields originate from sources within a volume.
Avoid these traps
Common Mistakes
- Applying the theorem to open surfaces without adding the missing 'cap'.
- Forgetting to use the outward-pointing unit normal vector.
- Failing to account for singularities in the vector field inside the volume.
One free problem
Practice Problem
Calculate the outward flux of the vector field F = x*i + y*j + z*k through the surface of a sphere of radius R = 1 centered at the origin.
Solve for: flux
Hint: The divergence of F = (x, y, z) is 3. Integrate this constant over the volume of the sphere.
The full worked solution stays in the interactive walkthrough.
References
Sources
- Stewart, J. (2015). Calculus: Early Transcendentals.
- Feynman, R. P. (1963). The Feynman Lectures on Physics.
- Stewart, J. (2015). Calculus: Early Transcendentals (8th ed.). Cengage Learning.