Divergence Theorem Calculator
Relates the outward flux of a vector field through a closed surface to its volume divergence.
Formula first
Overview
The Divergence Theorem, also known as Gauss's Theorem, equates the net outward flux of a vector field through a closed surface to the volume integral of the field's divergence within that surface. It transforms a boundary calculation into an interior accumulation calculation, acting as a 3D extension of the Fundamental Theorem of Calculus.
Symbols
Variables
\text{Concept-only} = Note
Apply it well
When To Use
When to use: Apply this theorem when calculating the total flux through a closed, piecewise smooth boundary where the volume integral of the divergence is easier to compute than the surface integral. It is specifically valid for vector fields with continuous first-order partial derivatives inside the region.
Why it matters: It is essential for deriving physical conservation laws, such as Gauss's Law in electromagnetism and the continuity equation in fluid mechanics. By relating local behavior (divergence) to global behavior (flux), it allows scientists to predict how substances or forces move through a boundary based on internal sources.
Avoid these traps
Common Mistakes
- Using for open surfaces.
- Flux direction (outward normal).
One free problem
Practice Problem
Calculate the total outward flux of the vector field F = (2x, 2y, 2z) through the surface of a cube with side length 3 units, centered at the origin.
Solve for:
Hint: Calculate the divergence of the field and multiply it by the volume of the cube.
The full worked solution stays in the interactive walkthrough.
References
Sources
- Calculus: Early Transcendentals by James Stewart
- Vector Calculus by Jerrold E. Marsden and Anthony J. Tromba
- Wikipedia: Divergence theorem
- Introduction to Electrodynamics by David J. Griffiths
- Calculus: Early Transcendentals, 8th Edition by James Stewart
- Mathematical Methods for Physicists, 7th Edition by George B. Arfken, Hans J. Weber, and Frank E. Harris
- Stewart Calculus: Early Transcendentals
- Standard curriculum — Vector Calculus