Divergence (concept) Calculator
Scalar measure of source or sink.
Formula first
Overview
Divergence is a differential operator that quantifies the net magnitude of a vector field's source or sink at a specific point. It represents the volume density of the outward flux of a vector field from an infinitesimal volume around a given point.
Symbols
Variables
\text{Concept-only} = Note
Apply it well
When To Use
When to use: Use divergence when you need to determine if a fluid or field is expanding, contracting, or maintaining a constant density at a point. It is the primary operator used in the Divergence Theorem to convert a surface flux integral into a volume integral over the enclosed region.
Why it matters: It is a fundamental concept in physics, forming the basis of Gauss's Law in electromagnetism and the continuity equation in fluid mechanics. Understanding divergence allows engineers and physicists to model mass conservation and predict how fields like heat or electricity propagate through space.
Avoid these traps
Common Mistakes
- Thinking result is a vector.
- Confusing notation with gradient.
One free problem
Practice Problem
Find the divergence of the vector field F = 4x i - 2y j + 7z k.
Solve for:
Hint: Take the partial derivative of each component with respect to its corresponding variable and sum them.
The full worked solution stays in the interactive walkthrough.
References
Sources
- Wikipedia: Divergence
- Calculus by James Stewart
- Halliday, Resnick, Walker - Fundamentals of Physics
- Griffiths, David J. - Introduction to Electrodynamics
- Calculus: Early Transcendentals by James Stewart
- Div, Grad, Curl, and All That: An Informal Text on Vector Calculus by H. M. Schey
- Mathematical Methods for Physicists by George B. Arfken, Hans J. Weber, and Frank E. Harris
- Standard curriculum — Vector Calculus