LaTeX
\iiint_V \nabla\cdot\mathbf F\,dV=\iint_{\partial V}\mathbf F\cdot\mathbf n\,dSVariables
- V: volume
- ∂V: boundary surface
- n: outward unit normal
How to use this formula
Relates volume divergence to outward flux across the boundary.
Important notes
- The field and region need suitable smoothness.
Quick example
Apply it to convert a closed-surface flux integral to a volume integral.
Applicability, worked calculation, and verification
Assumptions and domain checks
- The field and region need suitable smoothness.
- Preserve matrix order and verify dimension compatibility; multiplication and inversion are not generally commutative or always defined.
- Use a consistent orientation and coordinate system, and confirm the smoothness and boundary assumptions required by the theorem.
Worked example
Apply it to convert a closed-surface flux integral to a volume integral.
Common mistakes
- Before substituting values into Divergence Theorem, map each variable to its definition and preserve every sign, exponent, subscript, and grouping mark.
- Verify the result of Divergence Theorem with a known case, inverse operation, dimensional check, or independent calculation before publishing it.
Continue the workflow
Use Divergence Theorem in your own work
- Check the domainMatch the variables and assumptions to the problem before substituting values.
- Copy the exact notationPreserve grouping, signs, and exponents in
\iiint_V \nabla\cdot\mathbf F\,dV=\iint_{\partial V}\mathbf F\cdot\mathbf n\,dS. - Edit or convertOpen the expression in the LaTeX editor, then export it for your document or web page.
Review and verification
Last reviewed: 2026-07-23
Automated quality check: Kept noindex until the missing evidence is supplied.
Formula references
- Digital Library of Mathematical FunctionsNational Institute of Standards and Technology — Definitions, notation, identities, and reference material for mathematical functions.
Frequently asked questions
What is the Divergence Theorem used for?
Relates volume divergence to outward flux across the boundary.
Can I copy this formula as LaTeX?
Yes. Copy \iiint_V \nabla\cdot\mathbf F\,dV=\iint_{\partial V}\mathbf F\cdot\mathbf n\,dS or open it in the LaTeX editor.
What should I check before using it?
Confirm that each variable, unit, domain restriction, and assumption matches the problem.