Divergence Theorem
Flux through a closed surface equals integral of divergence.
Key idea
Flux through a closed surface equals integral of divergence.
In depth
∯_S F · dS = ∭_V (∇·F) dV. The total flux out of a closed surface equals the total divergence inside. Used to simplify flux calculations.
Worked example
F = ⟨x, y, z⟩ on a sphere of radius a: flux = 3 × (4/3)πa³ = 4πa³.
Formula
Real-world uses
- Calculating total charge inside a region from the electric field on its boundary
Common mistake
Applying it to a non-closed surface.
How to learn this
- Read the key idea and say it in your own words.
- Work through the worked example, then try a similar problem.
- Use the interactive lesson’s visual and mini-quiz to check the pattern.
Frequently asked questions
What is Divergence Theorem?
Flux through a closed surface equals integral of divergence. It is a topic at the expert level in Calculus.
What is the formula for Divergence Theorem?
The key formula is ∯ F·dS = ∭ ∇·F dV.
Where is Divergence Theorem used in real life?
Real-world examples: Calculating total charge inside a region from the electric field on its boundary.
What should I learn before Divergence Theorem?
Make sure you are comfortable with Stokes’ Theorem, the previous topic in Calculus.
What is a common mistake with Divergence Theorem?
Applying it to a non-closed surface.
How can I practise Divergence Theorem?
Open the interactive lesson, play with the visual, then answer the three-question mini-quiz. Getting two right marks the topic complete.
What comes after Divergence Theorem?
Move on to Differentiating Under the Integral next.
Related expert topics in Calculus
- Derivatives of Inverse Trig – Derivatives of arcsin, arccos, arctan.
- Implicit Differentiation – Differentiate both sides of F(x,y)=0 treating y as a function of x.
- Logarithmic Differentiation – Take logs before differentiating products and powers.
- Related Rates – Link rates of change of connected quantities.
- Linear Approximation – Approximate f near a using its tangent line.
- Differentials – Use dy = f′(x)dx for small changes.