Divergence and Curl
Measures of source strength and rotation.
Key idea
Measures of source strength and rotation.
In depth
Divergence ∇·F measures net outflow per unit volume, and curl ∇×F measures local rotation. A field with zero divergence is incompressible; zero curl means irrotational (and conservative on simple regions).
Worked example
F = ⟨x, y, z⟩ has divergence 3 and curl 0.
Formula
Real-world uses
- Measuring the source of airflow and the spin of water in weather and fluid dynamics
Common mistake
Taking a dot product when a cross product is needed.
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 and Curl?
Measures of source strength and rotation. It is a topic at the expert level in Calculus.
What is the formula for Divergence and Curl?
The key formula is ∇·F, ∇×F.
Where is Divergence and Curl used in real life?
Real-world examples: Measuring the source of airflow and the spin of water in weather and fluid dynamics.
What should I learn before Divergence and Curl?
Make sure you are comfortable with Green’s Theorem, the previous topic in Calculus.
What is a common mistake with Divergence and Curl?
Taking a dot product when a cross product is needed.
How can I practise Divergence and Curl?
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 and Curl?
Move on to Surface Integrals 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.