Stokes’ Theorem
Relates surface integral of curl to line integral.
Key idea
Relates surface integral of curl to line integral.
In depth
∮_C F · dr = ∬_S (∇×F) · dS. The circulation around a closed curve equals the total curl through any surface it bounds. It generalises Green’s theorem to 3D.
Worked example
If F = ⟨−y, x, 0⟩ and C is the unit circle in the xy-plane, circulation = 2π.
Formula
Real-world uses
- Linking the circulation of a fluid around a loop to the swirl over a surface in electromagnetism
Common mistake
Using inconsistent orientation between curve and surface normal.
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 Stokes’ Theorem?
Relates surface integral of curl to line integral. It is a topic at the expert level in Calculus.
What is the formula for Stokes’ Theorem?
The key formula is ∬ (∇×F)·dS = ∮ F·dr.
Where is Stokes’ Theorem used in real life?
Real-world examples: Linking the circulation of a fluid around a loop to the swirl over a surface in electromagnetism.
What should I learn before Stokes’ Theorem?
Make sure you are comfortable with Surface Integrals, the previous topic in Calculus.
What is a common mistake with Stokes’ Theorem?
Using inconsistent orientation between curve and surface normal.
How can I practise Stokes’ 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 Stokes’ Theorem?
Move on to Divergence Theorem 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.