Stokes' theorem connects the circulation of a vector field around a closed curve to the total curl passing through any surface that has that curve as its boundary. It is a central result of vector calculus because it turns a line integral into a surface integral, or the reverse, depending on which is easier. The theorem gives a precise mathematical link between local spinning behavior and global circulation.
It is used throughout physics, especially in fluid flow, electromagnetism, and field theory.
Key Facts
- Stokes' theorem: ∮∂S F · dr = ∬S (∇ × F) · n dS.
- The curve ∂S must be a closed, oriented boundary of the surface S.
- The surface orientation n and boundary direction must follow the right-hand rule.
- Curl measures local rotation of a vector field: ∇ × F.
- If ∇ × F = 0 everywhere on S, then ∮∂S F · dr = 0 for that surface.
- Different surfaces with the same boundary give the same integral if the field is smooth on and between them.
Vocabulary
- Vector field
- A vector field assigns a vector, such as velocity or force, to each point in space.
- Curl
- Curl is a vector that measures the local tendency of a vector field to rotate around a point.
- Line integral
- A line integral adds the tangential component of a vector field along a curve.
- Surface integral
- A surface integral adds a quantity over a surface, often using the surface normal direction.
- Orientation
- Orientation is the chosen direction of a surface normal and the matching direction around its boundary.
Common Mistakes to Avoid
- Using the wrong boundary direction, which changes the sign of the line integral because orientation controls positive circulation.
- Forgetting the dot product with the normal vector, which is wrong because Stokes' theorem uses only the component of curl passing through the surface.
- Applying Stokes' theorem to a nonclosed boundary curve, which is invalid because ∂S must form a closed loop.
- Choosing a complicated surface when a simpler one has the same boundary, which misses the main advantage that any compatible surface may be used.
Practice Questions
- 1 Let F = <-y, x, 0> and let C be the circle x^2 + y^2 = 4 in the xy-plane oriented counterclockwise viewed from above. Use Stokes' theorem to find ∮C F · dr.
- 2 Let F = <z, x, y> and let S be the triangle with vertices (0,0,0), (1,0,0), and (0,1,0), oriented upward. Compute ∬S (∇ × F) · n dS.
- 3 A vector field is smooth, and two different smooth surfaces share the same closed boundary curve with the same induced orientation. Explain why Stokes' theorem says the circulation around the boundary is the same for both surfaces.