A flux integral measures how much of a vector field passes through a surface. In physics, it can describe fluid flow through a membrane, electric field through a charged surface, or heat flow through a boundary. The key idea is that only the component of the vector field perpendicular to the surface contributes to flux.
This makes flux integrals a powerful bridge between geometry, vectors, and real physical flow.
Key Facts
- Flux through an oriented surface S is ∬_S F · n dS.
- If n is a unit normal vector, then F · n is the normal component of the field.
- For a parametrized surface r(u, v), flux is ∬_D F(r(u, v)) · (r_u × r_v) du dv.
- Reversing the orientation changes the sign of the flux: outward flux = - inward flux.
- If F is tangent to the surface everywhere, then F · n = 0 and the flux is 0.
- Divergence theorem: ∬_S F · n dS = ∭_V div F dV for a closed surface with outward orientation.
Vocabulary
- Flux
- Flux is the signed amount of a vector field passing through an oriented surface.
- Vector field
- A vector field assigns a vector to each point in a region of space.
- Unit normal vector
- A unit normal vector is a length 1 vector perpendicular to a surface at a point.
- Orientation
- Orientation is the chosen direction of the normal vector used to decide positive flux.
- Divergence theorem
- The divergence theorem relates total outward flux through a closed surface to the triple integral of divergence over the volume inside.
Common Mistakes to Avoid
- Using the tangent component of the field, which is wrong because flux depends only on F · n, the component perpendicular to the surface.
- Forgetting orientation, which is wrong because reversing the normal vector reverses the sign of the flux.
- Using dA when the surface is tilted or curved without correction, which is wrong because flux requires the actual surface area element dS or the vector area element r_u × r_v du dv.
- Applying the divergence theorem to an open surface, which is wrong unless the surface is first closed by adding missing boundary pieces.
Practice Questions
- 1 Compute the flux of F = <0, 0, 5> through the square 0 ≤ x ≤ 2, 0 ≤ y ≤ 3 in the plane z = 1 with upward orientation.
- 2 Let F = <x, y, z>. Use the divergence theorem to find the outward flux through the sphere x^2 + y^2 + z^2 = 4.
- 3 A vector field is tangent to a curved surface at every point. Explain what the flux through the surface is and why orientation does or does not matter.