A surface integral adds up values over a curved two-dimensional surface in three-dimensional space. It is the surface version of a double integral, but the small area pieces are tilted and stretched as the surface bends. Surface integrals matter in physics and engineering because they measure quantities spread across membranes, shells, fields, and boundaries.
They are used to compute mass on a curved sheet, heat flow through a surface, electric flux, and fluid flow through a barrier.
To compute a surface integral, the surface is usually parameterized by two variables, such as r(u, v) = <x(u, v), y(u, v), z(u, v)>. The vector r_u x r_v gives a normal direction, and its magnitude gives the local area stretch factor dS = |r_u x r_v| du dv. For a scalar field f, the integral adds f over area, while for a vector field F, the flux integral adds the component of F passing through the surface.
The sign of flux depends on the chosen orientation of the normal vector.
Key Facts
- Scalar surface integral: ∬_S f dS = ∬_D f(r(u, v)) |r_u x r_v| du dv.
- Surface element: dS = |r_u x r_v| du dv.
- Oriented vector area element: dS vector = (r_u x r_v) du dv.
- Flux integral: ∬_S F · n dS = ∬_D F(r(u, v)) · (r_u x r_v) du dv.
- For a graph z = g(x, y), dS = sqrt(1 + g_x^2 + g_y^2) dx dy.
- Changing the orientation of the surface normal changes the sign of flux but not the scalar surface integral.
Vocabulary
- Surface integral
- An integral that adds scalar values or vector flow over a surface in space.
- Parameterization
- A description of a surface using two input variables, usually written as r(u, v).
- Surface element
- The small area factor dS that accounts for how a parameter region is stretched onto a surface.
- Normal vector
- A vector perpendicular to a surface at a point, often used to define the orientation of the surface.
- Flux
- The amount of a vector field passing through a surface, computed using the dot product with the surface normal.
Common Mistakes to Avoid
- Using du dv instead of dS, which ignores the stretching caused by the surface parameterization and gives the wrong area scale.
- Forgetting the magnitude in scalar surface integrals, because ∬_S f dS requires |r_u x r_v| rather than the vector r_u x r_v.
- Using the wrong normal direction for flux, which reverses the sign of ∬_S F · n dS even when the magnitude is correct.
- Mixing parameter bounds with x and y bounds without converting the surface, which can integrate over the wrong region in the parameter plane.
Practice Questions
- 1 Let S be the square patch r(u, v) = <u, v, 2> for 0 ≤ u ≤ 3 and 0 ≤ v ≤ 4. Compute ∬_S 5 dS.
- 2 Let S be the plane patch r(u, v) = <u, v, u + v> for 0 ≤ u ≤ 1 and 0 ≤ v ≤ 2. Compute the surface area of S.
- 3 A vector field points mostly tangent to a surface at every point. Explain why its flux through the surface is small or zero, even if the vector field has large magnitude.