Surface Integral
A surface integral extends integration to a two-dimensional surface embedded in three-dimensional space, computing the total value of a scalar or vector field over that surface. Scalar surface integrals find quantities like surface area or total mass of a thin shell, while vector surface integrals (flux integrals) measure how much of a vector field passes through a surface. Surface integrals are central to Maxwell's equations, fluid dynamics, and the theorems of Gauss and Stokes.
Key Formula
∫∫_S F · dS = ∫∫_D F(r(u,v)) · (r_u × r_v) du dv
LaTeX: \iint_S \mathbf{F} \cdot d\mathbf{S} = \iint_D \mathbf{F}(\mathbf{r}(u,v)) \cdot (\mathbf{r}_u \times \mathbf{r}_v)\,du\,dv
| Symbol | Meaning | Unit |
|---|---|---|
| F | Vector field | varies (e.g., N/m²) |
| S | Surface of integration | m² |
| r(u,v) | Parametric surface representation | m |
| r_u × r_v | Normal vector to the surface (cross product of partial derivatives) | m² |
Worked Example
Problem
Find the flux of F(x,y,z) = (0, 0, 1) through the unit square S in the xy-plane (0≤x≤1, 0≤y≤1, z=0) with upward normal.
Solution
Step 1: Parametrize: r(x,y) = (x, y, 0), r_x = (1,0,0), r_y = (0,1,0). Step 2: Normal vector: r_x × r_y = (0,0,1) (pointing upward, correct orientation). Step 3: F(r(x,y)) = (0, 0, 1). Step 4: Dot product: F · (r_x × r_y) = (0)(0)+(0)(0)+(1)(1) = 1. Step 5: Integrate: ∫₀¹ ∫₀¹ 1 dx dy = 1.
Answer
Flux = 1 m² (or appropriate units)
Surface Integral Types and Physical Interpretations
| Integral Type | Formula | Physical Interpretation | Example Use |
|---|---|---|---|
| Scalar surface integral | ∫∫_S f dS | Sum of scalar values over surface | Surface area, mass of shell |
| Vector flux integral | ∫∫_S F·dS | Net flow of vector field through surface | Electric flux (Gauss's Law) |
| Closed surface integral | ∯ F·dS | Flux through closed surface | Divergence theorem application |
| Surface of revolution | Special parametrization | Integration over rotated curve | Area of cone, sphere |
| Graph surface | ∫∫_D f √(1+fₓ²+f_y²) dA | Surface over a flat domain | Area of z = g(x,y) |
Interactive Tools
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Related Terms
Line Integral
A line integral (also called a path integral or curve integral) computes the integral of a function along a curve or path in two or three dimensions. There are two types: scalar line integrals (integrating a scalar field along a curve) and vector line integrals (computing work done by a vector field along a path). Line integrals are fundamental in physics for calculating work, circulation, and flux in electromagnetic and fluid mechanics contexts.
Triple Integral
A triple integral extends the concept of integration to three dimensions, computing the accumulation of a function over a three-dimensional region. It is used to find volumes, masses, and other physical quantities distributed throughout a solid region in space. Applications include calculating mass of a solid with variable density, electric charge distributions, and gravitational potential fields.
From Latin "superficies" (surface, literally "over-face") and "integrare" (to make whole). Surface integrals were formalized in the 19th century by mathematicians including Gauss and Stokes in connection with their celebrated theorems.