| |

7: Surface Integrals

Integrating functions over surfaces in 3D.

Parametric Surfaces

A surface SS can be parametrized by:

r(u,v)=⟨x(u,v),y(u,v),z(u,v)⟩\mathbf{r}(u, v) = \langle x(u, v), y(u, v), z(u, v) \rangle

where (u,v)(u, v) varies over some region DD in the uvuv-plane.

Examples

Sphere of radius aa:

r(ϕ,θ)=⟨asin⁡ϕcos⁡θ,asin⁡ϕsin⁡θ,acos⁡ϕ⟩\mathbf{r}(\phi, \theta) = \langle a\sin\phi\cos\theta, a\sin\phi\sin\theta, a\cos\phi \rangle

Cylinder of radius rr:

r(θ,z)=⟨rcos⁡θ,rsin⁡θ,z⟩\mathbf{r}(\theta, z) = \langle r\cos\theta, r\sin\theta, z \rangle

Graph z=f(x,y)z = f(x, y):

r(x,y)=⟨x,y,f(x,y)⟩\mathbf{r}(x, y) = \langle x, y, f(x, y) \rangle

Tangent Planes and Normal Vectors

The tangent vectors to the surface:

ru=∂r∂u,rv=∂r∂v\mathbf{r}_u = \frac{\partial \mathbf{r}}{\partial u}, \quad \mathbf{r}_v = \frac{\partial \mathbf{r}}{\partial v}

The normal vector:

n=ru×rv\mathbf{n} = \mathbf{r}_u \times \mathbf{r}_v

This is perpendicular to the surface at each point.

For a Graph z=f(x,y)z = f(x, y)

n=⟨−fx,−fy,1⟩\mathbf{n} = \langle -f_x, -f_y, 1 \rangle

Surface Area

A=∬D∣ru×rv∣ dAA = \iint_D |\mathbf{r}_u \times \mathbf{r}_v| \, dA

For a Graph z=f(x,y)z = f(x, y)

A=∬D1+fx2+fy2 dAA = \iint_D \sqrt{1 + f_x^2 + f_y^2} \, dA

Surface Integrals of Scalar Functions

∬Sf dS=∬Df(r(u,v))∣ru×rv∣ dA\iint_S f \, dS = \iint_D f(\mathbf{r}(u, v)) |\mathbf{r}_u \times \mathbf{r}_v| \, dA

For a Graph

∬Sf dS=∬Df(x,y,f(x,y))1+fx2+fy2 dA\iint_S f \, dS = \iint_D f(x, y, f(x,y)) \sqrt{1 + f_x^2 + f_y^2} \, dA

Applications

  • Surface area: ∬S1 dS\iint_S 1 \, dS
  • Mass of a shell: ∬Sρ dS\iint_S \rho \, dS
  • Center of mass: xˉ=1m∬Sxρ dS\bar{x} = \frac{1}{m}\iint_S x\rho \, dS

Oriented Surfaces

An oriented surface has a chosen “positive” side (direction of normal).

  • Closed surface: Convention is outward normal
  • Surface with boundary: Use right-hand rule with boundary curve

Surface Integrals of Vector Fields (Flux)

The flux of F\mathbf{F} through surface SS:

∬SF⋅dS=∬SF⋅n dS\iint_S \mathbf{F} \cdot d\mathbf{S} = \iint_S \mathbf{F} \cdot \mathbf{n} \, dS

Computation

∬SF⋅dS=∬DF(r(u,v))⋅(ru×rv) dA\iint_S \mathbf{F} \cdot d\mathbf{S} = \iint_D \mathbf{F}(\mathbf{r}(u,v)) \cdot (\mathbf{r}_u \times \mathbf{r}_v) \, dA

For a Graph z=f(x,y)z = f(x, y) with Upward Normal

∬SF⋅dS=∬DF⋅⟨−fx,−fy,1⟩ dA\iint_S \mathbf{F} \cdot d\mathbf{S} = \iint_D \mathbf{F} \cdot \langle -f_x, -f_y, 1 \rangle \, dA

Or if F=⟨P,Q,R⟩\mathbf{F} = \langle P, Q, R \rangle:

∬SF⋅dS=∬D(−Pfx−Qfy+R) dA\iint_S \mathbf{F} \cdot d\mathbf{S} = \iint_D (-Pf_x - Qf_y + R) \, dA

Physical Interpretation of Flux

If F\mathbf{F} is a velocity field of a fluid:

Flux=∬SF⋅dS=volume of fluid passing through S per unit time\text{Flux} = \iint_S \mathbf{F} \cdot d\mathbf{S} = \text{volume of fluid passing through } S \text{ per unit time}
  • Positive flux: flow in direction of normal
  • Negative flux: flow opposite to normal
  • Zero flux: flow parallel to surface

Computing Flux Through Common Surfaces

Sphere of Radius aa (Outward Normal)

n=r∣r∣=⟨x,y,z⟩a\mathbf{n} = \frac{\mathbf{r}}{|\mathbf{r}|} = \frac{\langle x, y, z \rangle}{a} dS=a2sin⁡ϕ dϕ dθdS = a^2 \sin\phi \, d\phi \, d\theta

Cylinder (Outward Normal)

n=⟨x,y,0⟩r\mathbf{n} = \frac{\langle x, y, 0 \rangle}{r}

Plane

Normal is constant, dS=dAdS = dA.

Summary

ConceptFormula
Normal vectorn=ru×rv\mathbf{n} = \mathbf{r}_u \times \mathbf{r}_v
Surface area$\iint_D
Scalar surface integral$\iint_S f , dS = \iint_D f
Flux∬SF⋅dS=∬DF⋅(ru×rv) dA\iint_S \mathbf{F} \cdot d\mathbf{S} = \iint_D \mathbf{F} \cdot (\mathbf{r}_u \times \mathbf{r}_v) \, dA
Graph z=f(x,y)z=f(x,y)dS=1+fx2+fy2 dAdS = \sqrt{1 + f_x^2 + f_y^2} \, dA