Surface Integrals and Flux

1Introduction

This lecture distinguishes the surface integral of a scalar quantity from the flux of a vector field through a surface. The first accumulates a quantity distributed over area, whereas the second accumulates the component normal to an oriented surface.

2Terminology and Definitions

The flux of a vector field is the surface integral of its normal component and measures the net passage of the field through a surface.

A surface integral accumulates either a scalar quantity over a surface or the normal component of a vector field over that surface.

3Strategy

Represent the surface by $\mathit{r}(u,v)$\boldsymbol r(u,v) and construct a normal direction from the tangent vectors $r_u$\boldsymbol r_u and $r_v$\boldsymbol r_v. In a flux integral, only the component of the vector field in this normal direction contributes.

4Oriented Surfaces

A flux computation must distinguish the two sides of a surface. Reversing the normal direction reverses the sign of the flux. For a closed surface, the outward normal is the standard choice.

A surface is orientable if it admits a continuous unit normal field $\mathit{n}$\boldsymbol n over the entire surface. A connected orientable surface has the two orientations $\mathit{n}$\boldsymbol n and $-\mathit{n}$-\boldsymbol n. A globally consistent flux cannot be defined on a nonorientable surface. For a closed surface given as the boundary of a region, the outward orientation is standard.

5Precise Formulation

Let $\mathrm{\Omega }\subset R^2$\Omega\subset\mathbb R^2 be a bounded Jordan domain with piecewise $C^1$C^1 boundary, set $D=\stackrel{\_}{\mathrm{\Omega }}$D=\overline\Omega, and let $\mathit{r}$\boldsymbol r be $C^1$C^1 on an open neighborhood of $D$D. If $r_u\times r_v\ne 0$\boldsymbol r_u\times\boldsymbol r_v\ne\boldsymbol0 throughout $\mathrm{\Omega }$\Omega and $\mathit{r}$\boldsymbol r is injective on $\mathrm{\Omega }$\Omega, then $S=\mathit{r}\left(D\right)$S=\boldsymbol r(D) is a regular surface patch. Overlap restricted to the boundary has area zero and does not affect the integral. If a parameterization covers interior points more than once, the formula over its parameter domain represents an integral with multiplicity rather than the ordinary integral over the geometric surface.

For a continuous scalar field $g$g defined on a neighborhood of $S$S, define

This integral is independent of orientation. When $g=1$g=1, it equals the area of the surface.

For a continuous vector field $\mathit{F}$\boldsymbol F defined on a neighborhood of $S$S, suppose that $r_u\times r_v$\boldsymbol r_u\times\boldsymbol r_v agrees with the selected orientation. Define the flux by

At interior points of the patch, the corresponding unit normal is

Reversing the orientation reverses the sign of the flux. The vector $r_u\times r_v$\boldsymbol r_u\times\boldsymbol r_v simultaneously specifies a direction perpendicular to the tangent plane and the infinitesimal area scale. Thus $\parallel r_u\times r_v\parallel dudv$\|\boldsymbol r_u\times\boldsymbol r_v\|\,du\,dv is the unoriented area element, while the oriented flux uses $r_u\times r_{v}dudv$\boldsymbol r_u\times\boldsymbol r_v\,du\,dv without taking its norm.

If a compact piecewise regular surface of the type considered in this lecture cannot be represented regularly and one-to-one by a single patch, it is divided into finitely many regular patches whose overlaps are confined to boundaries of area zero. Its integral is the sum of the integrals over these patches. For flux, the normal on every patch must agree with the selected orientation of the surface. This construction defines the integrals over the spheres and closed surfaces used below.

Let ${\mathrm{\Omega }}^{\prime }$\Omega' be another bounded Jordan domain with piecewise $C^1$C^1 boundary and set $E=\stackrel{\_}{{\mathrm{\Omega }}^{\prime }}$E=\overline{\Omega'}. Let $\mathit{\phi }:E\to D$\boldsymbol\varphi:E\to D be the restriction of a $C^1$C^1 diffeomorphism between open neighborhoods of $E$E and $D$D, with $\mathit{\phi }\left(E\right)=D$\boldsymbol\varphi(E)=D, and set $\mathit{\rho }=\mathit{r}\circ \mathit{\phi }$\boldsymbol\rho=\boldsymbol r\circ\boldsymbol\varphi. The scalar surface integral is invariant under reparameterization because the change-of-variables formula uses the absolute value of the Jacobian. For flux, $\det D\mathit{\phi }>0$\det D\boldsymbol\varphi>0 preserves the orientation and the value. If $\det D\mathit{\phi }<0$\det D\boldsymbol\varphi<0 and the resulting cross product is used without correction, the parameterization represents the oppositely oriented surface and the flux changes sign. To preserve the previously selected geometric orientation, one must correct the sign of the cross product.

6Example 1: A Planar Patch

Consider the upward-oriented portion of the plane $z=0$z=0 with $0\text{[PARSE ERROR: Undefined("Command("le")")]}x\text{[PARSE ERROR: Undefined("Command("le")")]}1$0\le x\le1 and $0\text{[PARSE ERROR: Undefined("Command("le")")]}y\text{[PARSE ERROR: Undefined("Command("le")")]}1$0\le y\le1. For $\mathit{F}=(0,0,z+1)$\boldsymbol F=(0,0,z+1), the unit normal is $(0,0,1)$(0,0,1), and

Choosing the downward normal changes the value to $-1$-1.

7Example 2: A Sphere

Let $\mathit{F}=(x,y,z)$\boldsymbol F=(x,y,z) and orient the sphere of radius $R>0$R>0 centered at the origin outward. On the sphere, $\mathit{F}·\mathit{n}=R$\boldsymbol F\cdot\boldsymbol n=R. Since its area is $4\pi R^2$4\pi R^2, the flux is $4\pi R^3$4\pi R^3. The subsequent lecture verifies the same value by the divergence theorem: the divergence is $3$3, and the volume is $4\pi R^3/3$4\pi R^3/3.

8Example 3: A Cylindrical Side

Parameterize the lateral surface of a cylinder of radius $R$R and height $H$H by

where $0\text{[PARSE ERROR: Undefined("Command("le")")]}\theta \text{[PARSE ERROR: Undefined("Command("le")")]}2\pi$0\le\theta\le2\pi and $0\text{[PARSE ERROR: Undefined("Command("le")")]}z\text{[PARSE ERROR: Undefined("Command("le")")]}H$0\le z\le H. This parameterization excludes the top and bottom disks. The values $\theta =0$\theta=0 and $\theta =2\pi$\theta=2\pi describe the same seam, but this overlap lies on the boundary of the parameter domain and therefore does not affect the surface integral. The outward-oriented area vector is

For $\mathit{F}=(x,y,0)$\boldsymbol F=(x,y,0), the flux is

9Comparison: A Field Tangent to the Surface

On an upward-oriented patch of the plane $z=0$z=0, the field $\mathit{F}=(1,0,0)$\boldsymbol F=(1,0,0) is tangent to the plane. Consequently, $\mathit{F}·\mathit{n}=0$\boldsymbol F\cdot\boldsymbol n=0, and the flux is zero. A vector field may be nonzero along a surface while contributing no flux if its normal component vanishes.

10Connection to the Divergence Theorem

Let a bounded region $V$V have a piecewise $C^1$C^1 boundary, and suppose that $\mathit{F}$\boldsymbol F is $C^1$C^1 on a neighborhood of $\stackrel{\_}{V}$\overline V. The total outward flux through the closed surface $\mathit{\partial }V$\partial V equals the volume integral of the divergence over $V$V. This is the divergence theorem. The next lecture relates flux through the boundary to the local interior quantity $\nabla ·\mathit{F}$\nabla\cdot\boldsymbol F. The theorem does not apply under these hypotheses if $\mathit{F}$\boldsymbol F has a singularity inside $V$V.

11Common Errors

• Computing flux without specifying a normal orientation.
• Confusing the scalar area factor $\parallel r_u\times r_v\parallel$\|\boldsymbol r_u\times\boldsymbol r_v\| with the oriented area vector $r_u\times r_v$\boldsymbol r_u\times\boldsymbol r_v.
• Applying Stokes' theorem without checking the induced orientation of the boundary curve.

12Related Lectures