Surface Integral

The surface integral returns the surface area of a parametric surface, and can be combined with functions and/or vector fields in a similar way to its counterpart, the line integral. They are a generalisation of double integrals.

Definition §

But basically, integrating using a curved surface as the domain, instead of a flat rectangle like in double integrals.

Surface Integral (Scalar Function) ^definition-scalar

Let $O$ be a 3D parametric surface that is $C^1$ smooth (except possibly for a finite number of points), given by the parametric equation $\Phi :D\to O$ where $\Phi (s,t)=(u(s,t),v(s,t),w(s,t))$. If $f$ is a continuous scalar function defined over $O$, then the surface integral is given by:

$$\begin{array}{rl}\underset{\text{Surface Integral}}{\underbrace{{\iint }_{O}f(x,y,z)\text{d}S}}& ={\iint }_{D}f(\Phi (s,t))\Vert {\vec{T}}_s\times {\vec{T}}_t\Vert \text{d}s\text{d}t\\ & =\underset{\text{Double Integral}}{\underbrace{{\iint }_{D}f[\Phi (s,t)]\Vert {\Phi }_s\times {\Phi }_t\Vert \text{d}s\text{d}t}}\end{array}$$

• ${\vec{T}}_{s,t}$: The tangent vectors to the surface relative to $s$ and $t$ respectively.
• ${\iint }_D$ : The double integral over the ‘flat’ non-surface domain $D$
• $S:{\mathbb{R}}^2\to \mathbb{R}$

• $f=1$ returns the surface area of $O$
• The scalar surface integral is the same independent of the parameterisation of $O$

Surface/Flux Integral (Vector Field) ^definition-vector

Let $O$ be a 3D oriented parametric surface that is $C^1$ smooth (except possibly for a finite number of points), given by the parametric equation $\Phi :D\to O$ where $\Phi (s,t)=(u(s,t),v(s,t),w(s,t))$. Let $\vec{F}$ be a continuous vector field defined over $O$. Then, the surface integral (also called flux integral) is given by:

$$\underset{\text{Flux Integral}}{\underbrace{{\iint }_O\vec{F}\cdot \text{d}\vec{S}}}=\underset{\text{Surface Integral}}{\underbrace{{\iint }_O\vec{F}\cdot \hat{N}\text{d}S}}=\underset{\text{Double Integral}}{\underbrace{{\iint }_D\vec{F}\cdot ({\vec{T}}_s\times {\vec{T}}_t)\text{d}s\text{d}t}}$$

• ${\vec{T}}_{s,t}$: The tangent vectors to the surface relative to $s$ and $t$ respectively.
• $\hat{N}$: The unit normal vector field of $O$. This is what requires $O$ to be oriented.

• The flux integral is dependent on the orientation of $S$, since the negative orientation results in the flux integral giving the opposite sign as the positive orientated one.

Physical Interpretation §

For Scalar Functions §

• If $f$ is the area charge density $\sigma$, then the surface integral of $f$ over a surface $O$ gives the total charge of the surface.

For Vector Fields §

• If $\vec{F}$ is the fluid velocity vector field, then the flux integral returns the amount of fluid that flowed through a surface $S$, per unit time. For example, if $S$ is a a flat rectangle, the flux integral says how much fluid was perpendicular to $S$ per unit time.