Definition:Surface Integral

Definition

Let $S$ be a surface in a vector field $\mathbf F$.

Let $\d S$ be a small element of $S$.

Let $\mathbf v$ be the vector induced by $\mathbf F$ at the middle of $\d S$.

Let $\mathbf {\hat n}$ denote the positive unit normal to $S$ at $\d S$.

Let $\mathbf v$ make an angle $\theta$ with $\mathbf {\hat n}$.

Hence:

$\mathbf v \cdot \mathbf {\hat n} = v \cos \theta \rd S$

where:

$\cdot$ denotes dot product

$v$ denotes the magnitude of $\mathbf v$.

The surface integral of $\mathbf v$ over $S$ is therefore defined as:

$\ds \iint_S \mathbf v \cdot \mathbf {\hat n} \rd S = \iint_S v \cos \theta \rd S$

Also known as

A surface integral over a surface $S$ is also known as a total flux through $S$.

Physical Interpretation

Suppose $\mathbf v$ is interpreted as the velocity of some fluid in motion through a surface $S$.

Let $P$ be a point $P$ on $S$ at which a positive unit normal $\mathbf {\hat n}$ is constructed.

The expression $\mathbf v \cdot \mathbf {\hat n} \rd S$ denotes the amount of fluid passing through $\d S$ perpendicular to $S$ in unit time.

It is sufficient to consider this normal component, as the tangential component contributes nothing to the flow through $\d S$.

Hence the surface integral $I = \ds \iint_S \mathbf v \cdot \mathbf {\hat n} \rd S = \iint_S v \cos \theta \rd S$ expresses the total amount of fluid passing through $S$ in unit time.

If $I$ is positive, then this means there is a net outflow of fluid through $S$ from some source.

If $I$ is negative, then this means there is a net inflow of fluid through $S$ to some sink.

If $I$ is zero, the inflow equals the outflow, and either there are no sources or sinks within $S$, or that if there are some, their net inflow and outflow are equal.

Examples

Fluid in Motion

Let $\mathbf v$ be the velocity within a body of fluid $B$ as a point-function.

Let $S$ be a surface through which $B$ is in motion.

Let $\d S$ be a small element of $S$ whose center is at a point $P$.

Then the flow rate of $B$ through $S$ is given by the surface integral:

$\ds \iint_S \mathbf v \cdot \mathbf {\hat n} \rd S$

where $\mathbf {\hat n}$ denotes the unit normal to $S$ at $\d S$ in the direction of flow of $B$.

Electric Flux

Let $\mathbf E$ be an electric field acting over a region of space $R$.

Let $S$ be a surface through which $\mathbf E$ acts.

Let $\d S$ be a small element of $S$ whose center is at a point $P$.

Then the electric flux through $S$ to which $\mathbf E$ gives rise is given by the surface integral:

$\ds \iint_S \mathbf E \cdot \mathbf {\hat n} \rd S$

where $\mathbf {\hat n}$ denotes the unit normal to $S$ at $\d S$ in the direction of flow of $\mathbf E$.

Magnetic Flux

Let $\mathbf M$ be an magnetic field acting over a region of space $R$.

Let $S$ be a surface through which $\mathbf M$ acts.

Let $\d S$ be a small element of $S$ whose center is at a point $P$.

Then the magnetic flux through $S$ to which $\mathbf M$ gives rise is given by the surface integral:

$\ds \iint_S \mathbf M \cdot \mathbf {\hat n} \rd S$

where $\mathbf {\hat n}$ denotes the unit normal to $S$ at $\d S$ in the direction of flow of $\mathbf M$.

Flow of Heat

Let $\mathbf h$ be the flow of heat within a body $B$ as a point-function..

Let $S$ be a surface through which $\mathbf h$ acts.

Let $\d S$ be a small element of $S$ whose center is at a point $P$.

Then the heat flow through $S$ to which $\mathbf h$ gives rise is given by the surface integral:

$\ds \iint_S \mathbf h \cdot \mathbf {\hat n} \rd S$

where $\mathbf {\hat n}$ denotes the unit normal to $S$ at $\d S$ in the direction of flow of $\mathbf h$.

Sources

• 1951: B. Hague: An Introduction to Vector Analysis (5th ed.) ... (previous) ... (next): Chapter $\text {II}$: The Products of Vectors: $3$. Line and Surface Integrals: $(2.12)$