# Surface Integral/Examples

## Examples of Surface Integrals

### 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 $\mathbf h$.