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 $\mathbf h$.