Summary of Surface Integrals

Surfaces can be parameterized, just as curves can be parameterized. In general, surfaces must be parameterized with two parameters.
Surfaces can sometimes be oriented, just as curves can be oriented. Some surfaces, such as a Möbius strip, cannot be oriented.
A surface integral is like a line integral in one higher dimension. The domain of integration of a surface integral is a surface in a plane or space, rather than a curve in a plane or space.
The integrand of a surface integral can be a scalar function or a vector field. To calculate a surface integral with an integrand that is a function, use $\displaystyle\iint_{S} f(x,y,z)dS=\displaystyle\iint_{D}f ({\bf{r}}(u,v)) \Arrowvert{{\bf{t}}_{u}\times{\bf{t}}_{v}} \Arrowvert{dA}$ . To calculate a surface integral with an integrand that is a vector field, use $\displaystyle\iint_{S} {\bf{F}}\cdot {d{\bf{S}}}=\displaystyle\iint_{S} {\bf{F}}\cdot {\bf{N}}dS$
If $S$ is a surface, then the area of $S$ is $\displaystyle\iint_{S}dS$

Key Equations

Scalar surface integral
$\displaystyle\iint_{S} f(x,y,z)dS=\displaystyle\iint_{D}f ({\bf{r}}(u,v)) \Arrowvert{{\bf{t}}_{u}\times{\bf{t}}_{v}} \Arrowvert{dA}$
Flux integral
$\displaystyle\iint_{S} {\bf{F}}\cdot {\bf{N}}dS=\displaystyle\iint_{S} {\bf{F}}\cdot {d{\bf{S}}}=\displaystyle\iint_{S} {\bf{F}}({\bf{r}}(u,v))\cdot\left({\bf{t}}_{u}\times{\bf{t}}_{v}\right)dA$

flux integral: another name for a surface integral of a vector field; the preferred term in physics and engineering

grid curves: curves on a surface that are parallel to grid lines in a coordinate plane

heat flow: a vector field proportional to the negative temperature gradient in an object

mass flux: the rate of mass flow of a fluid per unit area, measured in mass per unit time per unit area

orientation of a surface: if a surface has an “inner” side and an “outer” side, then an orientation is a choice of the inner or the outer side; the surface could also have “upward” and “downward” orientations

parameter domain (parameter space): the region of the uv plane over which the parameters u and v vary for parameterization ${\bf{r}}(u, v) = {\langle} x (u, v), y (u, v), z (u, v) {\rangle}$

parameterized surface: a surface given by a description of the form ${\bf{r}}(u, v) = {\langle} x (u, v), y (u, v), z (u, v) {\rangle}$ , where the parameters u and v vary over a parameter domain in the uv-plane

regular parameterization: parameterization ${\bf{r}}(u, v) = {\langle} x (u, v), y (u, v), z (u, v) {\rangle}$ such that ${\bf{r}}_{u}{\times}{\bf{r}}_{v}$ is not zero for point $(u, v)$ in the parameter domain

surface area: the area of surface S given by the surface integral $\displaystyle{\int_{} {\int_{S} d{\bf{S}}}}$

surface integral of a scalar-valued function: a surface integral in which the integrand is a scalar function

surface integral of a vector field: a surface integral in which the integrand is a vector field