Summary of Surface Integrals

Essential Concepts

  • 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 [latex]\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}[/latex]. To calculate a surface integral with an integrand that is a vector field, use [latex]\displaystyle\iint_{S} {\bf{F}}\cdot {d{\bf{S}}}=\displaystyle\iint_{S} {\bf{F}}\cdot {\bf{N}}dS[/latex]
  • If [latex]S[/latex] is a surface, then the area of [latex]S[/latex] is [latex]\displaystyle\iint_{S}dS[/latex]

Key Equations

  • Scalar surface integral
    [latex]\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}[/latex]
  • Flux integral
    [latex]\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[/latex]

Glossary

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 [latex]{\bf{r}}(u, v) = {\langle} x (u, v), y (u, v), z (u, v) {\rangle}[/latex]
parameterized surface
a surface given by a description of the form [latex]{\bf{r}}(u, v) = {\langle} x (u, v), y (u, v), z (u, v) {\rangle}[/latex] , where the parameters u and v vary over a parameter domain in the uv-plane
regular parameterization
parameterization [latex]{\bf{r}}(u, v) = {\langle} x (u, v), y (u, v), z (u, v) {\rangle}[/latex] such that [latex]{\bf{r}}_{u}{\times}{\bf{r}}_{v}[/latex] is not zero for point [latex](u, v)[/latex] in the parameter domain
surface area
the area of surface S given by the surface integral [latex]\displaystyle{\int_{} {\int_{S} d{\bf{S}}}} [/latex]
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