Calculus Small Changes, Big Ideas

Section 19.5 Integrals over oriented surfaces

• For a surface to be oriented it has to have two sides. We take the tangent plane at each point and notice there are two unit normal vectors \(\uv n\) and \(-\uv n\text{.}\) As we move around the surface, these two normal vary smoothly. The surface is oriented if it is impossible to move around the surface following \(\uv n\) and end up with it overlapping \(-\uv n\text{.}\) Alternatively, we can make sure that a counterclockwise loop can’t be moved around the surface and end up with it going clockwise.

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• An example of a non-orientable surface is a Möbius strip.

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• If \(\bv{\cal{S}}\) is an oriented surface, then the surface integral of \(\bv F\) over \(\bv{\cal{S}}\) is

  \begin{equation*} \iint_{\bv{\cal{S}}}\bv F\cdot\dd{\bv{\cal{S}}}=\iint_{D} \bv F\cdot\qty(\pdv{\bv{r}}{u}\times\pdv{\bv{r}}{v})\dd{A}\text{.} \end{equation*}

  This can also be written as

  \begin{equation*} \iint_{D} \qty(\bv F\cdot\frac{\bv r'_{u}\times\bv r'_{v}}{\norm{\bv r'_u\times\bv r'_v}}\norm{\bv r'_u\times\bv r'_v})\dd{A}=\iint_{\cl S}\bv F\cdot\uv n\,\dd{\bv{\cal{S}}}\text{.} \end{equation*}

  That is, this surface integral represents the flux of \(\cl F\) through the surface.

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• For closed surfaces, positive orientation means the normals point out of the surface.

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• Can do examples with electric flux, heat flux, etc.

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