3: Surface Integrals

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• 3.1: Parametrized Surfaces
For many applications we will need to use integrals over surfaces. One obvious one is just computing surface areas. Another is computing the rate at which fluid traverses a surface. The first step is to simply specify surfaces carefully.
• 3.2: Tangent Planes
If you are confronted with a complicated surface and want to get some idea of what it looks like near a specific point, probably the first thing that you will do is find the plane that best approximates the surface near the point. That is, find the tangent plane to the surface at the point.
• 3.3: Surface Integrals
We are now going to define two types of integrals over surfaces.
• 3.4: Interpretation of Flux Integrals
We defined, in §3.3, two types of integrals over surfaces. We have seen, in §3.3.4, some applications that lead to integrals of the type $$\iint_S \rho\,\text{d}S\text{.}$$ We now look at one application that leads to integrals of the type $$\iint_S \vecs{F} \cdot\hat{\textbf{n}}\,\text{d}S\text{.}$$ Recall that integrals of this type are called flux integrals.
• 3.5: Orientation of Surfaces
One thing that made the flux integrals of the last section possible is that we could choose sensible unit normal vectors $$\hat{\textbf{n}}\text{.}$$ In this section, we explain this more carefully.

Thumbnail: The total flux through the surface is found by adding up for each patch. In the limit as the patches become infinitesimally small, this is the surface integral. (CC0; Chetvorno via Wikipedia)

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