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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)

    This page titled 3: Surface Integrals is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Joel Feldman, Andrew Rechnitzer and Elyse Yeager via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

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