surface integrals review - V9.1 Surface Integrals Surface...

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V9.1 Surface Integrals Surface integrals are a natural generalization of line integrals: instead of integrating over a curve, we integrate over a surface in 3-space. Such integrals are important in any of the subjects that deal with continuous media (solids, fluids, gases), as well as subjects that deal with force fields, like electromagnetic or gravitational fields. Though most of our work will be spent seeing how surface integrals can be calculated and what they are used for, we first want to indicate briefly how they are defined. The surface integral of the (continuous) function f ( x, y, z ) over the surface S is denoted by (1) f ( x, y, z ) dS . S You can think of dS as the area of an infinitesimal piece of the surface S . To define the integral (1), we subdivide the surface S into small pieces having area S i , pick a point ( x i , y i , z i ) in the i -th piece, and form the Riemann sum (2) f ( x i , y i , z i ) S i . w As the subdivision of S gets finer and finer, the corresponding sums (2) approach a limit which does not depend on the choice of the points or how the surface was subdivided. The
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