The surface-area element is actually the area of an "infinitesimal" parallelogram attached to the surface and formed by to "infinitesimal" tangent vectors along two intersecting coordinate curves. If the surface is represented via the position vector R, then and will be infinitesimal tangent vectors along the coordinate curves and , respectively. The cross product of these tangent vectors is normal to the surface, and its length is the area of the infinitesimal parallelogram formed by these tangent vectors at . Table 6.3.14(a) summarizes these relationships.
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Table 6.3.14(a) An infinitesimal normal N on a surface defined by the position vector R
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The surface-area element is the length of N, so = .