Surface integral

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surface integral

In rectangular Cartesian coordinates, the integral
where f(x, y, z) is a single-valued continuous function of x, y and z in a region in which S is a surface given by z = φ(x, y), and R is the projection of S on the xy-plane.

Similar surface integrals exist over surfaces x = φ(y, z) and y = φ(z, x). In vector notation the surface integral of a single-valued continuous vector F over the surface s can be written
where n is an outward directed unit vector normal to the surface. The surface integral for a closed surface may be related to a volume integral by the divergence theorem.
See line integral.