Surface integral

From Glossary of Meteorology

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.

Copyright 2022 American Meteorological Society (AMS). For permission to reuse any portion of this work, please contact Any use of material in this work that is determined to be “fair use” under Section 107 of the U.S. Copyright Act (17 U.S. Code § 107) or that satisfies the conditions specified in Section 108 of the U.S.Copyright Act (17 USC § 108) does not require AMS’s permission. Republication, systematic reproduction, posting in electronic form, such as on a website or in a searchable database, or other uses of this material, except as exempted by the above statement, require written permission or a license from AMS. Additional details are provided in the AMS Copyright Policy statement.