Surface integral Guide, Meaning , Facts, Information and Description

In mathematics, using a surface integral is a calculus technique found in differential geometry; from the start it has found major applications in physics, especially in the classical theory of electromagnetism. Surface integrals are definite integrals taken over some area that may be a curved set in space; they can be thought of as the double integral analog of the line integral.

Suppose there exists a vector field A, defined at all points on a surface S, which exists entirely within a Euclidean space.

Consider a unit of area on S, small enough to be approximated by a lamina. Let an axial vector, δσ, defined at all points on S, be a normal vector to the unit of area considered on S and have length equal to the magnitude of the unit of area considered.

The sum of all A.δσ is:

In the limit that δσ'' tends to 0, this sum tends to the surface integral:

Surface integrals also exist for scalar fields and are similarly defined, except that for a scalar function f, the integral is evaluated of the scalar product of f and δσ

Various useful results for surface integrals can be derived using differential geometry and vector calculus, such as the Divergence theorem.

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