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Concept
Surface integral
Summary
In mathematics, particularly multivariable calculus, a surface integral is a generalization of multiple integrals to integration over surfaces. It can be thought of as the double integral analogue of the line integral. Given a surface, one may integrate a scalar field (that is, a function of position which returns a scalar as a value) over the surface, or a vector field (that is, a function which returns a vector as value). If a region R is not flat, then it is called a surface as shown in the illustration.
Surface integrals have applications in physics, particularly with the theories of classical electromagnetism.
Assume that f is a scalar, vector, or tensor field defined on a surface S.
To find an explicit formula for the surface integral of f over S, we need to parameterize S by defining a system of curvilinear coordinates on S, like the latitude and longitude on a sphere. Let such a parameterization be r(s, t), where (s, t) varies in some region T in the plane. Then, the surface integral is given by
where the expression between bars on the right-hand side is the magnitude of the cross product of the partial derivatives of r(s, t), and is known as the surface element (which would, for example, yield a smaller value near the poles of a sphere. where the lines of longitude converge more dramatically, and latitudinal coordinates are more compactly spaced). The surface integral can also be expressed in the equivalent form
where g is the determinant of the first fundamental form of the surface mapping r(s, t).
For example, if we want to find the surface area of the graph of some scalar function, say z = f(x, y), we have
where r = (x, y, z) = (x, y, f(x, y)). So that , and . So,
which is the standard formula for the area of a surface described this way. One can recognize the vector in the second-last line above as the normal vector to the surface.
Because of the presence of the cross product, the above formulas only work for surfaces embedded in three-dimensional space.
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