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 over this surface a scalar field (that is, a function of position which returns a scalar as a value), 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 in the classical theories of electromagnetism and fluid mechanics.
Surface integrals of scalar fields 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
∬ S f d S = ∬ T f ( r ( s , t ) ) ‖ ∂ r ∂ s × ∂ r ∂ t ‖ d s d t {\displaystyle \iint _{S}f\,\mathrm {d} S=\iint _{T}f(\mathbf {r} (s,t))\left\|{\partial \mathbf {r} \over \partial s}\times {\partial \mathbf {r} \over \partial t}\right\|\mathrm {d} s\,\mathrm {d} t}
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
∬ S f d S = ∬ T f ( r ( s , t ) ) g d s d t {\displaystyle \iint _{S}f\,\mathrm {d} S=\iint _{T}f(\mathbf {r} (s,t)){\sqrt {g}}\,\mathrm {d} s\,\mathrm {d} t}
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
A = ∬ S d S = ∬ T ‖ ∂ r ∂ x × ∂ r ∂ y ‖ d x d y {\displaystyle A=\iint _{S}\,\mathrm {d} S=\iint _{T}\left\|{\partial \mathbf {r} \over \partial x}\times {\partial \mathbf {r} \over \partial y}\right\|\mathrm {d} x\,\mathrm {d} y}
where r = (x, y, z) = (x, y, f(x, y)). So that ∂ r ∂ x = ( 1 , 0 , f x ( x , y ) ) {\displaystyle {\partial \mathbf {r} \over \partial x}=(1,0,f_{x}(x,y))} , and ∂ r ∂ y = ( 0 , 1 , f y ( x , y ) ) {\displaystyle {\partial \mathbf {r} \over \partial y}=(0,1,f_{y}(x,y))} . So,
A
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