Surface integral

In mathematics, 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 its scalar fields (that is, functions which return scalars as values), and vector fields (that is, functions which return vectors as values). In mathematics, 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 its scalar fields (that is, functions which return scalars as values), and vector fields (that is, functions which return vectors as values). Surface integrals have applications in physics, particularly with the theories of classical electromagnetism. To find an explicit formula for the surface integral, we need to parameterize the surface of interest, S, by considering a system of curvilinear coordinates on S, like the latitude and longitude on a sphere. Let such a parameterization be x(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 x(s, t), and is known as the surface element. 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 x(s, t). For example, if we want to find the surface area of the graph of some scalar function, say z = f ( x , y ) {displaystyle z=f,(x,y)} , we have where r = ( x , y , z ) = ( x , y , f ( x , y ) ) {displaystyle mathbf {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, which is the standard formula for the area of a surface described this way. One can recognize the vector in the second line above as the normal vector to the surface.