Surface integral

In mathematics, a surface integral is a definite integral taken over some surface that may be a curved set in space; it can be thought of as the double integral analog of the path integral. Given a surface, one can integrate over it scalar fields (that is, functions which return numbers as values), and vector fields (that is, functions which return vectors as values).

Surface integrals have applications in physics, especially in the classical theory of electromagnetism.

Surface integrals of scalar fields

Consider a surface S on which a scalar field f is defined. If we think of S as made of some material, and for each x in S the number f(x) is the density of material at x, then the surface integral of f over S is the mass of S. One approach to calculating the surface integral is then to split the surface in many very small pieces, assume that on each piece the density is approximately constant, find the mass of each piece by multiplying the density of the piece by its area, and then sum up the resulting numbers to find the total mass of S.

To find an explicit formula for the surface integral, we need to parametrize S by considering on S a system of curvilinear coordinates, like the latitude and longitude on a sphere. Let such a parametrization be x(s, t), where (s, t) varies in some region T in the plane. Then, the surface integral is given by

$\displaystyle \int_S f \,dS = \iint_T f(\mathbf{x}(s, t)) \left|{\partial \mathbf{x} \over \partial s}\times {\partial \mathbf{x} \over \partial t}\right| ds\, dt$

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).

Surface integrals of vector fields

Consider a vector field v on S, that is, for each x in S, v(x) is a vector. Then the integral of v over S is called the flux. Imagine that we have a fluid flowing through S, such that v(x) determines the direction and velocity of the fluid at x. Then the flux is the quantity of fluid flowing through S in unit amount of time.

This illustration implies that if the vector field is tangent to S at each point, then the integral of the vector field is zero, because the fluid just flows in parallel to S, and neither in nor out. This also implies that if v does not just flow along S, that is, if v has both a tangential and normal component, then only the normal component contributes to the flow. Based on this reasoning, to find the surface integral of v over S, we need to take the dot product of v with the unit surface normal to S at each point, which will give us a scalar field, and integrate the obtained field as above. We find the formula

$\displaystyle \int_S {\mathbf v}\cdot \,{\mathbf {dS}} = \int_S ({\mathbf v}\cdot {\mathbf n})\,dS=\iint_T {\mathbf v}(\mathbf{x}(s, t))\cdot \left({\partial \mathbf{x} \over \partial s}\times {\partial \mathbf{x} \over \partial t}\right) ds\, dt.$

The cross product on the right-hand side of this expression is a surface normal determined by the parametrization.

Surface integrals of differential 2-forms

Let

$\displaystyle f=f_{1} dx \wedge dy + f_{2} dy \wedge dz + f_{3} dz \wedge dx$

be a differential 2-form defined on the surface S, and let

$\displaystyle \mathbf{x} (s,t)=( x(s,t), y(s,t), z(s,t))\!$

be an orientation preserving parametrization of S with $\displaystyle (s,t)$ in D. Then, the surface integral of f on S is given by

$\displaystyle \iint_D \left[ f_{1} ( \mathbf{x} (s,t)) \frac{\partial(x,y)}{\partial(s,t)} + f_{2} ( \mathbf{x} (s,t))\frac{\partial(y,z)}{\partial(s,t)} + f_{3} ( \mathbf{x} (s,t))\frac{\partial(z,x)}{\partial(s,t)} \right]\, ds dt$

where

$\displaystyle {\partial \mathbf{x} \over \partial s}\times {\partial \mathbf{x} \over \partial t}=\left(\frac{\partial(y,z)}{\partial(s,t)}, \frac{\partial(z,x)}{\partial(s,t)}, \frac{\partial(x,y)}{\partial(s,t))}\right)$

is the surface normal to S.

Let us note that the surface integral of this 2-form is the same as the surface integral of the vector field which has as components $\displaystyle f_1$ , $\displaystyle f_2$ and $\displaystyle f_3.$

Theorems involving surface integrals

Various useful results for surface integrals can be derived using differential geometry and vector calculus, such as the divergence theorem, and its generalization, Stokes' theorem.