Surface Integrals

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.

The definition of surface integral relies on splitting the surface into small surface elements.

An illustration of a single surface element. These elements are made infinitesimally small, by the limiting process, so as to approximate the surface.

Surface integrals of scalar fields

To find an explicit formula for the surface integral over a surface 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

$\iint _{S}f\,\mathrm {d} S=\iint _{T}f(\mathbf {r} (s,t))\left\|{\partial \mathbf {r} \over \partial s}\times {\partial \mathbf {r} \over \partial t}\right\|\mathrm {d} s\,\mathrm {d} t$

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

$\iint _{S}f\,\mathrm {d} S=\iint _{T}f(\mathbf {r} (s,t)){\sqrt {g}}\,\mathrm {d} s\,\mathrm {d} t$

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

$A=\iint _{S}\,\mathrm {d} S=\iint _{T}\left\|{\partial \mathbf {r} \over \partial x}\times {\partial \mathbf {r} \over \partial y}\right\|\mathrm {d} x\,\mathrm {d} y$

where r = (x, y, z) = (x, y, f(x, y)). So that ${\partial \mathbf {r} \over \partial x}=(1,0,f_{x}(x,y))$, and ${\partial \mathbf {r} \over \partial y}=(0,1,f_{y}(x,y))$. So,

${\begin{aligned}A&{}=\iint _{T}\left\|\left(1,0,{\partial f \over \partial x}\right)\times \left(0,1,{\partial f \over \partial y}\right)\right\|\mathrm {d} x\,\mathrm {d} y\\&{}=\iint _{T}\left\|\left(-{\partial f \over \partial x},-{\partial f \over \partial y},1\right)\right\|\mathrm {d} x\,\mathrm {d} y\\&{}=\iint _{T}{\sqrt {\left({\partial f \over \partial x}\right)^{2}+\left({\partial f \over \partial y}\right)^{2}+1}}\,\,\mathrm {d} x\,\mathrm {d} y\end{aligned}}$

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.

Note that because of the presence of the cross product, the above formulas only work for surfaces embedded in three-dimensional space.

This can be seen as integrating a Riemannian volume form on the parameterized surface, where the metric tensor is given by the first fundamental form of the surface.

Surface integrals of vector fields

Consider a vector field v on a surface S, that is, for each r = (x, y, z) in S, v(r) is a vector.

The surface integral can be defined component-wise according to the definition of the surface integral of a scalar field; the result is a vector. This applies for example in the expression of the electric field at some fixed point due to an electrically charged surface, or the gravity at some fixed point due to a sheet of material.

Alternatively, if we integrate the normal component of the vector field over the surface, the result is a scalar, usually called the flux passing through the surface. Imagine that we have a fluid flowing through S, such that v(r) determines the velocity of the fluid at r. The flux is defined as the quantity of fluid flowing through S per unit time.

This illustration implies that if the vector field is tangent to S at each point, then the flux 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 a normal component, then only the normal component contributes to the flux. Based on this reasoning, to find the flux, we need to take the dot product of v with the unit surface normal n to S at each point, which will give us a scalar field, and integrate the obtained field as above. We find the formula

${\begin{aligned}\iint _{S}{\mathbf {v} }\cdot \mathrm {d} {\mathbf {S} }&=\iint _{S}\left({\mathbf {v} }\cdot {\mathbf {n} }\right)\,\mathrm {d} S\\&{}=\iint _{T}\left({\mathbf {v} }(\mathbf {r} (s,t))\cdot {{\frac {\partial \mathbf {r} }{\partial s}}\times {\frac {\partial \mathbf {r} }{\partial t}} \over \left\|{\frac {\partial \mathbf {r} }{\partial s}}\times {\frac {\partial \mathbf {r} }{\partial t}}\right\|}\right)\left\|{\frac {\partial \mathbf {r} }{\partial s}}\times {\frac {\partial \mathbf {r} }{\partial t}}\right\|\mathrm {d} s\,\mathrm {d} t\\&{}=\iint _{T}{\mathbf {v} }(\mathbf {r} (s,t))\cdot \left({\frac {\partial \mathbf {r} }{\partial s}}\times {\frac {\partial \mathbf {r} }{\partial t}}\right)\mathrm {d} s\,\mathrm {d} t.\end{aligned}}$

The cross product on the right-hand side of this expression is a (not necessarily unital) surface normal determined by the parametrisation.

This formula defines the integral on the left (note the dot and the vector notation for the surface element).

We may also interpret this as a special case of integrating 2-forms, where we identify the vector field with a 1-form, and then integrate its Hodge dual over the surface. This is equivalent to integrating $\left\langle \mathbf {v} ,\mathbf {n} \right\rangle \mathrm {d} S$ over the immersed surface, where $\mathrm {d} S$ is the induced volume form on the surface, obtained by interior multiplication of the Riemannian metric of the ambient space with the outward normal of the surface.

Surface integrals of differential 2-forms

Let

$f=f_{z}\,\mathrm {d} x\wedge \mathrm {d} y+f_{x}\,\mathrm {d} y\wedge \mathrm {d} z+f_{y}\,\mathrm {d} z\wedge \mathrm {d} x$

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

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

be an orientation preserving parametrization of S with $(s,t)$ in D. Changing coordinates from $(x,y)$ to $(s,t)$, the differential forms transform as

$\mathrm {d} x={\frac {\partial x}{\partial s}}\mathrm {d} s+{\frac {\partial x}{\partial t}}\mathrm {d} t$

$\mathrm {d} y={\frac {\partial y}{\partial s}}\mathrm {d} s+{\frac {\partial y}{\partial t}}\mathrm {d} t$

So $\mathrm {d} x\wedge \mathrm {d} y$ transforms to ${\frac {\partial (x,y)}{\partial (s,t)}}\mathrm {d} s\wedge \mathrm {d} t$, where ${\frac {\partial (x,y)}{\partial (s,t)}}$ denotes the determinant of the Jacobian of the transition function from $(s,t)$ to $(x,y)$. The transformation of the other forms are similar.

Then, the surface integral of f on S is given by

$\iint _{D}\left[f_{z}(\mathbf {r} (s,t)){\frac {\partial (x,y)}{\partial (s,t)}}+f_{x}(\mathbf {r} (s,t)){\frac {\partial (y,z)}{\partial (s,t)}}+f_{y}(\mathbf {r} (s,t)){\frac {\partial (z,x)}{\partial (s,t)}}\right]\,\mathrm {d} s\,\mathrm {d} t$

where

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

is the surface element 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 $f_{x}$, $f_{y}$ and $f_{z}$.

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.