Surface integral

An integral over a surface. Let a surface $ S $ in the three-dimensional Euclidean space $ \mathbb{R}^{3} $, possibly with self-intersections, be specified by the vector representation $$ \mathbf{r} = \mathbf{r}(u,v) \qquad (1) $$ in Cartesian coordinates $ (x,y,z) $, where $ \mathbf{r}(u,v) = (x(u,v),y(u,v),z(u,v)) $ is a continuously-differentiable vector function defined on the closure $ \overline{G} $ of a two-dimensional Jordan-measurable domain $ G $ lying in the plane with Cartesian coordinates $ u $ and $ v $. Let $$ g_{11} \stackrel{\text{df}}{=} \left\langle \frac{\partial \mathbf{r}}{\partial u},\frac{\partial \mathbf{r}}{\partial u} \right\rangle, \qquad g_{12} \stackrel{\text{df}}{=} \left\langle \frac{\partial \mathbf{r}}{\partial u},\frac{\partial \mathbf{r}}{\partial v} \right\rangle, \qquad g_{22} \stackrel{\text{df}}{=} \left\langle \frac{\partial \mathbf{r}}{\partial v},\frac{\partial \mathbf{r}}{\partial v} \right\rangle $$ be the coefficients of the first fundamental form of $ S $. If $ F(x,y,z) $ is a function defined on $ S $, i.e. a function $ F(x(u,v),y(u,v),z(u,v)) $, then the following defines a surface integral of the first kind (or a integral over the surface area): $$ \iint_{S} F(x,y,z) ~ \mathrm{d}{S} \stackrel{\text{df}}{=} \iint_{G} F(x(u,v),y(u,v),z(u,v)) \sqrt{g_{11} g_{22} - g_{12}^{2}} ~ \mathrm{d}{u} ~ \mathrm{d}{v}. \qquad (2) $$

This definition is independent of the representation of the surface. A surface integral of the first kind is the limit of corresponding Riemann sums, which may be described in terms related to the surface. For example, if $ F(x(u,v),y(u,v),z(u,v)) $ is a Riemann-integrable function on $ \overline{G} $, if $ \tau = (S_{i})_{i = 1}^{k} $ is a decomposition of $ S $ into parts $ S_{i} $ that are images under the mapping (1) of sets $ E_{i} \subseteq \overline{G} $ forming a decomposition $ \tau_{0} = (E_{i})_{i = 1}^{k} $ of $ \overline{G} $ (see Multiple integral), and if $$ \operatorname{mes}(S_{i}) \stackrel{\text{df}}{=} \iint_{E_{i}} \sqrt{g_{11} g_{22} - g_{12}^{2}} ~ \mathrm{d}{u} ~ \mathrm{d}{v} $$ is the area of $ S_{i} $, then $$ \iint_{S} F(x,y,z) ~ \mathrm{d}{S} \stackrel{\text{df}}{=} \lim_{\delta_{\tau_{0}} \to 0} \sum_{i = 1}^{k} F(x(\xi_{i},\eta_{i}),y(\xi_{i},\eta_{i}),z(\xi_{i},\eta_{i})) \cdot \operatorname{mes}(S_{i}), $$ where $ \delta_{\tau_{0}} $ is the fineness (mesh) of $ \tau_{0} $ and $ (\xi_{i},\eta_{i}) \in E_{i} $. If $ S $ is explicitly specified in the form $ z = f(x,y) $, where $ (x,y) \in \overline{G} $, then (2) becomes $$ \iint_{S} F(x,y,z) ~ \mathrm{d}{S} = \iint_{G} f(x,y,f(x,y)) \sqrt{1 + \left( \frac{\partial f}{\partial x} \right)^{2} + \left( \frac{\partial f}{\partial y} \right)^{2}} ~ \mathrm{d}{x} ~ \mathrm{d}{y}. $$

If there are no singular points on a surface $ S $ with vector representation (1), i.e. if $ \mathbf{r}_{u} \times \mathbf{r}_{v} \neq \mathbf{0} $, then $ S $ can be oriented by selecting a continuous unit normal vector $ \mathbf{n} = (\cos(\alpha),\cos(\beta),\cos(\gamma)) $ on it, for example $$ \mathbf{n} = \frac{\mathbf{r}_{u} \times \mathbf{r}_{v}}{\| \mathbf{r}_{u} \times \mathbf{r}_{v} \|}. $$

For an oriented surface $ S^{+} $, one defines the surface integrals of the second kind by $$ \left. \begin{align} \iint_{S^{+}} F(x,y,z) ~ \mathrm{d}{x} ~ \mathrm{d}{y} & \stackrel{\text{df}}{=} \iint_{S} F(x,y,z) \cos(\gamma) ~ \mathrm{d}{S} \\ \iint_{S^{+}} F(x,y,z) ~ \mathrm{d}{y} ~ \mathrm{d}{z} & \stackrel{\text{df}}{=} \iint_{S} F(x,y,z) \cos(\alpha) ~ \mathrm{d}{S} \\ \iint_{S^{+}} F(x,y,z) ~ \mathrm{d}{z} ~ \mathrm{d}{x} & \stackrel{\text{df}}{=} \iint_{S} F(x,y,z) \cos(\beta) ~ \mathrm{d}{S} \end{align} \right\} \qquad (3) $$ where surface integrals of the first kind appear on the right-hand side. If $ S^{-} $ is the surface $ S $ with orientation opposite to that of $ S^{+} $, then $$ \iint_{S^{-}} F(x,y,z) ~ \mathrm{d}{x} ~ \mathrm{d}{y} = - \iint_{S^{+}} F(x,y,z) ~ \mathrm{d}{x} ~ \mathrm{d}{y}. $$ Similar equations apply to the other surface integrals of the second kind in (3). Surface integrals of the second kind resemble those of the first kind in being the limits of Riemann sums, which can be described in terms of the surface.

If $$ \mathbf{n} = \frac{\mathbf{r}_{u} \times \mathbf{r}_{v}}{\| \mathbf{r}_{u} \times \mathbf{r}_{v} \|}, $$ then $$ \iint_{S^{+}} F(x,y,z) ~ \mathrm{d}{x} ~ \mathrm{d}{y} = \iint_{G} F(x(u,v),y(u,v),z(u,v)) \frac{\partial(x,y)}{\partial(u,v)} ~ \mathrm{d}{u} ~ \mathrm{d}{v}. $$ Similar formulas apply to the other surface integrals of the second kind in (3).

In particular, for the case of a surface $ z = f(x,y) $, where $ (x,y) \in \overline{G} $, \begin{align} \iint_{S^{+}} F(x,y,z) ~ \mathrm{d}{x} ~ \mathrm{d}{y} &= \iint_{G} F(x,y,f(x,y)) ~ \mathrm{d}{x} ~ \mathrm{d}{y}, \\ \iint_{S^{-}} F(x,y,z) ~ \mathrm{d}{x} ~ \mathrm{d}{y} & = - \iint_{G} F(x,y,f(x,y)) ~ \mathrm{d}{x} ~ \mathrm{d}{y}. \end{align}

The first of these integrals is called an integral over the “upper” side of $ S $, and the second over the “lower” side. This terminology is used because the vector $$ \mathbf{n} = \frac{\mathbf{r}_{u} \times \mathbf{r}_{v}}{\| \mathbf{r}_{u} \times \mathbf{r}_{v} \|} $$ forms an acute angle with the $ z $-axis in the case of an explicit specification of $ S $, i.e. when $ x = u $, $ y = v $ and $ z = f(x,y) $, i.e. it is directed “upwards”, while in the second case, it forms an obtuse angle and is thus directed “downwards”.

If a smooth surface $ S $ is the boundary of a bounded domain and $ S^{+} $ denotes its orientation by means of the outward normal vector, hence $ S^{-} $ is determined by the inward normal vector, with respect to the domain, then the surface integrals of the second kind over the oriented surface $ S^{+} $ are called surface integrals with respect to the outside of the surface, while those over $ S^{-} $ are called surface integrals with respect to the inside.

The surface integrals for a piecewise-smooth surface that can be divided into a finite number of parts, each of which has a vector representation (1), are defined as the sums of the surface integrals over the corresponding parts. Thus, surface integrals defined with respect to piecewise-smooth surfaces are not dependent on the method of dividing the surfaces into those parts.

The Ostrogradski formula establishes a relationship between a triple integral over a three-dimensional bounded domain and the surface integrals over its boundary, while the Stokes formula gives a relationship between a surface integral and the curvilinear integral over the contour representing the boundary.

The surface integral $ \displaystyle \iint_{S} 1 ~ \mathrm{d}{S} $ is equal to the area of $ S $. If a mass with density $ F(x,y,z) $ is distributed on $ S $, then the surface integral $ \displaystyle \iint_{S} F(x,y,z) ~ \mathrm{d}{S} $ is equal to the entire mass. If $ \mathbf{a} = \mathbf{a}(x,y,z) $ is a vector function specified on a surface $ S $ that is oriented by means of the unit normal vector $ \mathbf{n} $, then the surface integral $$ \iint_{S} \langle \mathbf{a},\mathbf{n} \rangle ~ \mathrm{d}{S} $$ is called the flux of the vector field $ \mathbf{a} $ through $ S $. Clearly, this is independent of the choice of a coordinate system in $ \mathbb{R}^{3} $. One also uses surface integrals to describe a double-layer potential or a simple-layer potential.

If the surface $ S $ is a differentiable $ 2 $-manifold, if the continuously-differentiable non-negative functions $ {\phi_{j}}(x,y,z) $, where $ j \in \{ 1,\ldots,m \} $, form a partition of unity on $ S $, i.e. the support of each function is contained in some chart of $ S $ and $ \displaystyle \sum_{j = 1}^{m} {\phi_{j}}(x,y,z) = 1 $ for each point $ (x,y,z) \in S $, and if $ F(x,y,z) $ is a function defined on $ S $, then by definition, $$ \iint_{S} F(x,y,z) ~ \mathrm{d}{S} \stackrel{\text{df}}{=} \sum_{j = 1}^{m} \left[ \iint_{S} {\phi_{j}}(x,y,z) F(x,y,z) ~ \mathrm{d}{S} \right], \qquad (4) $$ where each integral on the right-hand side is understood in the sense of (2). If $ S^{+} $ is an oriented $ 2 $-manifold, then $$ \iint_{S^{+}} F(x,y,z) ~ \mathrm{d}{x} ~ \mathrm{d}{y} \stackrel{\text{df}}{=} \sum_{j = 1}^{m} \left[ \iint_{S^{+}} {\phi_{j}}(x,y,z) F(x,y,z) ~ \mathrm{d}{x} ~ \mathrm{d}{y} \right]. \qquad (5) $$ The other surface integrals of the second kind in (3) are defined similarly. The definitions (4) and (5) are independent of the choice of a partition of unity in $ S $.

References

[1]	V.A. Il’in, E.G. Poznyak, “Fundamentals of mathematical analysis”, 1–2, MIR (1982). (Translated from Russian)
[2]	L.D. Kudryavtsev, “Mathematical analysis”, Moscow (1973). (In Russian)
[3]	S.M. Nikol’skii, “A course of mathematical analysis”, 1–2, MIR (1977). (Translated from Russian)
[4]	B.A. Dubrovin, A.T. Fomenko, S.P. Novikov, “Modern geometry”, Springer (1987). (Translated from Russian)
[5]	A.S. Mishchenko, A.T. Fomenko, “A course of differential geometry and topology”, MIR (1988). (Translated from Russian)

Comments

Ostrogradski’s formula is usually called Gauss’ formula in the West. The phrase divergence theorem is also used.

In vector notation, one can write $$ g_{11} g_{22} - g_{12}^{2} = \left\| \frac{\partial \mathbf{r}}{\partial u} \times \frac{\partial \mathbf{r}}{\partial v} \right\|. $$ The distinction between surface integrals of the first and the second kind is not common in Western literature.

Integrals over differentiable $ 2 $-dimensional manifolds can be conveniently defined using $ 2 $-forms. See, e.g., [a4] and Integration on manifolds.

References

[a1]	G.E. Shilov, “Mathematical analysis”, 1–2, M.I.T. (1974). (Translated from Russian)
[a2]	T.M. Apostol, “Calculus”, 2, Waltham (1969).
[a3]	T.M. Apostol, “Mathematical analysis”, Addison-Wesley (1974).
[a4]	M. Berger, B. Gostiaux, “Differential geometry: manifolds, curves, and surfaces”, Springer (1988). (Translated from French)
[a5]	M. Spivak, “A comprehensive introduction to differential geometry”, 1979, Publish or Perish, pp. 1–5.
[a6]	J.J. Stoker, “Differential geometry”, Wiley (Interscience) (1969).
[a7]	D.J. Struik, “Lectures on classical differential geometry”, Addison-Wesley (1961).
[a8]	R.C. Buck, “Advanced calculus”, McGraw-Hill (1965).
[a9]	W. Fleming, “Functions of several variables”, Springer (1977).
[a10]	J. Marsden, A. Weinstein, “Calculus”, 3, Springer (1988).