View source for Surface integral

An [[Integral|integral]] over a [[Surface|surface]]. Let a surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091350/s0913501.png" /> in the three-dimensional Euclidean space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091350/s0913502.png" />, possibly with self-intersections, be specified by the vector representation

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091350/s0913503.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>

in Cartesian coordinates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091350/s0913504.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091350/s0913505.png" /> is a continuously-differentiable vector function defined on the closure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091350/s0913506.png" /> of a two-dimensional Jordan-measurable domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091350/s0913507.png" /> lying in the plane with Cartesian coordinates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091350/s0913508.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091350/s0913509.png" />. Let

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091350/s09135010.png" /></td> </tr></table>

be the coefficients of the [[First fundamental form|first fundamental form]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091350/s09135011.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091350/s09135012.png" /> is a function defined on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091350/s09135013.png" />, i.e. a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091350/s09135014.png" />, then the following defines the surface integral of the first kind (or the integral over the surface area):

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091350/s09135015.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091350/s09135016.png" /></td> </tr></table>

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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091350/s09135017.png" /> is a Riemann-integrable function, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091350/s09135018.png" /> is a decomposition of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091350/s09135019.png" /> into parts <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091350/s09135020.png" /> that are images under the mapping (1) of sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091350/s09135021.png" /> forming a decomposition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091350/s09135022.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091350/s09135023.png" /> (see [[Multiple integral|Multiple integral]]) and if

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091350/s09135024.png" /></td> </tr></table>

is the area of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091350/s09135025.png" />, then

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091350/s09135026.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091350/s09135027.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091350/s09135028.png" /> is the fineness (mesh) of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091350/s09135029.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091350/s09135030.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091350/s09135031.png" /> is explicitly specified in the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091350/s09135032.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091350/s09135033.png" />, (2) becomes

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091350/s09135034.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091350/s09135035.png" /></td> </tr></table>

If there are no singular points on a surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091350/s09135036.png" /> with vector representation (1), i.e. if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091350/s09135037.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091350/s09135038.png" /> can be oriented by selecting a continuous unit normal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091350/s09135039.png" /> on it, for example

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091350/s09135040.png" /></td> </tr></table>

For an oriented surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091350/s09135041.png" /> one defines the surface integrals of the second kind by

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091350/s09135042.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</td></tr></table>

where surface integrals of the first kind appear on the right-hand side. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091350/s09135043.png" /> is the surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091350/s09135044.png" /> with orientation opposite to that of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091350/s09135045.png" />, then

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091350/s09135046.png" /></td> </tr></table>

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

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091350/s09135047.png" /></td> </tr></table>

then

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091350/s09135048.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091350/s09135049.png" /></td> </tr></table>

Similar formulas apply to the other surface integrals of the second kind in (3).

In particular, for the case of a surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091350/s09135050.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091350/s09135051.png" />,

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091350/s09135052.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091350/s09135053.png" /></td> </tr></table>

The first of these integrals is called an integral over the  "upper"  side of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091350/s09135054.png" /> and the second over the  "lower"  side. This terminology is used because the vector

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091350/s09135055.png" /></td> </tr></table>

forms an acute angle with the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091350/s09135056.png" />-axis in the case of an explicit specification of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091350/s09135057.png" />, i.e. when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091350/s09135058.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091350/s09135059.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091350/s09135060.png" />, 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091350/s09135061.png" /> is the boundary of a bounded domain and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091350/s09135062.png" /> denotes its orientation by means of the outward normal, hence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091350/s09135063.png" /> is determined by the inward normal, with respect to the domain, then the surface integrals of the second kind over the oriented surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091350/s09135064.png" /> are called surface integrals with respect to the outside of the surface, while those over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091350/s09135065.png" /> 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|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|Stokes formula]] gives a relationship between a surface integral and the curvilinear integral over the contour representing the boundary.

The surface integral <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091350/s09135066.png" /> is equal to the [[Area|area]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091350/s09135067.png" />. If a mass with density <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091350/s09135068.png" /> is distributed on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091350/s09135069.png" />, then the surface integral <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091350/s09135070.png" /> is equal to the entire mass. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091350/s09135071.png" /> is a vector function specified on a surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091350/s09135072.png" /> that is oriented by means of the unit normal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091350/s09135073.png" />, then the surface integral

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091350/s09135074.png" /></td> </tr></table>

is called the flux of the vector field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091350/s09135075.png" /> through <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091350/s09135076.png" />. Clearly, this is independent of the choice of a coordinate system in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091350/s09135077.png" />. One also uses surface integrals to describe a [[Double-layer potential|double-layer potential]] or a [[Simple-layer potential|simple-layer potential]].

If the surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091350/s09135078.png" /> is a differentiable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091350/s09135079.png" />-manifold, if the continuously-differentiable non-negative functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091350/s09135080.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091350/s09135081.png" />, form a partition of unity on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091350/s09135082.png" />, i.e. the support of each function is contained in some [[Chart|chart]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091350/s09135083.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091350/s09135084.png" /> for each point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091350/s09135085.png" />, and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091350/s09135086.png" /> is a function defined on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091350/s09135087.png" />, then by definition

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091350/s09135088.png" /></td> <td valign="top" style="width:5%;text-align:right;">(4)</td></tr></table>

where each integral on the right-hand side is understood in the sense of (2). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091350/s09135089.png" /> is an oriented two-dimensional manifold, then

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091350/s09135090.png" /></td> <td valign="top" style="width:5%;text-align:right;">(5)</td></tr></table>

The other types of 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091350/s09135091.png" />.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  V.A. Il'in,   E.G. Poznyak,   "Fundamentals of mathematical analysis" , '''1–2''' , MIR  (1982)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  L.D. Kudryavtsev,   "Mathematical analysis" , Moscow  (1973)  (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  S.M. Nikol'skii,   "A course of mathematical analysis" , '''1–2''' , MIR  (1977)  (Translated from Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  B.A. Dubrovin,   A.T. Fomenko,   S.P. Novikov,   "Modern geometry" , Springer  (1987)  (Translated from Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  A.S. Mishchenko,   A.T. Fomenko,   "A course of differential geometry and topology" , MIR  (1988)  (Translated from Russian)</TD></TR></table>



====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

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091350/s09135092.png" /></td> </tr></table>

The distinction between surface integrals of the first and the second kind is not common in Western literature.

Integrals over differentiable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091350/s09135093.png" />-dimensional manifolds can be conveniently defined using <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091350/s09135094.png" />-forms. See, e.g., [[#References|[a4]]] and [[Integration on manifolds|Integration on manifolds]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  G.E. Shilov,   "Mathematical analysis" , '''1–2''' , M.I.T.  (1974)  (Translated from Russian)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  T.M. Apostol,   "Calculus" , '''2''' , Waltham  (1969)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  T.M. Apostol,   "Mathematical analysis" , Addison-Wesley  (1974)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  M. Berger,   B. Gostiaux,   "Differential geometry: manifolds, curves, and surfaces" , Springer  (1988)  (Translated from French)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  M. Spivak,   "A comprehensive introduction to differential geometry" , '''1979''' , Publish or Perish  pp. 1–5</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  J.J. Stoker,   "Differential geometry" , Wiley (Interscience)  (1969)</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  D.J. Struik,   "Lectures on classical differential geometry" , Addison-Wesley  (1961)</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  R.C. Buck,   "Advanced calculus" , McGraw-Hill  (1965)</TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top">  W. Fleming,   "Functions of several variables" , Springer  (1977)</TD></TR><TR><TD valign="top">[a10]</TD> <TD valign="top">  J. Marsden,   A. Weinstein,   "Calculus" , '''3''' , Springer  (1988)</TD></TR></table>