Editing Surface integral

{{Short description|Integration over a non-flat region in 3D space}}
[[Image:Surface integral illustration.svg|right|thumb|The definition of the surface integral relies on splitting the surface into small surface elements.]]
{{Calculus|Multivariable}}
{{CS1 config|mode=cs1}}
In [[mathematics]], particularly [[multivariable calculus]], a '''surface integral'''  is a generalization of [[multiple integral]]s to [[Integral|integration]] over [[surface (differential geometry)|surface]]s. It can be thought of as the [[double integral]] analogue of the [[line integral]].  Given a surface, one may integrate over this surface a [[scalar field]] (that is, a [[function (mathematics)|function]] of position which returns a [[Scalar (mathematics)|scalar]] as a value), or a [[vector field]] (that is, a function which returns a [[Vector (geometric)|vector]] as value). If a region R is not flat, then it is called a [[Differential geometry of surfaces|''surface'']] as shown in the illustration.

Surface integrals have applications in [[physics]], particularly in the [[classical physics|classical]] theories of [[electromagnetism]] and [[fluid mechanics]]. 

[[Image:Surface integral1.svg|right|thumb|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 ==
Assume that ''f'' is  a scalar, vector, or tensor field defined on a surface ''S''.
To find an explicit formula for the surface integral of ''f'' over ''S'', we need to [[Coordinate system|parameterize]] ''S'' by defining a system of [[curvilinear coordinates]] on ''S'', like the [[Geographic coordinate system|latitude and longitude]] on a [[sphere]]. Let such a parameterization be {{math|'''r'''(''s'', ''t'')}}, where {{math|(''s'', ''t'')}} varies in some region {{mvar|T}} in the [[Cartesian coordinate system#Cartesian coordinates in two dimensions|plane]]. Then, the surface integral is given by

:<math>
\iint_S f \,\mathrm dS
= \iint_T f(\mathbf{r}(s, t)) \left\|{\partial \mathbf{r} \over \partial s} \times {\partial \mathbf{r} \over \partial t}\right\| \mathrm ds\, \mathrm dt
</math>
where the expression between bars on the right-hand side is the [[Magnitude (mathematics)|magnitude]] of the [[cross product]] of the [[partial derivative]]s of {{math|'''r'''(''s'', ''t'')}}, and is known as the surface [[volume element#Area element of a 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
:<math>
\iint_S f \,\mathrm dS
= \iint_T f(\mathbf{r}(s, t)) \sqrt{g} \, \mathrm ds\, \mathrm dt
</math>
where {{mvar|g}} is the determinant of the [[first fundamental form]] of the surface mapping {{math|'''r'''(''s'', ''t'')}}.<ref>{{Cite book|title = Advanced Calculus of Several Variables|last = Edwards|first = C. H.|publisher = Dover|year = 1994|isbn = 0-486-68336-2|location = Mineola, NY|pages = 335}}</ref><ref>{{SpringerEOM|title =  Surface Integral|last = Hazewinkel|first = Michiel|oldid=39869}}</ref>

For example, if we want to find the [[surface area]] of the graph of some scalar function, say {{math|1=''z'' = ''f''(''x'', ''y'')}}, we have
:<math>
A = \iint_S \,\mathrm dS
= \iint_T \left\|{\partial \mathbf{r} \over \partial x} \times {\partial \mathbf{r} \over \partial y}\right\| \mathrm dx\, \mathrm dy
</math>
where {{math|1='''r''' = (''x'', ''y'', ''z'') = (''x'', ''y'', ''f''(''x'', ''y''))}}.  So that <math>{\partial \mathbf{r} \over \partial x}=(1, 0, f_x(x,y))</math>, and <math>{\partial \mathbf{r} \over \partial y}=(0, 1, f_y(x,y))</math>.  So,
:<math>\begin{align}
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 dx\, \mathrm dy \\
&{} = \iint_T \left\|\left(-{\partial f \over \partial x}, -{\partial f \over \partial y}, 1\right)\right\| \mathrm dx\, \mathrm dy \\
&{} = \iint_T \sqrt{\left({\partial f \over \partial x}\right)^2+\left({\partial f \over \partial y}\right)^2+1}\, \,  \mathrm dx\, \mathrm dy
\end{align}</math>
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 [[surface normal|normal vector]] to the surface.

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==<!-- This section is linked from [[Flux]] -->
{{multiple image
| align             = right
| direction         = vertical|header
| image1            = Surface integral - vector field thru a surface.svg
| caption1          = A curved surface <math>S</math> with a vector field <math>\mathbf{F}</math> passing through it. The red arrows (vectors) represent the magnitude and direction of the field at various points on the surface
| width1            = 300
| image2            = Surface integral - parametrized surface.svg
| caption2          = Surface divided into small patches <math>dS = du\,dv</math> by a parameterization of the surface <math>[u(\mathbf{x}),v(\mathbf{x})]</math>
| width2            = 300
| image3            = Surface integral - normal component of field.svg
| caption3          = The flux through each patch is equal to the normal (perpendicular) component of the field <math>F_n(\mathbf{x}) = F(\mathbf{x})\cos \theta</math> at the patch's location <math>\mathbf{x}</math> multiplied by the area <math>dS</math>. The normal component is equal to the [[dot product]] of <math>\mathbf{F}(\mathbf{x})</math> with the unit normal vector <math>\mathbf{n}(\mathbf{x})</math> ''(blue arrows)''
| width3            = 200
| image4            = Surface integral - definition.svg
| caption4          = The total flux through the surface is found by adding up <math>\mathbf{F} \cdot \mathbf{n}\;dS</math> for each patch.  In the limit as the patches become infinitesimally small, this is the surface integral<br/><math display="inline">\iint_S \mathbf {F\cdot n}\;dS</math>
| width4            = 300
| footer            = 
}}
Consider a vector field '''v''' on a surface ''S'', that is, for each {{math|1='''r''' = (''x'', ''y'', ''z'')}} in ''S'', '''v'''('''r''') is a vector.

The integral of '''v''' on ''S'' was defined in the previous section. Suppose now that it is desired to integrate only
the [[normal component]] of the vector field over the surface, the result being a scalar, usually called the [[flux of a vector field|flux]] passing through the surface. For example, 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, on the surface ''S'', the fluid just flows along ''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. In other words, we have to integrate '''v''' with respect to the vector surface element <math>\mathrm{d}\mathbf s = {\mathbf n} \mathrm{d}s</math>, which is the vector normal to ''S'' at the given point, whose magnitude is <math>\mathrm{d}s = \|\mathrm{d}{\mathbf s}\|.</math>

We find the formula
:<math>\begin{align}
\iint_S {\mathbf v}\cdot\mathrm d{\mathbf {s}} &= \iint_S \left({\mathbf v}\cdot {\mathbf n}\right)\,\mathrm ds\\
&{}= \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 ds\, \mathrm dt\\
&{}=\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 ds\, \mathrm dt.
\end{align}</math>

The cross product on the right-hand side of the last 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 <math>\left\langle \mathbf{v}, \mathbf{n} \right\rangle \mathrm dS </math> over the immersed surface, where <math>\mathrm dS</math> 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
:<math> f=\mathrm dx \mathrm dy\,f_{z}  + \mathrm dy \mathrm dz\,f_{x}  + \mathrm dz  \mathrm dx\,f_{y} </math>
be a [[differential form|differential 2-form]] defined on a surface ''S'', and let

:<math>\mathbf{r} (s,t)=( x(s,t), y(s,t), z(s,t))</math>

be an [[orientability|orientation preserving]] parametrization of ''S'' with <math>(s,t)</math> in ''D''. Changing coordinates from <math>(x, y)</math> to <math>(s, t)</math>, the differential forms transform as

:<math>\mathrm dx=\frac{\partial x}{\partial s}\mathrm ds+\frac{\partial x}{\partial t}\mathrm dt</math>
:<math>\mathrm dy=\frac{\partial y}{\partial s}\mathrm ds+\frac{\partial y}{\partial t}\mathrm dt</math>

So <math> \mathrm dx \mathrm dy </math> transforms to <math> \frac{\partial(x,y)}{\partial(s,t)}  \mathrm ds \mathrm dt </math>, where <math> \frac{\partial(x,y)}{\partial(s,t)} </math> denotes the [[determinant]] of the [[Jacobian matrix and determinant|Jacobian]] of the transition function from <math>(s, t)</math> to <math>(x,y)</math>. The transformation of the other forms are similar.

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

:<math>\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 ds\, \mathrm dt</math>

where
:<math>{\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)</math>
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 <math>f_x</math>, <math>f_y</math> and <math>f_z</math>.

== Theorems involving surface integrals ==
Various useful results for surface integrals can be derived using [[differential geometry]] and [[vector calculus]], such as the [[divergence theorem]], [[magnetic flux]], and its generalization, [[Stokes' theorem]].

== Dependence on parametrization ==
Let us notice that we defined the surface integral by using a parametrization of the surface ''S''. We know that a given surface might have several parametrizations. For example, if we move the locations of the North Pole and the South Pole on a sphere, the latitude and longitude change for all the points on the sphere. A natural question is then whether the definition of the surface integral depends on the chosen parametrization. For integrals of scalar fields, the answer to this question is simple; the value of the surface integral will be the same no matter what parametrization one uses.

For integrals of vector fields, things are more complicated because the surface normal is involved. It can be proven that given two parametrizations of the same surface, whose surface normals point in the same direction, one obtains the same value for the surface integral with both parametrizations. If, however, the normals for these parametrizations point in opposite directions, the value of the surface integral obtained using one parametrization is the negative of the one obtained via the other parametrization. It follows that given a surface, we do not need to stick to any unique parametrization, but, when integrating vector fields, we do need to decide in advance in which direction the normal will point and then choose any parametrization consistent with that direction.

Another issue is that sometimes surfaces do not have parametrizations which cover the whole surface. The obvious solution is then to split that surface into several pieces, calculate the surface integral on each piece, and then add them all up. This is indeed how things work, but when integrating vector fields, one needs to again be careful how to choose the normal-pointing vector for each piece of the surface, so that when the pieces are put back together, the results are consistent. For the cylinder, this means that if we decide that for the side region the normal will point out of the body, then for the top and bottom circular parts, the normal must point out of the body too.

Last, there are surfaces which do not admit a surface normal at each point with consistent results (for example, the [[Möbius strip]]).  If such a surface is split into pieces, on each piece a parametrization and corresponding surface normal is chosen, and the pieces are put back together, we will find that the normal vectors coming from different pieces cannot be reconciled. This means that at some junction between two pieces we will have normal vectors pointing in opposite directions. Such a surface is called [[Orientability|non-orientable]], and on this kind of surface, one cannot talk about integrating vector fields.

== See also ==
* [[Area element]]
* [[Divergence theorem]]
* [[Stokes' theorem]]
* [[Line integral]]
* [[Line element]]
* [[Volume element]]
* [[Volume integral]]
* [[Cartesian coordinate system]]
* [[Spherical coordinate system#Integration and differentiation in spherical coordinates|Volume and surface area elements in spherical coordinate systems]]
* [[Cylindrical coordinate system#Line and volume elements|Volume and surface area elements in cylindrical coordinate systems]]
* [[Holstein–Herring method]]

==References==
{{Reflist}}

== External links ==
* {{mathworld|id=SurfaceIntegral|title=Surface Integral}}

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[[Category:Multivariable calculus]]
[[Category:Area]]
[[Category:Surfaces|Integral]]