Surface Integrals - Revision history

Lila: /* Theorems involving surface integrals */

2021-11-10T18:57:12Z

Theorems involving surface integrals

Lila: /* Surface integrals of vector fields */

2021-11-10T18:56:05Z

Surface integrals of vector fields

Lila: /* Surface integrals of vector fields */

2021-11-10T18:54:45Z

Surface integrals of vector fields

Lila: /* Surface integrals of vector fields */

2021-11-10T18:54:23Z

Surface integrals of vector fields

Lila at 18:51, 10 November 2021

2021-11-10T18:51:35Z

Lila: Created page with "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. G..."

2021-11-10T18:51:02Z

Created page with "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. G..."

New page

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.

[[Image:Surface integral illustration.svg|right|thumb|The definition of surface integral relies on splitting the surface into small surface elements.]]
[[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 ==
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 {{math|'''r'''(''s'', ''t'')}}, where {{math|(''s'', ''t'')}} varies in some region {{mvar|T}} in the 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 of the cross product of the partial derivatives of {{math|'''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
:<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>{{Cite book|title = Encyclopedia of Mathematics|last = Hazewinkel|first = Michiel|publisher = Springer|year = 2001|isbn = 978-1-55608-010-4|pages = Surface Integral|url = https://www.encyclopediaofmath.org/index.php/Surface_integral}}</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 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 ==
{{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">\int_S \bold{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 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

:<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 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 <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=f_{z}\, \mathrm dx \wedge \mathrm dy + f_{x}\, \mathrm dy \wedge \mathrm dz + f_{y}\, \mathrm dz \wedge \mathrm dx</math>
be a 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 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 \wedge \mathrm dy </math> transforms to <math> \frac{\partial(x,y)}{\partial(s,t)} \mathrm ds \wedge \mathrm dt </math>, where <math> \frac{\partial(x,y)}{\partial(s,t)} </math> denotes the determinant of the 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|divergence theorem]], and its generalization, Stokes' theorem.

← Older revision		Revision as of 18:57, 10 November 2021
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	== Theorems involving surface integrals ==		== Theorems involving surface integrals ==
	Various useful results for surface integrals can be derived using differential geometry and vector calculus, such as the [[Divergence Theorem\|divergence theorem]], and its generalization, Stokes' theorem.		Various useful results for surface integrals can be derived using differential geometry and vector calculus, such as the [[Divergence Theorem\|divergence theorem]], and its generalization, Stokes' theorem.
		+
		+	==Licensing==
		+	Content obtained and/or adapted from:
		+	* [https://en.wikipedia.org/wiki/Surface_integral Surface integral, Wikipedia] under a CC BY-SA license

@@ Line 39: / Line 39: @@
 == Surface integrals of vector fields ==
-[[Surface integral - vector field thru a surface.svg|thumb|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]]
+[[File:Surface integral - vector field thru a surface.svg|thumb|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]]
 | image2   = Surface integral - parametrized surface.svg

@@ Line 41: / Line 41: @@
 [[File:Surface integral - vector field thru a surface.svg|thumb|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]]
-| image2   = Surface integral - parametrized surface.svg
+[[File:Surface integral - parametrized surface.svg|thumb|Surface divided into small patches <math>dS = du\,dv</math> by a parameterization of the surface <math>[u(\mathbf{x}),v(\mathbf{x})]</math>]]
-| 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
+[[File:Surface integral - normal component of field.svg|thumb|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)'']]
-| 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)''
+[[File:Surface integral - definition.svg|thumb|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">\int_S \bold{F\cdot n}\;dS</math>]]
 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.

@@ Line 18: / Line 18: @@
 = \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>{{Cite book|title = Encyclopedia of Mathematics|last = Hazewinkel|first = Michiel|publisher = Springer|year = 2001|isbn = 978-1-55608-010-4|pages = Surface Integral|url = https://www.encyclopediaofmath.org/index.php/Surface_integral}}</ref>
+where {{mvar|g}} is the determinant of the first fundamental form of the surface mapping {{math|'''r'''(''s'', ''t'')}}.
 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