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In [[differential geometry]], the '''first fundamental form''' is the [[inner product]] on the [[tangent space]] of a [[surface]] in three-dimensional [[Euclidean space]] which is induced [[canonical form|canonically]] from the [[dot product]] of '''R'''<sup>''3''</sup>.  It permits the calculation of [[curvature]] and metric properties of a surface such as length and area in a manner consistent with the [[ambient space]].  The first fundamental form is denoted by the Roman numeral I,
This is a preview for the new '''MathML rendering mode''' (with SVG fallback), which is availble in production for registered users.
:<math>\!\mathrm{I}(x,y)= \langle x,y \rangle.</math>


Let ''X''(''u'',&nbsp;''v'') be a [[parametric surface]]. Then the inner product of two [[tangent vector]]s is
If you would like use the '''MathML''' rendering mode, you need a wikipedia user account that can be registered here [[https://en.wikipedia.org/wiki/Special:UserLogin/signup]]
* Only registered users will be able to execute this rendering mode.
* Note: you need not enter a email address (nor any other private information). Please do not use a password that you use elsewhere.


:<math>
Registered users will be able to choose between the following three rendering modes:  
\begin{align}
& {} \quad \mathrm{I}(aX_u+bX_v,cX_u+dX_v) \\
& = ac \langle X_u,X_u \rangle + (ad+bc) \langle X_u,X_v \rangle + bd \langle X_v,X_v \rangle \\
& = Eac + F(ad+bc) + Gbd,
\end{align}
</math>


where ''E'', ''F'', and ''G'' are the '''coefficients of the first fundamental form'''.
'''MathML'''
:<math forcemathmode="mathml">E=mc^2</math>


The first fundamental form may be represented as a [[symmetric matrix]].
<!--'''PNG'''  (currently default in production)
:<math forcemathmode="png">E=mc^2</math>


:<math>\!\mathrm{I}(x,y) = x^T
'''source'''
\begin{pmatrix}
:<math forcemathmode="source">E=mc^2</math> -->
E & F \\
F & G
\end{pmatrix}y
</math>


==Further notation==
<span style="color: red">Follow this [https://en.wikipedia.org/wiki/Special:Preferences#mw-prefsection-rendering link] to change your Math rendering settings.</span> You can also add a [https://en.wikipedia.org/wiki/Special:Preferences#mw-prefsection-rendering-skin Custom CSS] to force the MathML/SVG rendering or select different font families. See [https://www.mediawiki.org/wiki/Extension:Math#CSS_for_the_MathML_with_SVG_fallback_mode these examples].
When the first fundamental form is written with only one argument, it denotes the inner product of that vector with itself.
:<math>\!\mathrm{I}(v)= \langle v,v \rangle = |v|^2</math>


The first fundamental form is often written in the modern notation of the [[metric tensor]].  The coefficients may then be written as <math>g_{ij}</math>:
==Demos==
:<math> \left(g_{ij}\right) = \begin{pmatrix}g_{11} & g_{12} \\g_{21} & g_{22}\end{pmatrix} =\begin{pmatrix}E & F \\F & G\end{pmatrix}</math>


The components of this tensor are calculated as the scalar product of tangent vectors ''X''<sub>1</sub> and ''X''<sub>2</sub>:
Here are some [https://commons.wikimedia.org/w/index.php?title=Special:ListFiles/Frederic.wang demos]:


:<math>g_{ij} = X_i \cdot X_j</math>


for ''i'', ''j'' = 1, 2. See example below.
* accessibility:
** Safari + VoiceOver: [https://commons.wikimedia.org/wiki/File:VoiceOver-Mac-Safari.ogv video only], [[File:Voiceover-mathml-example-1.wav|thumb|Voiceover-mathml-example-1]], [[File:Voiceover-mathml-example-2.wav|thumb|Voiceover-mathml-example-2]], [[File:Voiceover-mathml-example-3.wav|thumb|Voiceover-mathml-example-3]], [[File:Voiceover-mathml-example-4.wav|thumb|Voiceover-mathml-example-4]], [[File:Voiceover-mathml-example-5.wav|thumb|Voiceover-mathml-example-5]], [[File:Voiceover-mathml-example-6.wav|thumb|Voiceover-mathml-example-6]], [[File:Voiceover-mathml-example-7.wav|thumb|Voiceover-mathml-example-7]]
** [https://commons.wikimedia.org/wiki/File:MathPlayer-Audio-Windows7-InternetExplorer.ogg Internet Explorer + MathPlayer (audio)]
** [https://commons.wikimedia.org/wiki/File:MathPlayer-SynchronizedHighlighting-WIndows7-InternetExplorer.png Internet Explorer + MathPlayer (synchronized highlighting)]
** [https://commons.wikimedia.org/wiki/File:MathPlayer-Braille-Windows7-InternetExplorer.png Internet Explorer + MathPlayer (braille)]
** NVDA+MathPlayer: [[File:Nvda-mathml-example-1.wav|thumb|Nvda-mathml-example-1]], [[File:Nvda-mathml-example-2.wav|thumb|Nvda-mathml-example-2]], [[File:Nvda-mathml-example-3.wav|thumb|Nvda-mathml-example-3]], [[File:Nvda-mathml-example-4.wav|thumb|Nvda-mathml-example-4]], [[File:Nvda-mathml-example-5.wav|thumb|Nvda-mathml-example-5]], [[File:Nvda-mathml-example-6.wav|thumb|Nvda-mathml-example-6]], [[File:Nvda-mathml-example-7.wav|thumb|Nvda-mathml-example-7]].
** Orca: There is ongoing work, but no support at all at the moment [[File:Orca-mathml-example-1.wav|thumb|Orca-mathml-example-1]], [[File:Orca-mathml-example-2.wav|thumb|Orca-mathml-example-2]], [[File:Orca-mathml-example-3.wav|thumb|Orca-mathml-example-3]], [[File:Orca-mathml-example-4.wav|thumb|Orca-mathml-example-4]], [[File:Orca-mathml-example-5.wav|thumb|Orca-mathml-example-5]], [[File:Orca-mathml-example-6.wav|thumb|Orca-mathml-example-6]], [[File:Orca-mathml-example-7.wav|thumb|Orca-mathml-example-7]].
** From our testing, ChromeVox and JAWS are not able to read the formulas generated by the MathML mode.


==Calculating lengths and areas==
==Test pages ==


The first fundamental form completely describes the metric properties of a surface.  Thus, it enables one to calculate the lengths of curves on the surface and the areas of regions on the surface.  The [[line element]] ''ds'' may be expressed in terms of the coefficients of the first fundamental form as
To test the '''MathML''', '''PNG''', and '''source''' rendering modes, please go to one of the following test pages:
:<math>ds^2 = Edu^2+2Fdudv+Gdv^2 \,</math>. <!--- "\," improves the display of this formula in Wikipedia. Do not delete --->
*[[Displaystyle]]
*[[MathAxisAlignment]]
*[[Styling]]
*[[Linebreaking]]
*[[Unique Ids]]
*[[Help:Formula]]


The classical area element given by <math> dA = |X_u \times X_v| \ du\, dv</math> can be expressed in terms of the first fundamental form with the assistance of [[Lagrange's identity]],
*[[Inputtypes|Inputtypes (private Wikis only)]]
 
*[[Url2Image|Url2Image (private Wikis only)]]
:<math>dA = |X_u \times X_v| \ du\, dv= \sqrt{ \langle X_u,X_u \rangle \langle X_v,X_v \rangle - \langle X_u,X_v \rangle^2 } \ du\, dv = \sqrt{EG-F^2} \, du\, dv.</math>
==Bug reporting==
 
If you find any bugs, please report them at [https://bugzilla.wikimedia.org/enter_bug.cgi?product=MediaWiki%20extensions&component=Math&version=master&short_desc=Math-preview%20rendering%20problem Bugzilla], or write an email to math_bugs (at) ckurs (dot) de .
===Example===
 
The unit [[sphere]] in '''R'''<sup>''3''</sup> may be parametrized as
 
:<math>X(u,v) = \begin{pmatrix} \cos u \sin v \\ \sin u \sin v \\ \cos v \end{pmatrix},\ (u,v) \in [0,2\pi) \times [0,\pi).</math>
 
Differentiating <math>X(u,v)</math> with respect to u and v yields
 
:<math>X_u = \begin{pmatrix} -\sin u \sin v \\ \cos u \sin v \\ 0 \end{pmatrix},\ X_v = \begin{pmatrix} \cos u \cos v \\ \sin u \cos v \\ -\sin v \end{pmatrix}.</math>
 
The coefficients of the first fundamental form may be found by taking the dot product of the [[partial derivatives]].
 
:<math>E = X_u \cdot X_u = \sin^2 v</math>
:<math>F = X_u \cdot X_v = 0</math>
:<math>G = X_v \cdot X_v = 1</math>
 
====Length of a curve on the sphere====
 
The [[equator]] of the sphere is a parametrized curve given by <math>(u(t),v(t))=(t,\frac{\pi}{2})</math> with t ranging from 0 to <math>2\pi</math>. The line element may be used to calculate the length of this curve.
 
:<math>\int_0^{2\pi} \sqrt{ E\left(\frac{du}{dt}\right)^2 + 2F\frac{du}{dt}\frac{dv}{dt} + G\left(\frac{dv}{dt}\right)^2 } \,dt = \int_0^{2\pi} |\sin v| \,dt = 2\pi \sin \frac{\pi}{2} = 2\pi</math>
 
====Area of a region on the sphere====
 
The area element may be used to calculate the area of the sphere.
 
:<math>\int_0^{\pi} \int_0^{2\pi} \sqrt{ EG-F^2 } \ du\, dv = \int_0^{\pi} \int_0^{2\pi} \sin v \, du\, dv = 2\pi \left[-\cos v\right]_0^{\pi} = 4\pi</math>
 
==Gaussian curvature==
 
The [[Gaussian curvature]] of a surface is given by
 
:<math> K = \frac{\det \mathrm{I\!I}}{\det \mathrm{I}} = \frac{ LN-M^2}{EG-F^2 }, </math>
 
where ''L'', ''M'', and ''N'' are the coefficients of the [[second fundamental form]].
 
[[Theorema egregium]] of [[Carl Friedrich Gauss|Gauss]] states that the Gaussian curvature of a surface can be expressed solely in terms of the first fundamental form and its derivatives, so that ''K'' is in fact an intrinsic invariant of the surface. An explicit expression for the Gaussian curvature in terms of the first fundamental form is provided by the [[Gaussian curvature#Alternative_formulas|Brioschi formula]].
 
==See also==
*[[Metric tensor]]
*[[Second fundamental form]]
 
==External links==
*[http://mathworld.wolfram.com/FirstFundamentalForm.html First Fundamental Form &mdash; from Wolfram MathWorld]
*[http://planetmath.org/encyclopedia/FirstFundamentalForm.html PlanetMath: first fundamental form]
 
{{curvature}}
 
[[Category:Differential geometry of surfaces]]
[[Category:Differential geometry]]
[[Category:Surfaces]]

Latest revision as of 22:52, 15 September 2019

This is a preview for the new MathML rendering mode (with SVG fallback), which is availble in production for registered users.

If you would like use the MathML rendering mode, you need a wikipedia user account that can be registered here [[1]]

  • Only registered users will be able to execute this rendering mode.
  • Note: you need not enter a email address (nor any other private information). Please do not use a password that you use elsewhere.

Registered users will be able to choose between the following three rendering modes:

MathML

E=mc2


Follow this link to change your Math rendering settings. You can also add a Custom CSS to force the MathML/SVG rendering or select different font families. See these examples.

Demos

Here are some demos:


Test pages

To test the MathML, PNG, and source rendering modes, please go to one of the following test pages:

Bug reporting

If you find any bugs, please report them at Bugzilla, or write an email to math_bugs (at) ckurs (dot) de .

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