Metric tensor - Revision history

BrianMcMahon: Tidied translations.

2017-12-11T09:21:42Z

Tidied translations.

BrianMcMahon: Tidied translations.

2017-11-16T12:54:59Z

Tidied translations.

BrianMcMahon: Style edits to align with printed edition

2017-05-22T09:47:56Z

Style edits to align with printed edition

BrianMcMahon: Style edits to align with printed edition

2017-05-15T16:42:09Z

Style edits to align with printed edition

MassimoNespolo: Lang

2016-11-26T22:36:01Z

Lang

HerbertBernstein: Add explanation of what a metric tensor is making it clear the definition is for a restricted case; add common crystallographic presentations as 2 rows of 3 or as a G6 vector. remove stray comma.

2014-08-11T21:21:01Z

Add explanation of what a metric tensor is making it clear the definition is for a restricted case; add common crystallographic presentations as 2 rows of 3 or as a G6 vector. remove stray comma.

MassimoNespolo: /* See also */

2007-04-26T07:41:07Z

‎See also

MassimoNespolo at 11:22, 26 February 2007

2007-02-26T11:22:30Z

AndreAuthier at 14:23, 12 January 2007

2007-01-12T14:23:22Z

AndreAuthier at 14:23, 12 January 2007

2007-01-12T14:23:09Z

← Older revision		Revision as of 09:21, 11 December 2017
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−	<font color="blue">Tenseur métrique</font> (''Fr''). <font color="red">Metrischer Tensor</font> (''Ge''). <font color="black">Tensore metrico</font> (''It''). <font color="~~brown~~">~~Метрический тензор~~</font> (''Ru''). <font color="~~purple~~">~~計量テンソル~~</font> (''Ja''). <font color="green">Tensor métrico</font> (''Sp'').	+	<font color="blue">Tenseur métrique</font> (''Fr''). <font color="red">Metrischer Tensor</font> (''Ge''). <font color="black">Tensore metrico</font> (''It''). <font color="purple">計量テンソル</font> (''Ja''). <font color="brown">Метрический тензор</font> (''Ru''). <font color="green">Tensor métrico</font> (''Sp'').

← Older revision		Revision as of 12:54, 16 November 2017
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−	<~~Font~~ color="blue">Tenseur métrique</~~Font~~> (''Fr''). <~~Font~~ color="red">Metrischer Tensor</~~Font~~> (''Ge''). <~~Font~~ color="~~green~~">~~Tensor métrico~~</~~Font~~> (''Sp''). <~~Font~~ color="~~black~~">~~Tensore metrico~~</~~Font~~> (''It''). <~~Font~~ color="~~brown~~">~~Метрический тензор~~</~~Font~~> (''Ru''). <~~Font~~ color="~~purple~~">~~計量テンソル~~</~~Font~~> (''Ja'').	+	<font color="blue">Tenseur métrique</font> (''Fr''). <font color="red">Metrischer Tensor</font> (''Ge''). <font color="black">Tensore metrico</font> (''It''). <font color="brown">Метрический тензор</font> (''Ru''). <font color="purple">計量テンソル</font> (''Ja''). <font color="green">Tensor métrico</font> (''Sp'').
		+

	== Definition ==		== Definition ==

@@ Line 28: / Line 28: @@
 ( '''a<sup>2</sup>''', '''b<sup>2</sup>''', '''c<sup>2</sup>''', 2'''b . c''', 2'''a . c''', 2'''a . b''' )
-The  inverse matrix of ''g<sub>ij</sub>'', ''g<sup>ij</sup>'', relates the [[dual basis]], or [[reciprocal space]] vectors '''e<sup>i</sup>''' to the direct basis vectors '''e<sub>i</sub>''', through the relations:
+The  inverse matrix of ''g<sub>ij</sub>'', ''g<sup>ij</sup>'', relates the [[dual basis]], or [[reciprocal space]] vectors '''e'''<sup>''i''</sup> to the direct basis vectors '''e'''<sub>''i''</sub>, through the relations
-'''e<sup>j</sup>''' = ''g<sup>ij</sup>'' '''e<sub>j</sub>'''.
+'''e'''<sup>''j''</sup> = ''g<sup>ij</sup>'' '''e'''<sub>''j''</sub>.
 Note that ''g<sup>ik</sup>g<sub>kj</sub>'' = &#948;''<sup>k</sup><sub>j</sub>'', where &#948;''<sup>k</sup><sub>j</sub>'' is the Kronecker symbol, equal to 0 if ''i'' &#8800; ''j'', and equal to 1 if ''i'' = ''j''.
@@ Line 47: / Line 47: @@
 ''g<sup>12</sup>'' = ''g<sup>21</sup>'' = (''abc''<sup>2</sup>/ V<sup>2</sup>)(cos &#945; cos &#946; - cos &#947;);
 ''g<sup>23</sup>'' = ''g<sup>32</sup>'' = (''a<sup>2</sup>bc''/ V<sup>2</sup>)(cos &#946; cos &#947; - cos &#945;);
 ''g<sup>31</sup>'' = ''g<sup>13</sup>'' = (''ab<sup>2</sup>c''/ V<sup>2</sup>)(cos &#947; cos &#945; - cos &#946;)

@@ Line 1: / Line 1: @@
-<Font color="blue">Tenseur métrique</Font> (''Fr''); <Font color="red">Metrischer Tensor</Font> (''Ge''); <Font color="green">Tensor métrico</Font> (''SP'');  <Font color="black">Tensore metrico</Font> (''It''); <Font color="brown">Метрический тензор</Font> (''Ru''); <Font color="purple">計量テンソル</Font> (''Ja'').
+<Font color="blue">Tenseur métrique</Font> (''Fr''). <Font color="red">Metrischer Tensor</Font> (''Ge''). <Font color="green">Tensor métrico</Font> (''Sp'').  <Font color="black">Tensore metrico</Font> (''It''). <Font color="brown">Метрический тензор</Font> (''Ru''). <Font color="purple">計量テンソル</Font> (''Ja'').
 == Definition ==
@@ Line 17: / Line 17: @@
 ''g<sub>11</sub>'' = '''a<sup>2</sup>'''; ''g<sub>12</sub>'' = '''a . b'''; ''g<sub>13</sub>'' = '''a . c''';<br>
 ''g<sub>21</sub>'' = '''b . a'''; ''g<sub>22</sub>'' = '''b<sup>2</sup>'''; ''g<sub>23</sub>'' = '''b . c''';<br>
-''g<sub>31</sub>'' = '''c . a'''; ''g<sub>32</sub>'' = '''c . b'''; ''g<sub>33</sub>'' = '''c<sup>2</sup>''';
+''g<sub>31</sub>'' = '''c . a'''; ''g<sub>32</sub>'' = '''c . b'''; ''g<sub>33</sub>'' = '''c<sup>2</sup>'''.
 Because the metric tensor is symmetric, ''g<sub>12</sub>'' = ''g<sub>21</sub>'', ''g<sub>13</sub>'' = ''g<sub>31</sub>'', and   ''g<sub>13</sub>'' = ''g<sub>31</sub>''.  Thus there are only six unique elements, often presented as
@@ Line 26: / Line 26: @@
 or, multiplying the second row by 2,  as a so-called G<sup>6</sup> ("G" for Gruber) vector
-( '''a<sup>2</sup>''', '''b<sup>2</sup>''', '''c<sup>2</sup>''', 2 '''b . c''', 2 '''a . c''', 2 '''a . b''' )
+( '''a<sup>2</sup>''', '''b<sup>2</sup>''', '''c<sup>2</sup>''', 2'''b . c''', 2'''a . c''', 2'''a . b''' )
-The  inverse matrix of ''g<sub>ij</sub>'', ''g<sup>ij</sup>'', (''g<sup>ik</sup>g<sub>kj</sub>'' = &#948;''<sup>k</sup><sub>j</sub>'', Kronecker symbol, = 0 if ''i'' &#8800; ''j'', = 1 if ''i'' = ''j'') relates the [[dual basis]], or [[reciprocal space]] vectors '''e<sup>i</sup>''' to the direct basis vectors '''e<sub>i</sub>''' through the relations:
+'''e<sup>j</sup>''' = ''g<sup>ij</sup>'' '''e<sub>j</sub>'''.
-'''e<sup>j</sup>''' = ''g<sup>ij</sup>'' '''e<sub>j</sub>'''
+Note that ''g<sup>ik</sup>g<sub>kj</sub>'' = &#948;''<sup>k</sup><sub>j</sub>'', where &#948;''<sup>k</sup><sub>j</sub>'' is the Kronecker symbol, equal to 0 if ''i'' &#8800; ''j'', and equal to 1 if ''i'' = ''j''.
 In three-dimensional space, the dual basis vectors are identical to the [[reciprocal space]] vectors and the components of ''g<sup>ij</sup>'' are:
@@ Line 82: / Line 83: @@
 * The '''squares of the volumes''' ''V'' and ''V*'' of the direct space and reciprocal space unit cells are respectively equal to the determinants of the ''g<sub>ij</sub>'' 's and the ''g<sup>ij</sup>'' 's:
-''V''<sup> 2</sup> = &#916; (''g<sub>ij</sub>'') = ''abc''(1 - cos <sup>2</sup> &#945; - cos <sup>2</sup> &#946; - cos<sup>2</sup> &#947; + 2 cos &#945; cos &#945; cos &#945;)
+''V''<sup> 2</sup> = &#916; (''g<sub>ij</sub>'') = ''abc''(1 - cos <sup>2</sup> &#945; - cos <sup>2</sup> &#946; - cos<sup>2</sup> &#947; + 2 cos &#945; cos &#946; cos &#947;)
 ''V*''<sup>2</sup> = &#916; (''g<sup>ij</sup>'') = 1/ ''V''<sup> 2</sup>.
@@ Line 94: / Line 95: @@
 == See also ==
-*[[dual basis]]
+*[[Dual basis]]
-*[[reciprocal space]]
+*[[Reciprocal space]]
-*[http://www.iucr.org/iucr-top/comm/cteach/pamphlets/10/ Metric Tensor and Symmetry Operations in Crystallography]  (Teaching Pamphlet of the ''International Union of Crystallography'')
+*[http://www.iucr.org/iucr-top/comm/cteach/pamphlets/10/ ''Metric Tensor and Symmetry Operations in Crystallography'']  (Teaching Pamphlet No. 10 of the International Union of Crystallography)
-*Section 1.1.3 of ''International Tables of Crystallography, Volume B''
+*Chapter 1.1.3 of ''International Tables for Crystallography, Volume B''
-*Section 1.1.2 of ''International Tables of Crystallography, Volume D''
+*Chapter 1.1.2 of ''International Tables for Crystallography, Volume D''

← Older revision		Revision as of 22:36, 26 November 2016
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−	<Font color="blue">Tenseur métrique </Font>(''Fr''); <Font color="black"> Tensore metrico </Font>(''It''); <Font color="purple">計量テンソル</Font> (''Ja'').	+	<Font color="blue">Tenseur métrique</Font> (''Fr''); <Font color="red">Metrischer Tensor</Font> (''Ge''); <Font color="green">Tensor métrico</Font> (''SP''); <Font color="black">Tensore metrico</Font> (''It''); <Font color="brown">Метрический тензор</Font> (''Ru''); <Font color="purple">計量テンソル</Font> (''Ja'').

	== Definition ==		== Definition ==

@@ Line 2: / Line 2: @@
 == Definition ==
+A metric tensor is used to measure distances in a space.  In crystallography the spaces considered are vector spaces with
-Given a basis '''e<sub>i</sub>''' of a ''Euclidean space'', ''E<sup>n</sup>'', the metric tensor is a rank 2 tensor the components of
+''Euclidean'' metrics,  <i>i.e.</i> ones for which the rules of Euclidean geometry apply.  In that case,
 which are:
@@ Line 14: / Line 15: @@
 In a three-dimensional space with basis vectors '''a''', '''b''', '''c''', the coefficients ''g<sub>ij</sub>'' of the metric tensor are:
-''g<sub>11</sub>,'' = '''a<sup>2</sup>'''; ''g<sub>12</sub>'' = '''a . b'''; ''g<sub>13</sub>'' = '''a . c''';<br>
+''g<sub>11</sub>'' = '''a<sup>2</sup>'''; ''g<sub>12</sub>'' = '''a . b'''; ''g<sub>13</sub>'' = '''a . c''';<br>
 ''g<sub>21</sub>'' = '''b . a'''; ''g<sub>22</sub>'' = '''b<sup>2</sup>'''; ''g<sub>23</sub>'' = '''b . c''';<br>
 ''g<sub>31</sub>'' = '''c . a'''; ''g<sub>32</sub>'' = '''c . b'''; ''g<sub>33</sub>'' = '''c<sup>2</sup>''';
 The  inverse matrix of ''g<sub>ij</sub>'', ''g<sup>ij</sup>'', (''g<sup>ik</sup>g<sub>kj</sub>'' = &#948;''<sup>k</sup><sub>j</sub>'', Kronecker symbol, = 0 if ''i'' &#8800; ''j'', = 1 if ''i'' = ''j'') relates the [[dual basis]], or [[reciprocal space]] vectors '''e<sup>i</sup>''' to the direct basis vectors '''e<sub>i</sub>''' through the relations:

@@ Line 83: / Line 83: @@
 == See also ==
-[[dual basis]]<br>
+*[[dual basis]]
-[[reciprocal space]]<br>
+*[[reciprocal space]]
-[http://www.iucr.org/iucr-top/comm/cteach/pamphlets/10/ Metric Tensor and Symmetry Operations in Crystallography]  (Teaching Pamphlet of the ''International Union of Crystallography'')<br>
+*[http://www.iucr.org/iucr-top/comm/cteach/pamphlets/10/ Metric Tensor and Symmetry Operations in Crystallography]  (Teaching Pamphlet of the ''International Union of Crystallography'')
-Section 1.1.3 of ''International Tables of Crystallography, Volume B''<br>
+*Section 1.1.3 of ''International Tables of Crystallography, Volume B''
-Section 1.1.2 of ''International Tables of Crystallography, Volume D''<br>
+*Section 1.1.2 of ''International Tables of Crystallography, Volume D''
 [[Category:Fundamental crystallography]]<br>
 [[Category:Physical properties of crystals]]

← Older revision		Revision as of 11:22, 26 February 2007
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−	<Font color="blue">Tenseur métrique </Font>(''Fr''). <Font color="black"> Tensore metrico </Font>(''It'')	+	<Font color="blue">Tenseur métrique </Font>(''Fr''); <Font color="black"> Tensore metrico </Font>(''It''); <Font color="purple">計量テンソル</Font> (''Ja'').

	== Definition ==		== Definition ==

← Older revision		Revision as of 14:23, 12 January 2007
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	<Font color="blue">Tenseur métrique </Font>(''Fr''). <Font color="black"> Tensore metrico </Font>(''It'')		<Font color="blue">Tenseur métrique </Font>(''Fr''). <Font color="black"> Tensore metrico </Font>(''It'')
−
−
−

	== Definition ==		== Definition ==