Dual basis - Revision history

BrianMcMahon: Tidied translations and added German and Spanish (U. Mueller)

2017-11-10T13:56:31Z

Tidied translations and added German and Spanish (U. Mueller)

MassimoNespolo: Lang (Ar)

2017-09-14T09:15:43Z

Lang (Ar)

BrianMcMahon: Style edits to align with printed edition

2017-05-13T13:33:15Z

Style edits to align with printed edition

MassimoNespolo: better Japanese translation

2017-04-14T09:07:10Z

better Japanese translation

MassimoNespolo: /* See also */

2017-04-10T15:33:16Z

‎See also

MassimoNespolo: Ja

2016-10-04T11:16:51Z

BrianMcMahon at 12:12, 6 February 2012

2012-02-06T12:12:05Z

AndreAuthier at 07:30, 14 January 2007

2007-01-14T07:30:44Z

GiovanniFerraris: /* Other languages */

2006-03-17T15:29:40Z

‎Other languages

BrianMcMahon at 13:08, 25 January 2006

2006-01-25T13:08:25Z

← Older revision		Revision as of 13:56, 10 November 2017
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−	<font color = "blue">Base duale </font>(''Fr''); <font color="black"> Base duale </font>(''It''); <font color="purple">双対基底</font> (''Ja''); <font color="~~orange~~">~~أساس مزدوج~~</font> (''Ar'').	+	<font color="orange">أساس مزدوج</font> (''Ar''). <font color = "blue">Base duale</font> (''Fr''). <font color="red">Duale Basis</font> (''Ge''). <font color="black">Base duale</font> (''It''). <font color="purple">双対基底</font> (''Ja''). <font color="green">Base dual</font> (''Sp'').

	== Definition ==		== Definition ==

← Older revision		Revision as of 09:15, 14 September 2017
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−	<font color = "blue">Base duale </font>(''Fr''). <font color="black"> Base duale </font>(''It''). <font color="purple">双対基底</font> (''Ja'').	+	<font color = "blue">Base duale </font>(''Fr''); <font color="black"> Base duale </font>(''It''); <font color="purple">双対基底</font> (''Ja''); <font color="orange">أساس مزدوج</font> (''Ar'').

	== Definition ==		== Definition ==

@@ Line 1: / Line 1: @@
-<font color = "blue">Base duale </font>(''Fr''): <font color="black"> Base duale </font>(''It''); <font color="purple">双対基底</font> (''Ja'').
+<font color = "blue">Base duale </font>(''Fr''). <font color="black"> Base duale </font>(''It''). <font color="purple">双対基底</font> (''Ja'').
 == Definition ==
-The dual basis is a basis associated to the basis of a vector space. In three-dimensional space, it is isomorphous to the basis of the reciprocal lattice. It is mathematically defined as follows:
+The dual basis is a basis associated to the basis of a vector space. In three-dimensional space, it is isomorphous to the basis of the reciprocal lattice. It is mathematically defined as follows.
 Given a basis of ''n'' vectors '''e<sub>i</sub>''' spanning the [[direct space]] ''E<sup>n</sup>'', and a vector
@@ Line 12: / Line 13: @@
 where the ''g<sub>ji</sub>'' 's are the doubly covariant components of the [[metric tensor]].
-By solving these equations in terms of ''x<sup> j</sup>'', one gets:
+By solving these equations in terms of ''x<sup> j</sup>'', one gets
 ''x<sup> j</sup>'' = ''x<sub>i</sub> g<sup>ij</sup>''
-where the matrix of the  ''g<sup>ij</sup>'' 's is inverse of that of the ''g<sub>ij</sub>'' 's (''g<sup>ik</sup>g<sub>jk</sub>'' = &#948;<sup>i</sup><sub>j</sub>). The development of vector '''x''' with respect to basis vectors '''e<sub>i</sub>''' can now also be written:
+where the matrix of the  ''g<sup>ij</sup>'' 's is inverse of that of the ''g<sub>ij</sub>'' 's (''g<sup>ik</sup>g<sub>jk</sub>'' = &#948;<sup>i</sup><sub>j</sub>). The development of vector '''x''' with respect to basis vectors '''e<sub>i</sub>''' can now also be written
-'''x''' = ''x<sup> i</sup>'' '''e<sub>i</sub>''' = ''x<sub>i</sub> g<sup>ij</sup>'' '''e<sub>j</sub>'''
+'''x''' = ''x<sup> i</sup>'' '''e<sub>i</sub>''' = ''x<sub>i</sub> g<sup>ij</sup>'' '''e<sub>j</sub>'''.
-The set of ''n'' vectors '''e<sup>i</sup>''' = ''g<sup>ij</sup>'' '''e<sub>j</sub>''' that span the space ''E<sup>n</sup>'' forms a basis since vector '''x''' can be written:
+The set of ''n'' vectors '''e<sup>i</sup>''' = ''g<sup>ij</sup>'' '''e<sub>j</sub>''' that span the space ''E<sup>n</sup>'' forms a basis since vector '''x''' can be written
-'''x''' = ''x<sub>i</sub>'' '''e<sup>i</sup>'''
+'''x''' = ''x<sub>i</sub>'' '''e<sup>i</sup>'''.
 This basis is the ''dual basis'' and the ''n'' quantities ''x<sub>i</sub>'' defined above are the
 coordinates of '''x''' with respect to the dual basis. In a similar way one can express the direct basis vectors in terms of the dual basis vectors:
-'''e<sub>i</sub>''' = ''g<sub>ij</sub>'' '''e<sup>j</sup>'''
+'''e<sub>i</sub>''' = ''g<sub>ij</sub>'' '''e<sup>j</sup>'''.
-The scalar products of the basis vectors of the dual and direct bases are:
+The scalar products of the basis vectors of the dual and direct bases are
 ''g<sup>i</sup><sub>j</sub>'' = '''e<sup>i</sup>''' . '''e<sub>j</sub>''' = ''g<sup>ik</sup>'' '''e<sub>k</sub>''' . '''e<sub>j</sub>''' = ''g<sup>ik</sup>g<sub>jk</sub>'' = &#948;<sup>i</sup><sub>j</sub>.
-One has therefore, since the matrices ''g<sup>ik</sup>'' and ''g<sub>ij</sub>'' are inverse:
+One has therefore, since the matrices ''g<sup>ik</sup>'' and ''g<sub>ij</sub>'' are inverse,
 ''g<sup>i</sup><sub>j</sub>'' = '''e<sup>i</sup>''' . '''e<sub>j</sub>''' = &#948;<sup>i</sup><sub>j</sub>.
@@ Line 41: / Line 42: @@
 == Change of basis ==
-In a change of basis where the direct basis vectors and coordinates transform like:
+In a change of basis where the direct basis vectors and coordinates transform like
 '''e'<sub>j</sub>''' = ''A<sub>j</sub><sup>i</sup>'' '''e<sub>i</sub>'''; ''x'<sup>j</sup>'' = ''B<sub>i</sub><sup> j</sup>'' ''x<sup>i</sup>'',
 where ''A<sub>j</sub><sup>i</sup>'' and ''B<sub>i</sub><sup> j</sup>'' are transformation matrices, transpose of one another,
-the dual basis vectors '''e<sup>i</sup>''' and the coordinates ''x<sub>i</sub>'' transform according to:
+the dual basis vectors '''e<sup>i</sup>''' and the coordinates ''x<sub>i</sub>'' transform according to
 '''e'<sup>j</sup> ''' = ''B<sub>i</sub><sup> j</sup>'' '''e<sup>i</sup>'''; ''x'<sub>j</sub>'' = ''A<sub>j</sub><sup>i</sup>x<sub>i</sub>''.
@@ Line 54: / Line 55: @@
 == See also ==
-*[[metric tensor]]
+*[[Metric tensor]]
-*[[reciprocal space]]
+*[[Reciprocal space]]
-*[http://www.iucr.org/iucr-top/comm/cteach/pamphlets/4/ The Reciprocal Lattice]  (Teaching Pamphlet of the ''International Union of Crystallography'')
+*[http://www.iucr.org/iucr-top/comm/cteach/pamphlets/4/ ''The Reciprocal Lattice'']  (Teaching Pamphlet No. 4 of the International Union of Crystallography)
-*Section 1.1.2 of ''International Tables of Crystallography, Volume D''
+*Chapter 1.1.2 of ''International Tables for Crystallography, Volume D''
 [[Category:Fundamental crystallography]]<br>
 [[Category:Physical properties of crystals]]<br>

← Older revision		Revision as of 09:07, 14 April 2017
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−	<font color = "blue">Base duale </font>(''Fr''): <font color="black"> Base duale </font>(''It''); <font color="purple">~~ジュアル基底~~</font> (''Ja'').	+	<font color = "blue">Base duale </font>(''Fr''): <font color="black"> Base duale </font>(''It''); <font color="purple">双対基底</font> (''Ja'').
	== Definition ==		== Definition ==

@@ Line 52: / Line 52: @@
 The coordinates of a vector in reciprocal space are therefore ''covariant'' and the dual basis vectors (or reciprocal vectors) ''contravariant''.
-=== See also ===
+== See also ==
-[[metric tensor]]<br>
+*[[metric tensor]]
-[[reciprocal space]]<br>
+*[[reciprocal space]]
+*[http://www.iucr.org/iucr-top/comm/cteach/pamphlets/4/ The Reciprocal Lattice]  (Teaching Pamphlet of the ''International Union of Crystallography'')
-[http://www.iucr.org/iucr-top/comm/cteach/pamphlets/4/ The Reciprocal Lattice]  (Teaching Pamphlet of the ''International Union of Crystallography'')
+*Section 1.1.2 of ''International Tables of Crystallography, Volume D''
 [[Category:Fundamental crystallography]]<br>
 [[Category:Physical properties of crystals]]<br>

← Older revision		Revision as of 11:16, 4 October 2016
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−	<font color = "blue">Base duale </font>(''Fr''). <font color="black"> Base duale </font>(''It'')	+	<font color = "blue">Base duale </font>(''Fr''): <font color="black"> Base duale </font>(''It''); <font color="purple">ジュアル基底</font> (''Ja'').
	== Definition ==		== Definition ==

← Older revision		Revision as of 07:30, 14 January 2007
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−	= ~~Dual basis =~~	+	<Font color = "blue">Base duale </font>(''Fr''). <Font color="black"> Base duale </Font>(''It'')
−
−	~~=== Other languages ===~~
−
−	Base duale (''Fr'').<Font color="black"> Base duale </Font>(''It'')

	== Definition ==		== Definition ==

@@ Line 3: / Line 3: @@
 === Other languages ===
-Base duale (''Fr'').
+Base duale (''Fr'').<Font color="black"> Base duale </Font>(''It'')
 == Definition ==

@@ Line 10: / Line 10: @@
 Given a basis of ''n'' vectors '''e<sub>i</sub>''' spanning the [[direct space]] ''E<sup>n</sup>'', and a vector
-'''x''' = ''x<sup>i</sup>'' '''e<sub>i</sub>''', let us consider the ''n'' quantities defined by the
+'''x''' = ''x<sup> i</sup>'' '''e<sub>i</sub>''', let us consider the ''n'' quantities defined by the
 scalar products of '''x''' with the basis vectors, '''e<sub>i</sub>''':
-''x<sub>i</sub>'' = '''x''' . '''e<sub>i</sub>''' = ''x<sup>j</sup>'' '''e<sub>j</sub>''' . '''e<sub>i</sub>''' = ''x<sup>j</sup> g<sub>ji</sub>'',
+''x<sub>i</sub>'' = '''x''' . '''e<sub>i</sub>''' = ''x<sup> j</sup>'' '''e<sub>j</sub>''' . '''e<sub>i</sub>''' = ''x<sup> j</sup> g<sub>ji</sub>'',
 where the ''g<sub>ji</sub>'' 's are the doubly covariant components of the [[metric tensor]].
-By solving these equations in terms of ''x<sup>j</sup>'', one gets:
+By solving these equations in terms of ''x<sup> j</sup>'', one gets:
-''x^j^'' = ''x,,i,, g^ij^''
+''x<sup> j</sup>'' = ''x<sub>i</sub> g<sup>ij</sup>''
-where the matrix of the  ''g^ij^'' 's is inverse of that of the ''g,,ij,,'' 's (''g^ik^g,,jk,,'' = &#948;^i^,,j,,). The development of vector '''x''' with respect to basis vectors '''e,,i,,''' can now also be written:
+where the matrix of the  ''g<sup>ij</sup>'' 's is inverse of that of the ''g<sub>ij</sub>'' 's (''g<sup>ik</sup>g<sub>jk</sub>'' = &#948;<sup>i</sup><sub>j</sub>). The development of vector '''x''' with respect to basis vectors '''e<sub>i</sub>''' can now also be written:
- '''x''' = ''x^i^'' '''e,,i,,''' = ''x,,i,, g^ij^'' '''e,,j,,'''
+'''x''' = ''x<sup> i</sup>'' '''e<sub>i</sub>''' = ''x<sub>i</sub> g<sup>ij</sup>'' '''e<sub>j</sub>'''
-The set of ''n'' vectors '''e^i^''' = ''g^ij^'' '''e,,j,,''' that span the space ''E^n^'' forms a basis since vector '''x''' can be written:
+The set of ''n'' vectors '''e<sup>i</sup>''' = ''g<sup>ij</sup>'' '''e<sub>j</sub>''' that span the space ''E<sup>n</sup>'' forms a basis since vector '''x''' can be written:
- '''x''' = ''x,,i,,'' '''e^i^'''
+'''x''' = ''x<sub>i</sub>'' '''e<sup>i</sup>'''
-This basis is the ''dual basis'' and the ''n'' quantities ''x,,i,,'' defined above are the
+This basis is the ''dual basis'' and the ''n'' quantities ''x<sub>i</sub>'' defined above are the
 coordinates of '''x''' with respect to the dual basis. In a similar way one can express the direct basis vectors in terms of the dual basis vectors:
- '''e,,i,,''' = ''g,,ij,,'' '''e^j^'''
+'''e<sub>i</sub>''' = ''g<sub>ij</sub>'' '''e<sup>j</sup>'''
 The scalar products of the basis vectors of the dual and direct bases are:
- ''g^i^,,j,,'' = '''e^i^''' . '''e,,j,,''' = ''g^ik^'' '''e,,k,,''' . '''e,,j,,''' = ''g^ik^g,,jk,,'' = &#948;^i^,,j,,.
+''g<sup>i</sup><sub>j</sub>'' = '''e<sup>i</sup>''' . '''e<sub>j</sub>''' = ''g<sup>ik</sup>'' '''e<sub>k</sub>''' . '''e<sub>j</sub>''' = ''g<sup>ik</sup>g<sub>jk</sub>'' = &#948;<sup>i</sup><sub>j</sub>.
-One has therefore, since the matrices ''g^ik^'' and ''g,,ij,,'' are inverse:
+One has therefore, since the matrices ''g<sup>ik</sup>'' and ''g<sub>ij</sub>'' are inverse:
- ''g^i^,,j,,'' = '''e^i^''' . '''e,,j,,''' = &#948;^i^,,j,,.
+''g<sup>i</sup><sub>j</sub>'' = '''e<sup>i</sup>''' . '''e<sub>j</sub>''' = &#948;<sup>i</sup><sub>j</sub>.
-These relations show that the dual basis vectors satisfy the definition conditions of the reciprocal vectors. In a three-dimensional space the dual basis and the basis of ["reciprocal space"] are identical.
+These relations show that the dual basis vectors satisfy the definition conditions of the reciprocal vectors. In a three-dimensional space the dual basis and the basis of [[reciprocal space]] are identical.
 == Change of basis ==
@@ Line 48: / Line 48: @@
 In a change of basis where the direct basis vectors and coordinates transform like:
- '''e'<sub>j</sub>''' = ''A<sub>j</sub><sup>i</sup>'' '''e<sub>i</sub>'''; ''x'<sup>j</sup>'' = ''B<sub>i</sub?<sup>j</sup>'' ''x^i^'',
+'''e'<sub>j</sub>''' = ''A<sub>j</sub><sup>i</sup>'' '''e<sub>i</sub>'''; ''x'<sup>j</sup>'' = ''B<sub>i</sub><sup> j</sup>'' ''x<sup>i</sup>'',
-where ''A<sub>j</sub><sup>i</sup>'' and ''B<sub>i</sub><sup>j</sup>'' are transformation matrices, transpose of one another,
+where ''A<sub>j</sub><sup>i</sup>'' and ''B<sub>i</sub><sup> j</sup>'' are transformation matrices, transpose of one another,
 the dual basis vectors '''e<sup>i</sup>''' and the coordinates ''x<sub>i</sub>'' transform according to:
- '''e'<sup>j</sup> ''' = ''B<sub>i</sub>^j^'' '''e<sup>i</sup>'''; ''x'<sub>j</sub>'' = ''A<sub>j</sub><sup>i</sup>x<sub>i</sub>''.
+'''e'<sup>j</sup> ''' = ''B<sub>i</sub><sup> j</sup>'' '''e<sup>i</sup>'''; ''x'<sub>j</sub>'' = ''A<sub>j</sub><sup>i</sup>x<sub>i</sub>''.
 The coordinates of a vector in reciprocal space are therefore ''covariant'' and the dual basis vectors (or reciprocal vectors) ''contravariant''.