Orthotropic material - Revision history

207.230.192.98: correcting inconsistancies in notation

2015-01-12T19:07:00Z

correcting inconsistancies in notation

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	~~{{Dablink\|See [[scalar (disambiguation)\|scalar]] for an account of the broader concept also used in physics and computing.}}~~		Hi there, I am Alyson Boon even though it is not the title on my birth certification. Ohio is where his home is and his family members loves it. She is truly fond of caving but she doesn't have the time lately. I am an invoicing officer and I'll be promoted soon.<br><br>my site: [http://isaworld.pe.kr/?document_srl=392088 free psychic readings]

	~~In [[linear algebra]]~~, ~~[[real number]]s are called '''scalars''' and relate to vectors in a [[vector space]] through~~ the ~~operation of [[scalar multiplication]], in which a vector can be multiplied by a number to produce another vector.<ref>{{cite book \| last=Lay \| first=David C. \|~~ title=Linear Algebra and Its Applications \| publisher=[[Addison–Wesley]] \| year=2006 \| edition = 3rd \| isbn=0-321-28713-4}}</ref><ref>{{cite book \| last=Strang \| first=Gilbert \| authorlink=Gilbert Strang \| title=Linear Algebra and Its Applications \| publisher=[[Brooks Cole]] \| year=2006 \| edition = 4th \| isbn=0-03-010567-6}}</ref><ref>{{cite book \| last = Axler \| first = Sheldon \| title = Linear Algebra Done Right \| publisher = [[Springer Science+Business Media\|Springer]] \| year = 2002 \| edition = 2nd \| isbn = 0-387-98258-2}}</ref> More generally, a vector space may be defined by using any [[field (mathematics)\|field]] instead of real numbers, such as [[complex number]]s. Then the scalars of that vector space will be the elements of the associated field.

	~~A [[inner product\|scalar product]] operation (not to be confused with [[scalar multiplication]]) may be defined~~ on ~~a vector space, allowing two vectors to be multiplied to produce a scalar. A vector space equipped with a scalar product is called an [[inner product space]].~~

	~~The real component of a [[quaternion]] is also called its '''scalar part'''.~~

	The term is also sometimes used informally to mean a vector, [[matrix (mathematics)\|matrix]], [[tensor]], or other usually "compound" value that is actually reduced to a single component. Thus, for example, the product of a 1×''n'' matrix and an ''n''×1 matrix, which is formally a 1×1 matrix, is often said to be a '''scalar'''.

	~~The term '''[[scalar matrix]]'''~~ is ~~used to denote a matrix of the form ''kI''~~ where ~~''k''~~ is ~~a scalar~~ and ~~''I'' is the [[identity matrix]].~~

	~~==Etymology==~~

	The word ''scalar'' derives from the Latin word ''scalaris'', adjectival form from ''scala'' ([[Latin language\|Latin]] for "ladder"). The English word "scale" is also derived from ''scala''. The first recorded usage of the word "scalar" in mathematics was by [[François Viète]] in ''Analytic Art'' (''In artem analyticen isagoge'')(1591):<ref>http://math.ucdenver.edu/~wcherowi/courses/m4010/s08/lcviete.pdf Lincoln Collins. Biography Paper: Francois Viete</ref>

	~~:''Magnitudes that ascend or descend proportionally in keeping with their nature from one kind to another are called scalar terms.''~~
	~~:(Latin: ''Magnitudines quae ex genere ad genus sua vi proportionaliter adscendunt vel descendunt, vocentur Scalares.'')~~

	According to a citation in the ''[[Oxford English Dictionary]]'' the first recorded usage of the term in English was by [[William Rowan Hamilton\|W. R. Hamilton]] in 1846, to refer to the real part of a [[quaternion]]:

	:''The algebraically real part may receive, according to the question in which it occurs, all values contained on the one scale of progression of numbers from negative to positive infinity; we shall call it ~~therefore the scalar part~~.''

	~~==Definitions and properties==~~

	~~===Scalars of vector spaces===~~
	~~A [[vector space]]~~ is ~~defined as a set~~ of ~~vectors, a set of scalars, and a [[scalar multiplication]] operation that takes a scalar ''k'' and a vector '''v''' to another vector ''k'''''v''~~'~~. For example, in a [[coordinate space]],~~ the ~~scalar multiplication <math>k(v_1, v_2, \dots, v_n)</math> yields <math> (kv_1, kv_2, \dots, k v_n)</math>~~. ~~In a (linear) [[function space]], ''kƒ'' is the function ''x'' {{Mapsto}} ''k''(''ƒ''(''x'')).~~

	~~The scalars can be taken from any field, including the [[rational number\|rational]], [[algebraic number\|algebraic]], real,~~ and ~~complex numbers, as well as [[finite field]]s. a number by the elements inside the brackets.~~

	~~===Scalars as vector components===~~
	According to a fundamental theorem of linear algebra, every vector space has a [[basis (linear algebra)\|basis]]. It follows that every vector space over a scalar field ''K'' is [[isomorphism\|isomorphic]] to a [[coordinate vector space]] where the coordinates are elements of ''K''. For example, every real vector space of [[dimension (vector space)\|dimension]] ''n'' is isomorphic to ''n''-dimensional real space '''R'''<sup>''n''</sup>.

	~~===Scalars in normed vector spaces===~~
	Alternatively, a vector space ''V'' can be equipped with a [[norm (mathematics)\|norm]] function that assigns to every vector '''v''' in ''V'' a scalar \|\|'''v'''\|\|. By definition, multiplying '''v''' by a scalar ''k'' also multiplies its norm by \|''k''\|. If \|\|'''v'''\|\| is interpreted as the ''length'' of '''v'''~~, this operation can~~ be ~~described as '''scaling''' the length of '''v''' by ''k''~~. ~~A vector space equipped with a norm is called a [[normed vector space]] (or ''normed linear space'').~~

	~~The norm is usually defined to be an element of ''V''~~<~~nowiki~~>'<~~/nowiki~~>s scalar field ''K'', which restricts the latter to fields that support the notion of sign. Moreover, if ''V'' has dimension 2 or more, ''K'' must be closed under square root, as well as the four arithmetic operations; thus the rational numbers '''Q''' are excluded, but the [[surd field]] is acceptable. For this reason, not every scalar product space is a normed vector space.

	~~===Scalars in modules===~~
	When the requirement that the set of scalars form a field is relaxed so that it need only form a [[ring (mathematics)\|ring]] (so that, for example, the division of scalars need not be defined, or the scalars need not be [[commutative]]), the resulting more general algebraic structure is called a [[module (mathematics)\|module]].

	In this case the "scalars" may be complicated objects. For instance, if ''R'' is a ring, the vectors of the product space ''R''<sup>''n''</sup> can be made into a module with the ''n''×''n'' matrices with entries from ''R'' as the scalars. Another example comes from [[manifold\|manifold theory]], where the space of [[Section (fiber bundle)\|sections]] of the [[tangent bundle]] forms a module over the [[algebra]] of real functions on the manifold.

	~~===Scaling transformation===~~
	~~The scalar multiplication of vector spaces and modules is a special case of [[scaling (geometry)\|scaling]], a kind of [[linear transformation]].~~

	~~===Scalar operations (computer science)===~~
	~~Operations that apply to a single value at a time.~~

	* [[Scalar processor]]

	~~==See also==~~
	* [[Scalar (physics)]]

	~~==References==~~
	~~{{reflist}}~~

	~~== External links ==~~
	* {{springer\|title=Scalar\|id=p/s083240}}
	* {{MathWorld \|urlname=Scalar \|title=Scalar}}
	* [http://~~www~~.~~mathwords~~.~~com~~/~~s/scalar.htm Mathwords.com – Scalar]~~

	~~{{Linear algebra}}~~

	~~{{DEFAULTSORT:Scalar (Mathematics)}}~~
	~~[[Category:Abstract algebra]]~~
	~~[[Category:Linear algebra]]~~
	~~[[Category:Analytic geometry]~~]

en>Mark Arsten: Removing link(s) to "Material": deleted via Afd. (TW)

2014-01-16T03:02:08Z

Removing link(s) to "Material": deleted via Afd. (TW)

← Older revision		Revision as of 05:02, 16 January 2014
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	~~Nice~~ to ~~satisfy you~~, I ~~am Marvella Shryock~~. ~~My working day occupation~~ is a ~~meter reader~~. ~~To collect cash~~ is ~~what his family~~ and ~~him enjoy~~. ~~California~~ is where ~~her home~~ is ~~but she requirements~~ to ~~move simply because~~ of ~~her family~~.<br><br>~~Feel free~~ to ~~surf~~ to ~~my homepage~~ [~~https~~://www.~~epapyrus~~.com/xe/~~Purchase/5258960 home std test kit~~]		{{Dablink\|See [[scalar (disambiguation)\|scalar]] for an account of the broader concept also used in physics and computing.}}

			In [[linear algebra]], [[real number]]s are called '''scalars''' and relate to vectors in a [[vector space]] through the operation of [[scalar multiplication]], in which a vector can be multiplied by a number to produce another vector.<ref>{{cite book \| last=Lay \| first=David C. \| title=Linear Algebra and Its Applications \| publisher=[[Addison–Wesley]] \| year=2006 \| edition = 3rd \| isbn=0-321-28713-4}}</ref><ref>{{cite book \| last=Strang \| first=Gilbert \| authorlink=Gilbert Strang \| title=Linear Algebra and Its Applications \| publisher=[[Brooks Cole]] \| year=2006 \| edition = 4th \| isbn=0-03-010567-6}}</ref><ref>{{cite book \| last = Axler \| first = Sheldon \| title = Linear Algebra Done Right \| publisher = [[Springer Science+Business Media\|Springer]] \| year = 2002 \| edition = 2nd \| isbn = 0-387-98258-2}}</ref> More generally, a vector space may be defined by using any [[field (mathematics)\|field]] instead of real numbers, such as [[complex number]]s. Then the scalars of that vector space will be the elements of the associated field.

			A [[inner product\|scalar product]] operation (not to be confused with [[scalar multiplication]]) may be defined on a vector space, allowing two vectors to be multiplied to produce a scalar. A vector space equipped with a scalar product is called an [[inner product space]].

			The real component of a [[quaternion]] is also called its '''scalar part'''.

			The term is also sometimes used informally to mean a vector, [[matrix (mathematics)\|matrix]], [[tensor]], or other usually "compound" value that is actually reduced to a single component. Thus, for example, the product of a 1×''n'' matrix and an ''n''×1 matrix, which is formally a 1×1 matrix, is often said to be a '''scalar'''.

			The term '''[[scalar matrix]]''' is used to denote a matrix of the form ''kI'' where ''k'' is a scalar and ''I'' is the [[identity matrix]].

			==Etymology==

			The word ''scalar'' derives from the Latin word ''scalaris'', adjectival form from ''scala'' ([[Latin language\|Latin]] for "ladder"). The English word "scale" is also derived from ''scala''. The first recorded usage of the word "scalar" in mathematics was by [[François Viète]] in ''Analytic Art'' (''In artem analyticen isagoge'')(1591):<ref>http://math.ucdenver.edu/~wcherowi/courses/m4010/s08/lcviete.pdf Lincoln Collins. Biography Paper: Francois Viete</ref>

			:''Magnitudes that ascend or descend proportionally in keeping with their nature from one kind to another are called scalar terms.''
			:(Latin: ''Magnitudines quae ex genere ad genus sua vi proportionaliter adscendunt vel descendunt, vocentur Scalares.'')

			According to a citation in the ''[[Oxford English Dictionary]]'' the first recorded usage of the term in English was by [[William Rowan Hamilton\|W. R. Hamilton]] in 1846, to refer to the real part of a [[quaternion]]:

			:''The algebraically real part may receive, according to the question in which it occurs, all values contained on the one scale of progression of numbers from negative to positive infinity; we shall call it therefore the scalar part.''

			==Definitions and properties==

			===Scalars of vector spaces===
			A [[vector space]] is defined as a set of vectors, a set of scalars, and a [[scalar multiplication]] operation that takes a scalar ''k'' and a vector '''v''' to another vector ''k'''''v'''. For example, in a [[coordinate space]], the scalar multiplication <math>k(v_1, v_2, \dots, v_n)</math> yields <math> (kv_1, kv_2, \dots, k v_n)</math>. In a (linear) [[function space]], ''kƒ'' is the function ''x'' {{Mapsto}} ''k''(''ƒ''(''x'')).

			The scalars can be taken from any field, including the [[rational number\|rational]], [[algebraic number\|algebraic]], real, and complex numbers, as well as [[finite field]]s. a number by the elements inside the brackets.

			===Scalars as vector components===
			According to a fundamental theorem of linear algebra, every vector space has a [[basis (linear algebra)\|basis]]. It follows that every vector space over a scalar field ''K'' is [[isomorphism\|isomorphic]] to a [[coordinate vector space]] where the coordinates are elements of ''K''. For example, every real vector space of [[dimension (vector space)\|dimension]] ''n'' is isomorphic to ''n''-dimensional real space '''R'''<sup>''n''</sup>.

			===Scalars in normed vector spaces===
			Alternatively, a vector space ''V'' can be equipped with a [[norm (mathematics)\|norm]] function that assigns to every vector '''v''' in ''V'' a scalar \|\|'''v'''\|\|. By definition, multiplying '''v''' by a scalar ''k'' also multiplies its norm by \|''k''\|. If \|\|'''v'''\|\| is interpreted as the ''length'' of '''v''', this operation can be described as '''scaling''' the length of '''v''' by ''k''. A vector space equipped with a norm is called a [[normed vector space]] (or ''normed linear space'').

			The norm is usually defined to be an element of ''V''<nowiki>'</nowiki>s scalar field ''K'', which restricts the latter to fields that support the notion of sign. Moreover, if ''V'' has dimension 2 or more, ''K'' must be closed under square root, as well as the four arithmetic operations; thus the rational numbers '''Q''' are excluded, but the [[surd field]] is acceptable. For this reason, not every scalar product space is a normed vector space.

			===Scalars in modules===
			When the requirement that the set of scalars form a field is relaxed so that it need only form a [[ring (mathematics)\|ring]] (so that, for example, the division of scalars need not be defined, or the scalars need not be [[commutative]]), the resulting more general algebraic structure is called a [[module (mathematics)\|module]].

			In this case the "scalars" may be complicated objects. For instance, if ''R'' is a ring, the vectors of the product space ''R''<sup>''n''</sup> can be made into a module with the ''n''×''n'' matrices with entries from ''R'' as the scalars. Another example comes from [[manifold\|manifold theory]], where the space of [[Section (fiber bundle)\|sections]] of the [[tangent bundle]] forms a module over the [[algebra]] of real functions on the manifold.

			===Scaling transformation===
			The scalar multiplication of vector spaces and modules is a special case of [[scaling (geometry)\|scaling]], a kind of [[linear transformation]].

			===Scalar operations (computer science)===
			Operations that apply to a single value at a time.

			* [[Scalar processor]]

			==See also==
			* [[Scalar (physics)]]

			==References==
			{{reflist}}

			== External links ==
			* {{springer\|title=Scalar\|id=p/s083240}}
			* {{MathWorld \|urlname=Scalar \|title=Scalar}}
			* [http://www.mathwords.com/s/scalar.htm Mathwords.com – Scalar]

			{{Linear algebra}}

			{{DEFAULTSORT:Scalar (Mathematics)}}
			[[Category:Abstract algebra]]
			[[Category:Linear algebra]]
			[[Category:Analytic geometry]]

en>JYBot: r2.7.1) (Robot: Modifying fa:مواد ارثوتروپیک

2012-08-15T15:04:13Z

r2.7.1) (Robot: Modifying fa:مواد ارثوتروپیک

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