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A [[Quantity|quantity]] all values of which can be expressed by one (real) number. More generally, a scalar is an element of some [[Field|field]].
 
A [[Quantity|quantity]] all values of which can be expressed by one (real) number. More generally, a scalar is an element of some [[Field|field]].
  
 
====Comments====
 
====Comments====
Generally, the setting is that of a (ground) field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083240/s0832401.png" /> (more generally, a ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083240/s0832402.png" />) and a vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083240/s0832403.png" /> (of functions, vectors, matrices, tensors, etc.) over it (more generally, a module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083240/s0832404.png" />). The elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083240/s0832405.png" /> (respectively, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083240/s0832406.png" />) are called scalars. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083240/s0832407.png" /> (respectively, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083240/s0832408.png" />) is an algebra with unit element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083240/s0832409.png" />, the elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083240/s08324010.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083240/s08324011.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083240/s08324012.png" /> (respectively, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083240/s08324013.png" />) are also called scalars. For example, one sometimes speaks of the (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083240/s08324014.png" />)-matrices <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083240/s08324015.png" /> as scalar matrices. The scalar multiples of an element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083240/s08324016.png" /> (respectively, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083240/s08324017.png" />) are the elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083240/s08324018.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083240/s08324019.png" /> (respectively, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083240/s08324020.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083240/s08324021.png" />).
+
Generally, the setting is that of a (ground) field $  F $(
 +
more generally, a ring $  R $)  
 +
and a vector space $  V $(
 +
of functions, vectors, matrices, tensors, etc.) over it (more generally, a module $  M $).  
 +
The elements of $  F $(
 +
respectively, $  R $)  
 +
are called scalars. If $  V $(
 +
respectively, $  M $)  
 +
is an algebra with unit element $  e $,  
 +
the elements $  \lambda e $,  
 +
$  \lambda $
 +
in $  F $(
 +
respectively, $  R $)  
 +
are also called scalars. For example, one sometimes speaks of the ( $  n \times n $)-
 +
matrices $  \mathop{\rm diag} ( \lambda \dots \lambda ) $
 +
as scalar matrices. The scalar multiples of an element $  v \in V $(
 +
respectively, $  m \in M $)  
 +
are the elements $  \lambda v $,  
 +
$  \lambda \in F $(
 +
respectively, $  \lambda m $,  
 +
$  \lambda \in R $).
  
 
The term  "scalar"  comes from the original meaning as a quantity which can be completely specified by one (real) number.
 
The term  "scalar"  comes from the original meaning as a quantity which can be completely specified by one (real) number.
  
A scalar field on a manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083240/s08324022.png" /> is a function on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083240/s08324023.png" />; that is, a scalar field, or field of scalars, is a tensor field (cf. [[Tensor bundle|Tensor bundle]]) of rank <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083240/s08324024.png" />. These are the scalars in the algebra of tensor fields on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083240/s08324025.png" /> over the ring of functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083240/s08324026.png" />.
+
A scalar field on a manifold $  M $
 +
is a function on $  M $;  
 +
that is, a scalar field, or field of scalars, is a tensor field (cf. [[Tensor bundle|Tensor bundle]]) of rank $  ( 0, 0) $.  
 +
These are the scalars in the algebra of tensor fields on $  M $
 +
over the ring of functions on $  M $.
  
 
A scalar operator on, say, a complex Banach space is a scalar multiple of the identity operator.
 
A scalar operator on, say, a complex Banach space is a scalar multiple of the identity operator.
  
Given a left module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083240/s08324027.png" /> over a ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083240/s08324028.png" /> and an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083240/s08324029.png" />-algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083240/s08324030.png" />, one forms the [[Tensor product|tensor product]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083240/s08324031.png" />. This is a module over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083240/s08324032.png" />. The module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083240/s08324033.png" /> is said to be obtained from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083240/s08324034.png" /> by extension of scalars.
+
Given a left module $  M $
 +
over a ring $  R $
 +
and an $  R $-
 +
algebra $  S $,  
 +
one forms the [[Tensor product|tensor product]] $  S \otimes _ {R} M $.  
 +
This is a module over $  S $.  
 +
The module $  S \otimes _ {R} M $
 +
is said to be obtained from $  M $
 +
by extension of scalars.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  P.M. Cohn,  "Algebra" , '''1''' , Wiley  (1982)  pp. 70</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  K. Rektorys (ed.) , ''Applicable mathematics'' , Iliffe  (1969)  pp. 270; 290</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  P.M. Cohn,  "Algebra" , '''1''' , Wiley  (1982)  pp. 70</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  K. Rektorys (ed.) , ''Applicable mathematics'' , Iliffe  (1969)  pp. 270; 290</TD></TR></table>

Latest revision as of 08:12, 6 June 2020


A quantity all values of which can be expressed by one (real) number. More generally, a scalar is an element of some field.

Comments

Generally, the setting is that of a (ground) field $ F $( more generally, a ring $ R $) and a vector space $ V $( of functions, vectors, matrices, tensors, etc.) over it (more generally, a module $ M $). The elements of $ F $( respectively, $ R $) are called scalars. If $ V $( respectively, $ M $) is an algebra with unit element $ e $, the elements $ \lambda e $, $ \lambda $ in $ F $( respectively, $ R $) are also called scalars. For example, one sometimes speaks of the ( $ n \times n $)- matrices $ \mathop{\rm diag} ( \lambda \dots \lambda ) $ as scalar matrices. The scalar multiples of an element $ v \in V $( respectively, $ m \in M $) are the elements $ \lambda v $, $ \lambda \in F $( respectively, $ \lambda m $, $ \lambda \in R $).

The term "scalar" comes from the original meaning as a quantity which can be completely specified by one (real) number.

A scalar field on a manifold $ M $ is a function on $ M $; that is, a scalar field, or field of scalars, is a tensor field (cf. Tensor bundle) of rank $ ( 0, 0) $. These are the scalars in the algebra of tensor fields on $ M $ over the ring of functions on $ M $.

A scalar operator on, say, a complex Banach space is a scalar multiple of the identity operator.

Given a left module $ M $ over a ring $ R $ and an $ R $- algebra $ S $, one forms the tensor product $ S \otimes _ {R} M $. This is a module over $ S $. The module $ S \otimes _ {R} M $ is said to be obtained from $ M $ by extension of scalars.

References

[a1] P.M. Cohn, "Algebra" , 1 , Wiley (1982) pp. 70
[a2] K. Rektorys (ed.) , Applicable mathematics , Iliffe (1969) pp. 270; 290
How to Cite This Entry:
Scalar. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Scalar&oldid=48613