Tensor - Revision history

SamHB: use more familiar term

2018-08-04T19:15:42Z

use more familiar term

SamHB: Don't use "suffix notation". The correct term is "index notation".

2018-08-04T04:13:31Z

Don't use "suffix notation". The correct term is "index notation".

SamHB at 23:03, August 3, 2018

2018-08-03T23:03:04Z

FredericBernard: Added some examples, references and wikify

2017-09-11T15:19:20Z

Added some examples, references and wikify

DavidB4-bot: /* See also */Category

2016-07-29T20:18:20Z

‎See also: Category

DavidB4-bot: /* See also */clean up & uniformity

2016-07-13T20:13:27Z

‎See also: clean up & uniformity

SamHB: Dewikify "tensor analysis", and add stuff. I have some things to say about this. See talk.

2010-02-18T04:42:40Z

Dewikify "tensor analysis", and add stuff. I have some things to say about this. See talk.

Ed Poor: Tensor analysis

2010-02-13T20:10:25Z

Tensor analysis

Aschlafly: better intro, added cat vector analysis

2009-12-25T18:16:38Z

better intro, added cat vector analysis

Henry8th: New page: Vectors and matrices are examples of more general objects called tensors. Tensors are defined via their transformation properties: suppose we have a set of numbers
2008-05-11T21:39:21Z

New page: Vectors and matrices are examples of more general objects called tensors. Tensors are defined via their transformation properties: suppose we have a set of numbers <m...

New page
[[vector|Vectors]] and [[matrix|matrices]] are examples of more general objects called tensors. Tensors are defined via their transformation properties: suppose we have a set of numbers <math>v_i</math>, and we want to know how their values change under rotation of Cartesian axes. If the values in the new co-ordinate system <math>v'_i</math> can be written

<math>
v'_i=L_{ij}v_j
</math>

where <math>L_{ij}</math> are the elements of a rotation matrix then the <math>v_i</math> are said to be the components of a rank one tensor. Similarly, the components of a rank two tensor satisfy

<math>
a'_{ij}=L_{im}L_{jn}a_{mn}
</math>

and for higher order tensors, we just keep adding more of the <math>L_{ij}</math> rotation matrices. Scalars, vectors and matrices are rank zero, rank one and rank two tensors respectively.

==See also==
[[Suffix notation]]

[[category:mathematics]]

@@ Line 3: / Line 3: @@
 Briefly, a tensor is some kind of linear function involving vectors.  The "rank" of a tensor indicates how many vectors are involved as arguments to that function.  A simple number (that is, a "[[scalar]]") is a function of no arguments and just returns a number, and is called a "zeroth rank" tensor.  (Lest the reader think that scalars are utterly trivial, consider that "scalar fields", "vector fields", and "tensor fields" are assignments of scalars, vectors and tensors at every point in space, and that the [[gradient]] of a scalar field is a first-rank, or [[vector]] covariant tensor field.)
-Tensors may be "covariant", "contravariant", or "mixed", depending on just how they deal with their arguments.  They can usually be converted into each other.  A [[vector]] is a first rank contravariant tensor.  A linear function that maps an input vector to an output vector (sometimes called a "linear operator") is a second rank mixed tensor.  (Such a thing is often called a [[matrix]], but this glosses over a distinction between a thing and the numbers that describe it.)
+Tensors may be "covariant", "contravariant", or "mixed", depending on just how they deal with their arguments.  They can usually be converted into each other.  A [[vector]] is a first rank contravariant tensor.  A linear function that maps an input vector to an output vector (sometimes called a "linear transformation") is a second rank mixed tensor.  (Such a thing is often called a [[matrix]], but this glosses over a distinction between a thing and the numbers that describe it.)
 Tensors are often defined via their transformation properties, that is, by how their components change when one rotates the coordinate axes.  Suppose we have a set of numbers <math>v_i</math>, and we want to know how their values change under rotation of Cartesian axes. If the values in the new co-ordinate system <math>v'_i</math> can be written as:

@@ Line 51: / Line 51: @@
 ==See also==
-[[Suffix notation]]
+*[[Tensor index notation]]
 [[Category:Mathematics]]
 [[Category:Physics]]
 [[Category:Vector Analysis]]

@@ Line 3: / Line 3: @@
 Briefly, a tensor is some kind of linear function involving vectors.  The "rank" of a tensor indicates how many vectors are involved as arguments to that function.  A simple number (that is, a "[[scalar]]") is a function of no arguments and just returns a number, and is called a "zeroth rank" tensor.  (Lest the reader think that scalars are utterly trivial, consider that "scalar fields", "vector fields", and "tensor fields" are assignments of scalars, vectors and tensors at every point in space, and that the [[gradient]] of a scalar field is a first-rank, or [[vector]] covariant tensor field.)
-Tensors may be "covariant", "contravariant", or "mixed", depending on just how they deal with their arguments.  They can usually be converted into each other.  A [[vector]] is a first rank contravariant tensor.  A linear function that maps an input vector to an output vector is a second rank mixed tensor.  (Such a thing is often called a [[matrix]], but this glosses over a distinction between a thing and the numbers that describe it.)
+Tensors may be "covariant", "contravariant", or "mixed", depending on just how they deal with their arguments.  They can usually be converted into each other.  A [[vector]] is a first rank contravariant tensor.  A linear function that maps an input vector to an output vector (sometimes called a "linear operator") is a second rank mixed tensor.  (Such a thing is often called a [[matrix]], but this glosses over a distinction between a thing and the numbers that describe it.)
 Tensors are often defined via their transformation properties, that is, by how their components change when one rotates the coordinate axes.  Suppose we have a set of numbers <math>v_i</math>, and we want to know how their values change under rotation of Cartesian axes. If the values in the new co-ordinate system <math>v'_i</math> can be written as:

@@ Line 1: / Line 1: @@
-'''Tensors''' are mathematical objects consisting of indices and components, which obey rules of transformation.  Tensors, and tensor fields, are useful in mechanical engineering, electromagnetic theory, differential geometry, and the [[general theory of relativity]].  The study of tensors is variously called "tensor algebra", "tensor calculus", or "tensor analysis".
+'''Tensors''' are mathematical objects consisting of indices and components, which obey rules of transformation.<ref>[http://mathworld.wolfram.com/Tensor.html Tensors on mathworld.wolfram.com]</ref>  Tensors, and tensor fields, are useful in mechanical engineering, [[Electromagnetism|electromagnetic theory]], [[differential geometry]], and the [[general theory of relativity]].  The study of tensors is variously called "tensor algebra", "tensor calculus", or "tensor analysis".
-Briefly, a tensor is some kind of linear function involving vectors.  The "rank" of a tensor indicates how many vectors are involved as arguments to that function.  A simple number (that is, a "scalar") is a function of no arguments and just returns a number, and is a "zeroth rank" tensor.  (Lest the reader think that scalars are utterly trivial, consider that "scalar fields", "vector fields", and "tensor fields" are assignments of scalars, vectors and tensors at every point in space, and that the [[gradient]] of a scalar field is a first-rank covariant tensor field.)
+Briefly, a tensor is some kind of linear function involving vectors.  The "rank" of a tensor indicates how many vectors are involved as arguments to that function.  A simple number (that is, a "[[scalar]]") is a function of no arguments and just returns a number, and is called a "zeroth rank" tensor.  (Lest the reader think that scalars are utterly trivial, consider that "scalar fields", "vector fields", and "tensor fields" are assignments of scalars, vectors and tensors at every point in space, and that the [[gradient]] of a scalar field is a first-rank, or [[vector]] covariant tensor field.)
 Tensors may be "covariant", "contravariant", or "mixed", depending on just how they deal with their arguments.  They can usually be converted into each other.  A [[vector]] is a first rank contravariant tensor.  A linear function that maps an input vector to an output vector is a second rank mixed tensor.  (Such a thing is often called a [[matrix]], but this glosses over a distinction between a thing and the numbers that describe it.)
-Tensors are often defined via their transformation properties, that is, by how their components change when one rotates the coordinate axes.  Suppose we have a set of numbers <math>v_i</math>, and we want to know how their values change under rotation of Cartesian axes. If the values in the new co-ordinate system <math>v'_i</math> can be written
+Tensors are often defined via their transformation properties, that is, by how their components change when one rotates the coordinate axes.  Suppose we have a set of numbers <math>v_i</math>, and we want to know how their values change under rotation of Cartesian axes. If the values in the new co-ordinate system <math>v'_i</math> can be written as:
-<math>v'_i=L_{ij}v_j\,</math>
+:<math>v'_i=L_{ij}v_j\,</math>
-where <math>L_{ij}\,</math> are the elements of the rotation matrix then the <math>v_i\,</math> are said to be the components of a first rank contravariant tensor, that is, a vector.  Similarly, the components of a second rank contravariant tensor satisfy
+where <math>L_{ij}\,</math> are the elements of the rotation matrix then the <math>v_i\,</math> are said to be the components of a first rank contravariant tensor, that is, a vector. Using the [[Suffix notation|Einstein summation convention]], the repeated <math>j</math> index is summed over. This process of summing over a repeated index is known as "contraction", or in this case "contracting the j index". The components of a second rank contravariant tensor satisfy
-<math>a'_{ij}=L_{im}L_{jn}a_{mn}\,</math>
+:<math>a'_{ij}=L_{im}L_{jn}a_{mn}\,</math>
 and for higher order tensors, we just keep adding more of the <math>L_{ij}\,</math> rotation matrices, or their inverses for covariant tensors.
 ==See also==

@@ Line 20: / Line 20: @@
 [[Category:Mathematics]]
 [[Category:Physics]]
-[[Category:Vector analysis]]
+[[Category:Vector Analysis]]

← Older revision		Revision as of 20:10, February 13, 2010
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−	'''Tensors''' are mathematical objects consisting of indices and components, which obey rules of transformation. Tensor analysis is useful in mechanical engineering, electromagnetic theory, differential geometry, and the [[general theory of relativity]].	+	'''Tensors''' are mathematical objects consisting of indices and components, which obey rules of transformation. [[Tensor analysis]] is useful in mechanical engineering, electromagnetic theory, differential geometry, and the [[general theory of relativity]].

	[[vector\|Vectors]] and [[matrix\|matrices]] are examples of more general objects called tensors. Tensors are defined via their transformation properties: suppose we have a set of numbers <math>v_i</math>, and we want to know how their values change under rotation of Cartesian axes. If the values in the new co-ordinate system <math>v'_i</math> can be written		[[vector\|Vectors]] and [[matrix\|matrices]] are examples of more general objects called tensors. Tensors are defined via their transformation properties: suppose we have a set of numbers <math>v_i</math>, and we want to know how their values change under rotation of Cartesian axes. If the values in the new co-ordinate system <math>v'_i</math> can be written

← Older revision		Revision as of 18:16, December 25, 2009
Line 1:		Line 1:
		+	'''Tensors''' are mathematical objects consisting of indices and components, which obey rules of transformation. Tensor analysis is useful in mechanical engineering, electromagnetic theory, differential geometry, and the [[general theory of relativity]].
		+
	[[vector\|Vectors]] and [[matrix\|matrices]] are examples of more general objects called tensors. Tensors are defined via their transformation properties: suppose we have a set of numbers <math>v_i</math>, and we want to know how their values change under rotation of Cartesian axes. If the values in the new co-ordinate system <math>v'_i</math> can be written		[[vector\|Vectors]] and [[matrix\|matrices]] are examples of more general objects called tensors. Tensors are defined via their transformation properties: suppose we have a set of numbers <math>v_i</math>, and we want to know how their values change under rotation of Cartesian axes. If the values in the new co-ordinate system <math>v'_i</math> can be written

Line 17:		Line 19:

	[[category:mathematics]]		[[category:mathematics]]
		+	[[category:vector analysis]]