Tensor index notation - Revision history

DavidB4-bot: /* Einstein Summation Convention */clean up, typos fixed: so called → so-called

2018-09-28T02:27:15Z

Einstein Summation Convention: clean up, typos fixed: so called → so-called

SamHB: This stuff is tricky. Saying just how you say things that no one would say in any case, is hard.

2018-08-28T20:39:18Z

This stuff is tricky. Saying just how you say things that no one would say in any case, is hard.

SamHB at 17:08, August 28, 2018

2018-08-28T17:08:39Z

SamHB: explain ambiguity resolution

2018-08-28T17:07:37Z

explain ambiguity resolution

SamHB at 00:09, August 5, 2018

2018-08-05T00:09:34Z

SamHB: This should just about do it. Most readers still won't know why all this is true.

2018-08-05T00:08:07Z

This should just about do it. Most readers still won't know why all this is true.

SamHB: more scrupulous correctness

2018-08-04T23:17:46Z

more scrupulous correctness

SamHB at 19:48, August 4, 2018

2018-08-04T19:48:11Z

SamHB: better symbols

2018-08-04T17:37:09Z

better symbols

SamHB: Put out-of-sidebar material back into "familiar" notation.

2018-08-04T17:26:54Z

Put out-of-sidebar material back into "familiar" notation.

@@ Line 113: / Line 113: @@
 |}
 ==Einstein Summation Convention==
-Index notation greatly simplifies the manipulation of [[tensor]]s. A commonly used convention when dealing with tensors is the so called "Einstein summation convention." Any index that is repeated exactly twice is assumed to be summed over and the sum symbols are not written.<ref>[http://mathworld.wolfram.com/EinsteinSummation.html Einstein Summation Convention at mathworld.wolfram.com]</ref> In the example above, the sum:
+Index notation greatly simplifies the manipulation of [[tensor]]s. A commonly used convention when dealing with tensors is the so-called "Einstein summation convention." Any index that is repeated exactly twice is assumed to be summed over and the sum symbols are not written.<ref>[http://mathworld.wolfram.com/EinsteinSummation.html Einstein Summation Convention at mathworld.wolfram.com]</ref> In the example above, the sum:
 :<math>

@@ Line 14: / Line 14: @@
 So, the components of the above vector could be written <math>\vec{\boldsymbol{x}}=(x^1,x^2,x^3)</math>.
-But doesn't writing the vector components as superscripts lead to ambiguity with raising something to a power?  Not really.  Proper tensor/vector equations '''almost never raise a vector component to a power.'''  When one must do so, the convention is to put the power in parentheses.  So <math>x^2</math> is the second component of a vector, <math>x^{(2)}</math> is the number x squared, and <math>x^{2^{(2)}}</math> is the second component squared.
+But doesn't writing the vector components as superscripts lead to ambiguity with raising something to a power?  Not really.  Proper tensor/vector equations '''almost never raise a vector component to a power.'''  When one must do so, the convention is to put the power in parentheses.  So <math>x^2</math> is the second component of a vector, <math>x^{(2)}</math> is the number x squared, and <math>x^{2^{(2)}}</math> or <math>(x^2)^2</math> is the second component squared.
 </div>
 |}

@@ Line 14: / Line 14: @@
 So, the components of the above vector could be written <math>\vec{\boldsymbol{x}}=(x^1,x^2,x^3)</math>.
-But doesn't writing the vector components as superscripts lead to ambiguity with raising something to a power?  Not really.  Proper tensor vector equations '''almost never raise a vector component to a power.'''  When one must do so, the convention is to put the power in parentheses.  So <math>x^2</math> is the second component of a vector, <math>x^{(2)}</math> is the number x squared, and <math>x^{2^{(2)}}</math> is the second component squared.
+But doesn't writing the vector components as superscripts lead to ambiguity with raising something to a power?  Not really.  Proper tensor/vector equations '''almost never raise a vector component to a power.'''  When one must do so, the convention is to put the power in parentheses.  So <math>x^2</math> is the second component of a vector, <math>x^{(2)}</math> is the number x squared, and <math>x^{2^{(2)}}</math> is the second component squared.
 </div>
 |}

@@ Line 13: / Line 13: @@
 So, the components of the above vector could be written <math>\vec{\boldsymbol{x}}=(x^1,x^2,x^3)</math>.
 </div>
 |}

@@ Line 128: / Line 128: @@
 The rules for really correct tensor notation, including Einstein Summation, are as follows:
 :Einstein summation must be over one subscript and one superscript.
 :Indices that appear in terms to be added or subtracted, or on opposite sides of and equals sign, must match: subscript for subscript, and superscript for superscript.
 :For any index left over after summation, there is an implicit "for all" in the equation.
 So the above equation becomes just

@@ Line 122: / Line 122: @@
 :Each index cannot appear more than twice in any term, so <math>a_{ij}=b_{ik}c_{kl}d_{lj}</math> is valid but <math>a_{ij}=b_{ik}c_{kk}d_{kj}</math> is not as it has more than two k's.
 :Each term must contain the same non-repeated indices, so <math>a_{ij}x_j + b_{ij}x_j</math> is valid but <math>a_{ij}x_j + b_{lj}x_j</math> is not as the first term does not have the index l.
-In general, the indices may subscript or superscript depending on whether the tensors are covariant or contravariant, respectively.
+In general, the indices are subscript or superscript depending on whether the tensors are covariant or contravariant, respectively.
 ==References==

@@ Line 28: / Line 28: @@
 a_{31}x_1+a_{32}x_2+a_{33}x_3 = b_3
 </math>
 This can be rewritten in a matrix/vector form as equation <math>\boldsymbol{A}\,\vec{\boldsymbol{x}}=\vec{\boldsymbol{b}}</math>:
@@ Line 72: / Line 71: @@
 When you see such an equation, remember that it is a shorthand notation for writing three equations at once, for <math>i=1,2,3</math> (in 3D). The major advantage with index notation is that objects such as <math>a_{ij}</math> as [[scalar]]s and therefore [[Commutative property|commute]] (ab=ba). Tensors themselves however can represent objects such as [[matrices]] and do not in general commute.
 ==Einstein Summation Convention==
 Index notation greatly simplifies the manipulation of [[tensor]]s. A commonly used convention when dealing with tensors is the so called "Einstein summation convention." Any index that is repeated exactly twice is assumed to be summed over and the sum symbols are not written.<ref>[http://mathworld.wolfram.com/EinsteinSummation.html Einstein Summation Convention at mathworld.wolfram.com]</ref> In the example above, the sum:

@@ Line 9: / Line 9: @@
 :'''Indices of vectors are written as superscripts.'''
 Get used to seeing things written that way.
 So, the components of the above vector could be written <math>\vec{\boldsymbol{x}}=(x^1,x^2,x^3)</math>.

@@ Line 13: / Line 13: @@
 </div>
 |}
-More concisely, we could write <math>x^i</math> for the components of the vector, where <math>i=1,2,3</math>. To motivate this notation, we will consider the equation <math>Ax=b</math> for some matrix <math>A</math> and vectors <math>x</math> and <math>b</math>. We will use the convention that if <math>A</math> is a matrix, then <math>(A)_{ij}=a_{ij}</math> is the element of that matrix in the ith row and jth column. One way of thinking of vector equations is as a shorthand for a set of simultaneous equations - each component of the vectors gives an equation. Explicitly, consider the set of three equations for the three unknowns <math>x^1, x^2, x^3</math>:
+More concisely, we could write <math>x_i</math> for the components of the vector, where <math>i=1,2,3</math>. To motivate this notation, we will consider the equation <math>Ax=b</math> for some matrix <math>A</math> and vectors <math>\vec{\boldsymbol{x}}</math> and <math>\vec{\boldsymbol{b}}</math>. We will use the convention that if <math>A</math> is a matrix, then <math>(A)_{ij}=a_{ij}</math> is the element of that matrix in the ith row and jth column. One way of thinking of vector equations is as a shorthand for a set of simultaneous equations - each component of the vectors gives an equation. Explicitly, consider the set of three equations for the three unknowns <math>x_1, x_2, x_3</math>:
 :<math>
-a_{11} x^1+a_{12} x^2+a_{13} x^3 = b^1
+a_{11} x_1+a_{12} x_2+a_{13} x_3 = b_1
 </math>
 :<math>
-a_{21} x^1+a_{22} x^2+a_{23} x^3 = b^2
+a_{21}x_1+a_{22}x_2+a_{23}x_3 = b_2
 </math>
 :<math>
-a_{31} x^1+a_{32} x^2+a_{33} x^3 = b^3
+a_{31}x_1+a_{32}x_2+a_{33}x_3 = b_3
 </math>