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more scrupulous correctness
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>
</math>
This can be rewritten in a matrix/vector form as equation <math>\boldsymbol{A}\,\vec{\boldsymbol{x}}=\vec{\boldsymbol{b}}</math>:
This can be rewritten in a matrix/vector form as equation <math>\boldsymbol{A}\,\vec{\boldsymbol{x}}=\vec{\boldsymbol{b}}</math>:
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.
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.
::{| cellspacing="0" cellpadding="0" style="margin:2em 2em 1em 0em; width:80%"
| style="vertical-align:top; border:1px solid Indigo; background-color:Lavender;" |
<div style="border-bottom:1px solid Indigo; background-color:#c0c0fa; padding:0.0em 0.5em 0.0em 0.5em; font-size:110%; font-weight:bold;">'''REALLY CORRECT TENSOR INDEX NOTATION'''</div>
<div style="padding:0.4em 1em 0.3em 1em;">
We can rewrite the preceding paragraphs in strict tensor form, by using the convention that if <math>\boldsymbol{A}</math> is a matrix, then <math>\boldsymbol{A}^i_j</math> is the element of that matrix in the ith row and jth column. So the equations, in terms of the correct vector components are:
:<math>
a^1_1 x^1+a^1_2 x^2+a^1_3 x^3 = b^1
</math>
:<math>
a^2_1 x^1+a^2_2 x^2+a^2_3 x^3 = b^2
</math>
:<math>
a^3_1 x^1+a^3_2 x^2+a^3_3 x^3 = b^3
</math>
Or, more succinctly:
:<math>
\sum_{j=1}^3 a^1_j x^j = b^1
</math>
:<math>
\sum_{j=1}^3 a^2_j x^j = b^2
</math>
:<math>
\sum_{j=1}^3 a^3_j x^j = b^3
</math>
Or just
:<math>
\sum_{j=1}^3 a^i_j x^j = b^i
</math>
for all <math>i</math> from 1 to 3.
</div>
|}
==Einstein Summation Convention==
==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: