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Tensor index notation (view source)
Revision as of 02:27, September 28, 2018
, 02:27, September 28, 2018→Einstein Summation Convention: clean up, typos fixed: so called → so-called
:'''Indices of vectors are written as superscripts.'''
:'''Indices of vectors are written as superscripts.'''
Get used to seeing things written that way.
Get used to seeing things written that way.
Vectors are "contravariant first-rank tensors". Contravariant tensors have their indices written as superscripts, while covariant tensors have their indices written as subscripts.
So, the components of the above vector could be written <math>\vec{\boldsymbol{x}}=(x^1,x^2,x^3)</math>.
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> or <math>(x^2)^2</math> is the second component squared.
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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.
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<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.
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==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:
:<math>
:<math>
: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 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.
: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.
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<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;">
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
:<math>
b^i = a^i_j x^j
</math>
That is the really proper way to express that vector <math>\vec{\boldsymbol{b}}</math> is the result of applying the linear transformation <math>\boldsymbol{A}</math> to the vector <math>\vec{\boldsymbol{x}}</math>
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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==
==References==