Difference between revisions of "Tensor index notation"

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m (Get rid of the "suffix" stuff.)
(Try to make things strictly correct, even if reader doesn't understand just why. It's really more straightforward. In progress.)
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'''Tensor index notation''' is a method of notation which is of use when dealing with [[tensor]]s.<ref>[http://www.brown.edu/Departments/Engineering/Courses/En221/Notes/Index_notation/Index_notation.htm Tensor index notation] from brown.edu</ref> Particular examples of tensors include [[vector]]s and [[matrix|matrices]], and index notation can greatly simplify algebraic manipulations involving these types of mathematical object.
 
'''Tensor index notation''' is a method of notation which is of use when dealing with [[tensor]]s.<ref>[http://www.brown.edu/Departments/Engineering/Courses/En221/Notes/Index_notation/Index_notation.htm Tensor index notation] from brown.edu</ref> Particular examples of tensors include [[vector]]s and [[matrix|matrices]], and index notation can greatly simplify algebraic manipulations involving these types of mathematical object.
  
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The components of a vector (with respect to some co-ordinate system) might be written <math>\boldsymbol{x}=(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>x,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>:
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The components of a vector (with respect to some co-ordinate system) might be written <math>\boldsymbol{x}=(x_1,x_2,x_3)</math>.
 +
But before we go any further, we should note that proper tensor index notation is quite fussy about which indices are subscripts and which are superscripts.  Elementary treatments don't bother with this, and just use a "subscripts everywhere" philosophy.  But proper placement of indices as subscripts and superscripts is important in proper tensor algebra.  Even if one doesn't understand the theoretical reasons for this, things are notationally simpler once one gets used to it.  The fundamental principle, for now, is that:
 +
:'''Indices of vectors are written as superscripts.'''
 +
Just get used to that.
 +
 
 +
So, the components of the above vector could be written <math>\boldsymbol{x}=(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>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>:
  
 
:<math>
 
:<math>
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a_{11} x_1+a_{12} x_2+a_{13} x_3 = b_1
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a_{11} x^1+a_{12} x^2+a_{13} x^3 = b^1
 
</math>
 
</math>
  
 
:<math>
 
:<math>
−
a_{21}x_1+a_{22}x_2+a_{23}x_3 = b_2
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a_{21} x^1+a_{22} x^2+a_{23} x^3 = b^2
 
</math>
 
</math>
  
 
:<math>
 
:<math>
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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>
  

Revision as of 05:12, August 4, 2018

Tensor index notation is a method of notation which is of use when dealing with tensors.[1] Particular examples of tensors include vectors and matrices, and index notation can greatly simplify algebraic manipulations involving these types of mathematical object.

The components of a vector (with respect to some co-ordinate system) might be written . But before we go any further, we should note that proper tensor index notation is quite fussy about which indices are subscripts and which are superscripts. Elementary treatments don't bother with this, and just use a "subscripts everywhere" philosophy. But proper placement of indices as subscripts and superscripts is important in proper tensor algebra. Even if one doesn't understand the theoretical reasons for this, things are notationally simpler once one gets used to it. The fundamental principle, for now, is that:

Indices of vectors are written as superscripts.

Just get used to that.

So, the components of the above vector could be written .

More concisely, we could write for the components of the vector, where . To motivate this notation, we will consider the equation for some matrix and vectors and . We will use the convention that if is a matrix, then 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 :


This can be rewritten in a matrix/vector form as equation :

Comparison of these two forms should convince you that the "go along the column and down the rows" rule for multiplying a matrix and a vector is sensible. We can also write the above system of equations more succinctly in index notation. We notice that in any of the three equations, the first index on the elements is fixed whilst the second varies from 1 to 3. Thus:

Even more succinctly, we can write this as the single expression

When you see such an equation, remember that it is a shorthand notation for writing three equations at once, for (in 3D). The major advantage with index notation is that objects such as as scalars and therefore 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 tensors. 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.[2] In the example above, the sum:

can be written simply as . There are three rules when using this convention:

Any repeated index is implicitly summed over, as above.
Each index cannot appear more than twice in any term, so is valid but is not as it has more than two k's.
Each term must contain the same non-repeated indices, so is valid but 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.

References

  1. ↑ Tensor index notation from brown.edu
  2. ↑ Einstein Summation Convention at mathworld.wolfram.com

See also

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