Difference between revisions of "Tensor index notation"
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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. |
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>: | 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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</math> | </math> | ||
| − | 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 | + | 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 <math>a_{ij}</math> elements is fixed whilst the second varies from 1 to 3. Thus: |
:<math> | :<math> | ||
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</math> | </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 | + | 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== | ==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: | |
:<math> | :<math> | ||
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can be written simply as <math>a_{ij}x_j=b_i</math>. There are three rules when using this convention: | can be written simply as <math>a_{ij}x_j=b_i</math>. There are three rules when using this convention: | ||
| − | :Any repeated | + | :Any repeated index is implicitly summed over, as above. |
| − | :Each | + | :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 | + | :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 | + | In general, the indices may subscript or superscript depending on whether the tensors are covariant or contravariant, respectively. |
==References== | ==References== | ||
Revision as of 04:24, 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 
. 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 
. 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.
