Einstein summation convention
The convention was introduced by Einstein in 1916, who later jested to a
friend, "I have made a great discovery in mathematics; I have suppressed
the summation sign every time that the summation must be made over an
index which occurs twice..."
According to this convention, when an index variable appears twice in a
single term, it implies that we are summing over all of its possible values. In
typical applications, the index values are 1,2,3 (representing the three
dimensions of physical Euclidean space), or 0,1,2,3 or 1,2,3,4 (representing
the four dimensions of space-time, or Minkowski space), but they can have
any range, even (in some applications) an infinite set.
The starting point for the index notation is the concept of a basis of vectors.
A basis is a set of linearly independent vectors that span the vector space.
For example, in three-dimensional space,
basis vectors
Griffiths
x1=x, x2=y, x3=z
summation convention rules
http://ocw.mit.edu/NR/rdonlyres/Physics/8-07Fall-2005/
Summation Convention Rule #1
Repeated, doubled indices in quantities multiplied together are implicitly
summed.
Doubled indices in quantities multiplied together are sometimes called
paired indices. If the writer’s intent is not to have the repeated index
summed over, then this must be made explicit.
Example: The index i has a particular value
When the index appears only once, the index i is called a free index: it is
free to take any value, and the equation must hold for all values.
Indices that are summed over are called dummy indices. The names of
dummy indices are arbitrary.
Summation Convention Rule #2
Indices that are not summed over (free indices) are allowed to take all
possible values unless stated otherwise.
Summation Convention Rule #3
It is illegal to use the same dummy index more than twice in a term
unless its meaning is made explicit.
free indices
paired indices
illegal
Vector operations
Linear superposition:
The dot product of two vectors:
By Rule #1, there is an implied sum on both i and j where they
occur paired. By Rule #3, it is mandatory that different indices be
used for the expansion of A and B. Here we used the
orthonormality property of basis vectors:
Kronecker delta
squared length of
vector A
The curl of two vectors:
let us write:
vector expansion
in the vector basis
Levi-Civita symbol (tensor)
A permutation of (123) is defined to be a rearrangement of them
obtained by exchanging elements of the set. Even permutations
have an even number of exchanges; odd permutations have an odd
number. For example, (213) is an odd permutation of (123) but
(231) is an even permutation. The even permutations (231) and
(312) are often called cyclic permutations of (123) because they are
obtained by rolling around in a cycle like links on a bicycle chain.
There are three odd permutations of (123): (213), (132), and (321).
Thus, the Levi-Civita symbol is zero aside from 6 terms.
Swapping any two indices gives a sign-change:
which explains why the Levi-Civita tensor is sometimes called the
completely anti-symmetric tensor.
Compare with Griffiths:
Important identity:
Example 1: calculate
Since
Example 2: calculate
In particular:
Example 3: calculate
Matrix (tensor) operations
Second-rank tensor:
trace of matrix M
product of two matrices
symmetric
antisymmetric