The e-System Notation of Tensor Analysis

San José State University Department of Economics

applet-magic.com Thayer Watkins Silicon Valley & Tornado Alley USA

E-systems can be vastly generalized beyond the case used here but let us start with indices i,j and k ranging from 1 to 3. First the notion of a permuation of the indices must be considered and the even-ness or odd-ness defined. A permutation of 123 is any rearrangement of the numbers such as 213 or 321. An odd permutation is one that can be achieved with an odd number of transpositions. For example, 213 can be achieved from 123 by transposing the 1 and 2. If the 1 and 3 in 213 are transposed then the result is 231 and thus 231 is an even permutation of 123.

Now the 3x3x3 array e_ijk can be defined as

e_ijk = 0 if i=j, j=k or i=k
e_ijk = +1 if ijk is an even permuatation of 123
e_ijk = −1 if ijk is an odd permuatation of 123

A primary use of e-systems is for operations concerning determinants.
In conventional notation the determinant of matrix A whose elements are { a_ij} is expressed as
|A| = Σ_ja_ijCof(a_ij)

where Cof(a_ij), the cofactor of a_ij, is obtained by evaluating the determinant of the submatrix formed by deleting the i-th row and j-th column of the matrix A and multiplying by (−1)^i+j. For a 3x3 matrix A the cofactor of a₁₁ is (−1)²[a₂₂a₃₃−a₁₂a₂₁. For higher order matrices the formula for the cofactor of an element is impossibly complex.
The determinant in term of e-systems is a marvel of simplicity. It is
|A| = Σ_i,j,k,p,q,r e_ijka_pia_qja_rk
where in tensor analysis, because of the repeated indices,
the Summation sign Σ is understood and not displayed
|A| = e_ijka_pia_qja_rk

Thus mathematical manipulations, such as taking the derivative of a determinant, are easy to express.
The generalization of e-systems to the case in which the indices range from 1 to n is obvious.
(To be continued.)

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eijk = 0 if i=j, j=k or i=k eijk = +1 if ijk is an even permuatation of 123 eijk = −1 if ijk is an odd permuatation of 123

|A| = ΣjaijCof(aij)

|A| = Σi,j,k,p,q,r eijkapiaqjark where in tensor analysis, because of the repeated indices, the Summation sign Σ is understood and not displayed |A| = eijkapiaqjark

e_ijk = 0 if i=j, j=k or i=k
e_ijk = +1 if ijk is an even permuatation of 123
e_ijk = −1 if ijk is an odd permuatation of 123

|A| = Σ_ja_ijCof(a_ij)

|A| = Σ_i,j,k,p,q,r e_ijka_pia_qja_rk
where in tensor analysis, because of the repeated indices,
the Summation sign Σ is understood and not displayed
|A| = e_ijka_pia_qja_rk