Magic of tensor algebra: Part 8 - On convolutions of the Levi-Civita tensor

Content

Introduction

In the last article, we ran into the construction of the form $\ varepsilon ^ {\, mqi} \, \ varepsilon _ {\, ijk}$ - product of the contravariant Levi-Civita tensor on the covariant one. And, I must say, I simplified it not too elegant, but rather clumsy. In addition, the final expression of the Rodrigue formula, which is in the component form, which in the non-component form turned out to be extremely inconvenient in terms of further transformation. But I promised the reader to show how to obtain the angular velocity of a rigid body from the expression of the rotation matrix through the parameters of the final rotation, therefore, the questions presented below will be crucial in applying the tensor approach to the kinematics and dynamics of a solid body. At the same time I will once again recommend a rather old site “What Mathematics Looks Like” , although it was created on the engine of the people.ru, but it contains information that has already pushed me in the right direction several times when solving problems in studying tensor algebra.

So, let's talk about convolutions of the Levi-Civita tensor.

1. Symbols of Veblen

The reader probably already noticed that all components of the Levi-Civita tensor have a common factor: $\ sqrt g$ in a covariant representation and $\ frac {1} {\ sqrt g}$ in contravariant. It is logical to present this tensor in some more simplified form.

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$\ varepsilon _ {\, ijk} = \ sqrt g \, E _ {\, ijk} \ quad (1)$

$\ varepsilon _ {\, ijk} = \ frac {1} {\ sqrt g} \, E ^ {\, ijk} \ quad (2)$

Where

$E _ {\, ijk} = E ^ {\, ijk} = \ begin {cases} +1, \ quad P (i, j, k) = 1, \ -1, \ quad P (i, j, k ) = -1, \\ quad 0, i = j \ vee i = k \ vee j = k \ end {cases} \ quad (3)$

- an expression that determines the value of the so-called Veblen symbols , in which $P (i, j, k)$ - the function that determines the parity or oddness of the permutation of indices.

Thus, the procedure for multiplying the Levi-Civita tensors is reduced to operating with the symbols of Veblen

$\ varepsilon ^ {\, mnp} \, \ varepsilon _ {\, ijk} = \ frac {1} {\ sqrt g} \, \ sqrt g \, E ^ {\, mnp} \, E _ {\, ijk} = \ delta_ {ijk} ^ {\, mnp} \ quad (4)$

Where $\ delta_ {ijk} ^ {\, mnp}$ called the generalized Kronecker delta. The generalized Kronecker delta - three times contravariant and three times the covariant tensor of the sixth rank. For three-dimensional space, this construction has 3 ⁶ = 729 components. Too much say whatever. Besides, to imagine an array of components of the sixth (!) Rank tensor, using our three-dimensional thinking, is quite problematic. But this is usually not required - (4) participates in transformations where it collapses with other tensors. Therefore, it is useful to study the convolutions of the tensor (4), which will give us a way to fold and transform expressions in which the Levi-Civita tensor is involved.

2. Veblen's symbol as a determinant

Is it possible to approach more formally the definition of the value of expression (3) for any value of the indices? You can, if you pay attention to what

$E _ {\, 123} = \ left | \ begin {matrix} 1 & amp; & amp; 0 & amp; & amp; 0 \\ 0 & amp; & amp; 1 & amp; & amp; 0 \\ 0 & amp; & amp; 0 & amp; & amp; 1 \ end {matrix} \ right | = 1 \ quad (5)$

This is nothing more than the value of the Veblen symbol for a set of indices (1,2,3). Now rearrange in (5) a pair of columns.

$E _ {\, 213} = \ left | \ begin {matrix} 0 & amp; & amp; 1 & amp; & amp; 0 \\ 1 & amp; & amp; 0 & amp; & amp; 0 \\ 0 & amp; & amp; 0 & amp; & amp; 1 \ end {matrix} \ right | = -1$

Hmm ... Well, a few more columns will be swapped

$E _ {\, 231} = \ left | \ begin {matrix} 0 & amp; & amp; 0 & amp; & amp; 1 \\ 1 & amp; & amp; 0 & amp; & amp; 0 \\ 0 & amp; & amp; 1 & amp; & amp; 0 \ end {matrix} \ right | = 1$

Well, of course, the value of the Veblen symbol is equal to the determinant of a matrix made up of a unit matrix whose columns are taken in the order dictated by the order of the indices of the symbol! To get the expression of a general form, we represent the identity matrix, as is customary in tensor calculus through the Kronecker delta

$\ mathbf E = \ begin {bmatrix} \ delta_ {1} ^ {\, 1} & amp; & amp; \ delta_ {2} ^ {\, 1} & amp; & amp; \ delta_ {3} ^ {\, 1} \\ \ delta_ {1} ^ {\, 2} & amp; & amp; \ delta_ {2} ^ {\, 2} & amp; & amp; \ delta_ {3} ^ {\, 2} \\ \ delta_ {1} ^ {\, 3} & amp; & amp; \ delta_ {2} ^ {\, 3} & amp; & amp; \ delta_ {3} ^ {\, 3} \\ \ end {bmatrix} \ quad (6)$

Let me remind you that the Kronecker delta is equal to one when the indices coincide and zero if the indices are different. Now we will make a determinant for any symbol of Veblen

$E _ {\, 312} = \ left | \ begin {matrix} \ delta_ {3} ^ {\, 1} & amp; & amp; \ delta_ {1} ^ {\, 1} & amp; & amp; \ delta_ {2} ^ {\, 1} \\ \ delta_ {3} ^ {\, 2} & amp; & amp; \ delta_ {1} ^ {\, 2} & amp; & amp; \ delta_ {2} ^ {\, 2} \\ \ delta_ {3} ^ {\, 3} & amp; & amp; \ delta_ {1} ^ {\, 3} & amp; & amp; \ delta_ {2} ^ {\, 3} \\ \ end {matrix} \ right | = \ left | \ begin {matrix} 0 & amp; & amp; 1 & amp; & amp; 0 \\ 0 & amp; & amp; 0 & amp; & amp; 1 \\ 1 & amp; & amp; 0 & amp; & amp; 0 \ end {matrix} \ right | = -1$

that's right, which means it is not difficult to write in general

$E _ {\, ijk} = \ left | \ begin {matrix} \ delta_ {i} ^ {\, 1} & amp; & amp; \ delta_ {j} ^ {\, 1} & amp; & amp; \ delta_ {k} ^ {\, 1} \\ \ delta_ {i} ^ {\, 2} & amp; & amp; \ delta_ {j} ^ {\, 2} & amp; & amp; \ delta_ {k} ^ {\, 2} \\ \ delta_ {i} ^ {\, 3} & amp; & amp; \ delta_ {j} ^ {\, 3} & amp; & amp; \ delta_ {k} ^ {\, 3} \\ \ end {matrix} \ right | \ quad (7)$

Expression (7) is a general expression for an arbitrary symbol of Veblen, which allows us to derive

3. Analytical expression of the component of the generalized Kronecker delta

Take and multiply one Veblen symbol by another

$E ^ {\, mnp} \, E _ {\, ijk} = \ left | \ begin {matrix} \ delta_ {m} ^ {\, 1} & amp; & amp; \ delta_ {n} ^ {\, 1} & amp; & amp; \ delta_ {p} ^ {\, 1} \\ \ delta_ {m} ^ {\, 2} & amp; & amp; \ delta_ {n} ^ {\, 2} & amp; & amp; \ delta_ {p} ^ {\, 2} \\ \ delta_ {m} ^ {\, 3} & amp; & amp; \ delta_ {n} ^ {\, 3} & amp; & amp; \ delta_ {p} ^ {\, 3} \\ \ end {matrix} \ right | \ left | \ begin {matrix} \ delta_ {i} ^ {\, 1} & amp; & amp; \ delta_ {j} ^ {\, 1} & amp; & amp; \ delta_ {k} ^ {\, 1} \\ \ delta_ {i} ^ {\, 2} & amp; & amp; \ delta_ {j} ^ {\, 2} & amp; & amp; \ delta_ {k} ^ {\, 2} \\ \ delta_ {i} ^ {\, 3} & amp; & amp; \ delta_ {j} ^ {\, 3} & amp; & amp; \ delta_ {k} ^ {\, 3} \\ \ end {matrix} \ right | \ quad (8)$

To calculate (8) we have to remember that the determinant of the matrix product is the product of determinants from each matrix (regardless of the order of the factors), and therefore we calculate the product of matrices composed of the elements of the determinants included in (8)

$\ begin {bmatrix} \ delta_ {m} ^ {\, 1} & amp; & amp; \ delta_ {n} ^ {\, 1} & amp; & amp; \ delta_ {p} ^ {\, 1} \\ \ delta_ {m} ^ {\, 2} & amp; & amp; \ delta_ {n} ^ {\, 2} & amp; & amp; \ delta_ {p} ^ {\, 2} \\ \ delta_ {m} ^ {\, 3} & amp; & amp; \ delta_ {n} ^ {\, 3} & amp; & amp; \ delta_ {p} ^ {\, 3} \\ \ end {bmatrix} \ begin {bmatrix} \ delta_ {i} ^ {\, 1} & & &; \ delta_ {j} ^ {\, 1} & amp; & amp; \ delta_ {k} ^ {\, 1} \\ \ delta_ {i} ^ {\, 2} & amp; & amp; \ delta_ {j} ^ {\, 2} & amp; & amp; \ delta_ {k} ^ {\, 2} \\ \ delta_ {i} ^ {\, 3} & amp; & amp; \ delta_ {j} ^ {\, 3} & amp; & amp; \ delta_ {k} ^ {\, 3} \\ \ end {bmatrix} = \ begin {bmatrix} \ delta_ {1} ^ {\, m} & &; \ delta_ {2} ^ {\, m} & amp; & amp; \ delta_ {3} ^ {\, m} \\ \ delta_ {1} ^ {\, n} & amp; & amp; \ delta_ {2} ^ {\, n} & amp; & amp; \ delta_ {3} ^ {\, n} \\ \ delta_ {1} ^ {\, p} & amp; & amp; \ delta_ {2} ^ {\, p} & amp; & amp; \ delta_ {3} ^ {\, p} \\ \ end {bmatrix} \ begin {bmatrix} \ delta_ {i} ^ {\, 1} & & &; \ delta_ {j} ^ {\, 1} & amp; & amp; \ delta_ {k} ^ {\, 1} \\ \ delta_ {i} ^ {\, 2} & amp; & amp; \ delta_ {j} ^ {\, 2} & amp; & amp; \ delta_ {k} ^ {\, 2} \\ \ delta_ {i} ^ {\, 3} & amp; & amp; \ delta_ {j} ^ {\, 3} & amp; & amp; \ delta_ {k} ^ {\, 3} \\ \ end {bmatrix} =$

$= \ begin {bmatrix} \ delta _ {\ alpha} ^ {\, m} \, \ delta_ {i} ^ {\ alpha} & amp; & amp; \ delta _ {\ alpha} ^ {\, m} \, \ delta_ {j} ^ {\ alpha} & amp; & amp; \ delta _ {\ alpha} ^ {\, m} \, \ delta_ {k} ^ {\ alpha} \\ \ delta _ {\ alpha} ^ {\, n} \, \ delta_ {i} ^ {\ alpha} & amp; & amp; \ delta _ {\ alpha} ^ {\, n} \, \ delta_ {j} ^ {\ alpha} & amp; & amp; \ delta _ {\ alpha} ^ {\, n} \, \ delta_ {k} ^ {\ alpha} \\ \ delta _ {\ alpha} ^ {\, p} \, \ delta_ {i} ^ {\ alpha} & amp; & amp; \ delta _ {\ alpha} ^ {\, p} \, \ delta_ {j} ^ {\ alpha} & amp; & amp; \ delta _ {\ alpha} ^ {\, p} \, \ delta_ {k} ^ {\ alpha} \ end {bmatrix} \ quad (9)$

Here we had to use, firstly, the symmetry of the Kronecker delta $\ delta_ {j} ^ {\, i} = \ delta_ {i} ^ {\, j}$ and, secondly, when performing the matrix product in the generated amounts for each element of the result, only the terms in which the upper and lower silent indices are repeated are repeated, again because of the Kronecker delta properties. Let's transform (9) taking into account one more property of the Kronecker delta

$\ delta_ {k} ^ {\, i} \, \ delta_ {j} ^ {\, k} = \ delta_ {j} ^ {\, i}$

and get the final expression of the components of the generalized Kronecker delta

$E ^ {\, mnp} \, E _ {\, ijk} = \ delta_ {ijk} ^ {\, mnp} = \ left | \ begin {matrix} \ delta_ {i} ^ {\, m} & amp; & amp; \ delta_ {j} ^ {\, m} & amp; & amp; \ delta_ {k} ^ {\, m} \\ \ delta_ {i} ^ {\, n} & amp; & amp; \ delta_ {j} ^ {\, n} & amp; & amp; \ delta_ {k} ^ {\, n} \\ \ delta_ {i} ^ {\, p} & amp; & amp; \ delta_ {j} ^ {\, p} & amp; & amp; \ delta_ {k} ^ {\, p} \ end {matrix} \ right | \ quad (10)$

All 729 components are described by one compact formula (10). This is very good and extremely useful for practical purposes. For example, it is now very easy to write the product of contravariant and covariant Levi-Civita tensors

$\ varepsilon ^ {\, mnp} \, \ varepsilon _ {\, ijk} = \ delta_ {ijk} ^ {\, mnp} = \ left | \ begin {matrix} \ delta_ {i} ^ {\, m} & amp; & amp; \ delta_ {j} ^ {\, m} & amp; & amp; \ delta_ {k} ^ {\, m} \\ \ delta_ {i} ^ {\, n} & amp; & amp; \ delta_ {j} ^ {\, n} & amp; & amp; \ delta_ {k} ^ {\, n} \\ \ delta_ {i} ^ {\, p} & amp; & amp; \ delta_ {j} ^ {\, p} & amp; & amp; \ delta_ {k} ^ {\, p} \ end {matrix} \ right | \ quad (11)$

4. Convolution of the product of Levi-Civita tensors for a different number of pairs of indices

Using (10) we can easily and unconditionally calculate the reconciliation (11). We turn (11) one pair of indices

$\ varepsilon ^ {\, mnk} \, \ varepsilon _ {\, ijk} = \ delta_ {ij} ^ {\, mn} = \ left | \ begin {matrix} \ delta_ {i} ^ {\, m} & amp; & amp; \ delta_ {j} ^ {\, m} & amp; & amp; \ delta_ {k} ^ {\, m} \\ \ delta_ {i} ^ {\, n} & amp; & amp; \ delta_ {j} ^ {\, n} & amp; & amp; \ delta_ {k} ^ {\, n} \\ \ delta_ {i} ^ {\, k} & amp; & amp; \ delta_ {j} ^ {\, k} & amp; & amp; \ delta_ {k} ^ {\, k} \ end {matrix} \ right | \ quad (12)$

We decompose (12) on the last line

$\ varepsilon ^ {\, mnk} \, \ varepsilon _ {\, ijk} = \ delta_ {i} ^ {\, k} \, \ left | \ begin {matrix} \ delta_ {j} ^ {\, m} & amp; & amp; \ delta_ {k} ^ {\, m} \\ \ delta_ {j} ^ {\, n} & amp; & amp; \ delta_ {k} ^ {\, n} \ end {matrix} \ right | - \ delta_ {j} ^ {\, k} \ left | \ begin {matrix} \ delta_ {i} ^ {\, m} & amp; & amp; \ delta_ {k} ^ {\, m} \\ \ delta_ {i} ^ {\, n} & amp; & amp; \ delta_ {k} ^ {\, n} \ end {matrix} \ right | + \ delta_ {k} ^ {\, k} \, \ left | \ begin {matrix} \ delta_ {i} ^ {\, m} & amp; & amp; \ delta_ {j} ^ {\, m} \\ \ delta_ {i} ^ {\, n} & amp; & amp; \ delta_ {j} ^ {\, n} \ end {matrix} \ right | =$

$= \ left | \ begin {matrix} \ delta_ {j} ^ {\, m} & amp; & amp; \ delta_ {i} ^ {\, k} \, \ delta_ {k} ^ {\, m} \\ \ delta_ {j} ^ {\, n} & &; \ delta_ {i} ^ {\, k} \, \ delta_ {k} ^ {\, n} \ end {matrix} \ right | - \ left | \ begin {matrix} \ delta_ {i} ^ {\, m} & amp; & amp; \ delta_ {j} ^ {\, k} \, \ delta_ {k} ^ {\, m} \\ \ delta_ {i} ^ {\, n} & &; \ delta_ {j} ^ {\, k} \, \ delta_ {k} ^ {\, n} \ end {matrix} \ right | + \ delta_ {k} ^ {\, k} \, \ left | \ begin {matrix} \ delta_ {i} ^ {\, m} & amp; & amp; \ delta_ {j} ^ {\, m} \\ \ delta_ {i} ^ {\, n} & amp; & amp; \ delta_ {j} ^ {\, n} \ end {matrix} \ right | \ quad (13)$

In (13) in the first two terms, we multiplied the second column by the factor before the determinant, in accordance with the rules for calculating them. Convert further

$\ varepsilon ^ {\, mnk} \, \ varepsilon _ {\, ijk} = \ left | \ begin {matrix} \ delta_ {j} ^ {\, m} & amp; & amp; \ delta_ {i} ^ {\, m} \\ \ delta_ {j} ^ {\, n} & amp; & amp; \ delta_ {i} ^ {\, n} \ end {matrix} \ right | - \ left | \ begin {matrix} \ delta_ {i} ^ {\, m} & amp; & amp; \ delta_ {j} ^ {\, m} \\ \ delta_ {i} ^ {\, n} & amp; & amp; \ delta_ {j} ^ {\, n} \ end {matrix} \ right | + \ delta_ {k} ^ {\, k} \, \ left | \ begin {matrix} \ delta_ {i} ^ {\, m} & amp; & amp; \ delta_ {j} ^ {\, m} \\ \ delta_ {i} ^ {\, n} & amp; & amp; \ delta_ {j} ^ {\, n} \ end {matrix} \ right | =$

$= - \ left | \ begin {matrix} \ delta_ {i} ^ {\, m} & amp; & amp; \ delta_ {j} ^ {\, m} \\ \ delta_ {i} ^ {\, n} & amp; & amp; \ delta_ {j} ^ {\, n} \ end {matrix} \ right | - \ left | \ begin {matrix} \ delta_ {i} ^ {\, m} & amp; & amp; \ delta_ {j} ^ {\, m} \\ \ delta_ {i} ^ {\, n} & amp; & amp; \ delta_ {j} ^ {\, n} \ end {matrix} \ right | + \ delta_ {k} ^ {\, k} \, \ left | \ begin {matrix} \ delta_ {i} ^ {\, m} & amp; & amp; \ delta_ {j} ^ {\, m} \\ \ delta_ {i} ^ {\, n} & amp; & amp; \ delta_ {j} ^ {\, n} \ end {matrix} \ right | =$

$= - 2 \, \ left | \ begin {matrix} \ delta_ {i} ^ {\, m} & amp; & amp; \ delta_ {j} ^ {\, m} \\ \ delta_ {i} ^ {\, n} & amp; & amp; \ delta_ {j} ^ {\, n} \ end {matrix} \ right | + 3 \, \ left | \ begin {matrix} \ delta_ {i} ^ {\, m} & amp; & amp; \ delta_ {j} ^ {\, m} \\ \ delta_ {i} ^ {\, n} & amp; & amp; \ delta_ {j} ^ {\, n} \ end {matrix} \ right | = \ left | \ begin {matrix} \ delta_ {i} ^ {\, m} & amp; & amp; \ delta_ {j} ^ {\, m} \\ \ delta_ {i} ^ {\, n} & amp; & amp; \ delta_ {j} ^ {\, n} \ end {matrix} \ right |$

Here we took (-1) for the bracket in the first term, rearranging the columns, and also used the fact that $\ delta_ {k} ^ {\, k} = \ delta_ {1} ^ {\, 1} + \ delta_ {2} ^ {\, 2} + \ delta_ {3} ^ {\, 3} = 3$ - convolution of the Kronecker delta, that is, its trace.

That is, the final we get

$\ varepsilon ^ {mnk} \, \ varepsilon_ {ijk} = \ left | \ begin {matrix} \ delta_ {i} ^ {\, m} & amp; & amp; \ delta_ {j} ^ {\, m} \\ \ delta_ {i} ^ {\, n} & amp; & amp; \ delta_ {j} ^ {\, n} \ end {matrix} \ right | = \ delta_ {i} ^ {\, m} \, \ delta_ {j} ^ {\, n} - \ delta_ {j} ^ {\, m} \, \ delta_ {i} ^ {\, n} \ quad (14)$

Now we turn the product of Levi-Civita tensors in two pairs of indices. To do this, we fold (14)

$\ varepsilon ^ {mjk} \, \ varepsilon_ {ijk} = \ delta_ {i} ^ {\, m} \, \ delta_ {j} ^ {\, j} - \ delta_ {j} ^ {\, m} \, \ delta_ {i} ^ {\, j} = 3 \, \ delta_ {i} ^ {\, m} - \ delta_ {i} ^ {\, m} = 2 \, \ delta_ {i} ^ {\, m} \ quad (15)$

Well, finally, perform a convolution of three pairs of indices

$\ varepsilon ^ {ijk} \, \ varepsilon_ {ijk} = 2 \, \ delta_ {i} ^ {\, i} = 2 \ cdot 3 = 6 \ quad (16)$

Expressions (14) - (16) clearly show that the “crocodile” from the product of Levi-Civita tensors is still quite tame, and it is not necessary to memorize these formulas, it is enough to remember (11), which is not so difficult. Using (11) one can very effectively simplify tensor expressions.

5. Back to the rotation tensor

Let us return to the expression for the rotation tensor, but we will not introduce an intermediate contravariant antisymmetric tensor, but just like that, with the Levi-Civita tensors, we write it down

$B_k ^ {\, m} = u ^ {\, m} \, g_ {jk} \, u ^ {\, j} - \ cos \ varphi \, \ varepsilon ^ {\, mqi} \, u_ {q } \, \ varepsilon _ {\, ijk} \, u ^ j + \ sin \ varphi \, g ^ {mi} \, U_ {ik} \ right \ quad (17)$

Using (11) we will execute as it is necessary convolution

$\ varepsilon ^ {\, mqi} \ varepsilon _ {\, ijk} = \ left | \ begin {matrix} \ delta_ {i} ^ {\, m} & amp; & amp; \ delta_ {j} ^ {\, m} & amp; & amp; \ delta_ {k} ^ {\, m} \\ \ delta_ {i} ^ {\, q} & amp; & amp; \ delta_ {j} ^ {\, q} & amp; & amp; \ delta_ {k} ^ {\, q} \\ \ delta_ {i} ^ {\, i} & amp; & amp; \ delta_ {j} ^ {\, i} & amp; & amp; \ delta_ {k} ^ {\, i} \ end {matrix} \ right | = \ delta_ {i} ^ {\, i} \, \ left | \ begin {matrix} \ delta_ {j} ^ {\, m} & amp; & amp; \ delta_ {k} ^ {\, m} \\ \ delta_ {j} ^ {\, q} & amp; & amp; \ delta_ {k} ^ {\, q} \ end {matrix} \ right | - \ delta_ {j} ^ {\, i} \, \ left | \ begin {matrix} \ delta_ {i} ^ {\, m} & amp; & amp; \ delta_ {k} ^ {\, m} \\ \ delta_ {i} ^ {\, q} & amp; & amp; \ delta_ {k} ^ {\, q} \ end {matrix} \ right | + \ delta_ {k} ^ {\, i} \, \ left | \ begin {matrix} \ delta_ {i} ^ {\, m} & amp; & amp; \ delta_ {j} ^ {\, m} \\ \ delta_ {i} ^ {\, q} & amp; & amp; \ delta_ {j} ^ {\, q} \ end {matrix} \ right | =$

$= 3 \, \ left | \ begin {matrix} \ delta_ {j} ^ {\, m} & amp; & amp; \ delta_ {k} ^ {\, m} \\ \ delta_ {j} ^ {\, q} & amp; & amp; \ delta_ {k} ^ {\, q} \ end {matrix} \ right | - \, \ left | \ begin {matrix} \ delta_ {j} ^ {\, m} & amp; & amp; \ delta_ {k} ^ {\, m} \\ \ delta_ {j} ^ {\, q} & amp; & amp; \ delta_ {k} ^ {\, q} \ end {matrix} \ right | + \ left | \ begin {matrix} \ delta_ {k} ^ {\, m} & amp; & amp; \ delta_ {j} ^ {\, m} \\ \ delta_ {k} ^ {\, q} & amp; & amp; \ delta_ {j} ^ {\, q} \ end {matrix} \ right | = \ left | \ begin {matrix} \ delta_ {j} ^ {\, m} & amp; & amp; \ delta_ {k} ^ {\, m} \\ \ delta_ {j} ^ {\, q} & amp; & amp; \ delta_ {k} ^ {\, q} \ end {matrix} \ right | =$

$= \ delta_ {j} ^ {\, m} \, \ delta_ {k} ^ {\, q} - \ delta_ {k} ^ {\, m} \, \ delta_ {j} ^ {\, q} \ quad (18)$

Using (18) will allow us to rewrite (17) in a more digestible form.

$B_k ^ {\, m} = u ^ {\, m} \, g_ {jk} \, u ^ {\, j} - \ cos \ varphi \, \ left (\ delta_ {j} ^ {\, m } \, \ delta_ {k} ^ {\, q} - \ delta_ {k} ^ {\, m} \, \ delta_ {j} ^ {\, q} \ right) \, u_ {q} \, u ^ j + \ sin \ varphi \, g ^ {mi} \, U_ {ik} \ right \ quad (19)$

which will allow us to work with the rotation tensor more effectively than as long as I have

Conclusion

Again, a small insight into the theory. Such a somewhat ragged rhythm of the cycle of tensors is explained by the fact that the articles are the results of the author’s own “digging” into the topic. It was impossible not to tell about the generalized Kronecker delta - it is still useful to us more than once, in those cases where it will be necessary to transform expressions containing a double vector product and having a Levi-Civita tensor product in combination with convolution.

In view of the foregoing, the seventh article of the cycle will be subject to some adjustment.

Thank you all for your attention and see you soon!

PS: I would like to say a special thank you to V. G. Rechkalov, based on the book “Vector and Tensor Algebra for Future Physicists and Technicians” , a website was created for the people, the link to which I already cited twice.

To be continued…

Source: https://habr.com/ru/post/262497/

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