A simple algorithm for calculating Christoffel symbols, a covariant projection of the result of the Laplace operator's action on the vector, vector curl and other similar operations in an arbitrary oblique base are proposed. For an arbitrary base with ortho
- curvilinear basis
- orthonormal basis
- covariant derivative
- Christoffel symbol
Our great interest in calculating the result of the action of the Laplace operator and the curl on a contravariant vector function arose mainly after reading the monographs [1, 2]. There, the results of covariant differentiation of a contravariant vector in spherical and cylindrical coordinate systems are presented in a purely formal way, with no references to the primary sources where they could be considered in detail.
This raises a quite natural question, the answer to which we have not found in the monographs mentioned above. The question is how to calculate the results of the action of the Laplace operator and curl from a vector function in arbitrary (but not only orthogonal) coordinates; for example, in parabolic or elliptic ones.
Perhaps, such a question, according to the authors of classical works [1, 2], is so simple that they did not even consider it necessary to indicate a link to the source where the answer could be found. We found no answer to this question, neither in monograph , nor in monograph .
As this problem is not given in detail in the literature, we will consider it here in this paper.
It is worth noting that the effect of the second covariant derivative on the vector function is extremely important from the point of view of its application to the problems of the theory of elasticity and hydrodynamics.
This circumstance prompted us to write this paper, where in great detail, we describe the basic concepts of calculating the projections of vectors (Δ
To show examples that prove the correctness of the common approach described below, we demonstrate the corresponding calculations in the cases of spherical and cylindrical coordinate systems with results that automatically coincide with the commonly accepted ones.
Anticipating a possible question concerning the word ‘alternative’, which we have introduced, let us immediately note that the approach that will be discussed later does not include such concepts as Lame coefficients anywhere, provided that we are considering an orthogonal coordinate system. Our approach is based on the use of particular differentials introduced in a certain way, with the help of which all mathematical calculations are carried out due to the common calculation method proposed below.
The spatial metric in its standard form is 
The relation of the contravariant metric tensor to the covariant (2) is determined by the inverse ratio
We present Eq. (1) as follows:
By introducing intermediate differentials
Clearly, the square of an element of length Eq. (6) is written in the orthogonal basis. By omitting the summation sign, we have
As a rule, in all practically solved problems, only the orthogonal coordinate system is used. Therefore, expression (7) will be the main representation of the metric that we will use.
As we can see from Eq. (7), in this representation, the metric tensor, both covariant and contravariant, is simply a Kronecker symbol, that is
This means that in such a representation, there is a determination of Christoffel symbols of the first and second kinds, which are obviously also equal to each other (see Ref.  and also papers [5, 6]); their determination is in the form of the standard formula (see Refs [3, 4])
This simply means that we cannot use Eq. (9), but we must proceed from a direct determination, which is written as (see Ref. )
Since we know development
It should be noted that calculation of the Christoffel symbols by Eq. (11) automatically results in their correct dimension
Let there be some arbitrary continuous contravariant vector function
Then its projections on the orts
To calculate this expression, we first find the first and second covariant derivatives of vector
For the first covariant derivative, we have
The second covariant derivative will be as follows:
By applying determination Eqs (13) and (14) for tensor
By collapsing expression (15) by indexes
Highlighting here the Laplace operator in an explicit form, that is, as
Thus, the covariant projection according to Eq. (12) will be
Let us now consider another problem. Let there be a covariant vector
As above, in the case of a contravariant vector, we find the first two covariant derivatives of vector
For the second covariant derivative (representing this tensor in factorised form, as
By applying determination Eqs (22) and (23) for tensor
By collapsing this expression by the indices
Comparing Eq. (25) with Eq. (20), we can see that, in general, vectors
Due to Ricci's lemma, according to which the covariant derivative of the metric tensor is zero, that is, we have
By collapsing expression (26) by the indices
Taking the arithmetic mean of Eqs (20) and (25), we obtain the desired common determination for calculating the result of the action of the Laplace operator on a vector function in an arbitrary curved coordinate system as
In the case of an orthonormal basis, where, as we know, the formulas are satisfied
It immediately follows that
Eq. (31) is the final formula that allows us to calculate the projections of vector Δ
We now calculate the projections of the curl of vector
On the other hand, the same expression can be represented as
It is clear that they should both lead to the same result. Therefore, taking their half-sum, we find
Carrying out simple transformations here, we finally come to the general invariant expression for the curl of a contravariant vector function, which is correct in any curved coordinate system as
In the case of orthogonal coordinates, when using the already familiar orthonormal basis, where the conditions are always satisfied (which is extremely convenient for various kinds of calculations)
Since the length element in this case is
By applying here the Christoffel symbols from Eq. (39), with the intermediate differentials of the considered orthonormal basis Eq. (38) applied to the denominators, we finally find
For the projection on the ort
By applying here the non-zero values of the Christoffel symbols according to Eq. (39) and the intermediate differentials from Eq. (38), we get
Finally, the projection on the ort
Using Eqs (38) and (39), we have from here
Putting Eqs (40)–(42) together, we find
Comparing projections (44) with the formulas given, for example, in monograph  on page 615 (see also Ref. ), we see their complete coincidence. At the same time, we emphasise that the authors of both monographs do not specify the source of information in which the corresponding calculations are given.
this, there are non-zero Christoffel symbols
Using the general Eq. (31) for projections of vector
For the projection on the ort
By applying Eqs (45) and (46) here, we will finally have
For the projection on the ort
Putting Eqs (47)–(49) together, we have
Thus, we made certain that the general Eq. (31) is correct, and it can always be used in calculations in any orthogonal coordinate system.
Let us now proceed for verification of Eq. (36).
Let us consider spherical coordinates, where, according to Eq. (38), the intermediate differentials are
Then, for the projections of the curl on the unit orts
By applying the differentials Eq. (38) to the denominators, according to Eq. (51), and using the values of the Christoffel symbols Eq. (39), we get
The projections of the curl of vector Eq. (52) are the well-known expressions that are given in every source that is somehow associated with curved coordinates (see, for example, the already mentioned monograph ).
In the case of a cylindrical SC, the result will also be correct.
Thus, using the general Eq. (36), we can calculate the expressions for vector curl in any orthogonal coordinate system (and not just in spherical and cylindrical SC, as is usually done) thanks to the method of transition to an orthonormal basis demonstrated above.
In conclusion, we emphasise several important factors.
An algorithm for calculating the projections of vectors (Δ
General expressions for (Δ
Special cases of calculation (Δ