Journal Details
First Published
01 Jan 2016
Publication timeframe
2 times per year
access type Open Access

About one method of calculation in the arbitrary curvilinear basis of the Laplace operator and curl from the vector function

Published Online: 14 May 2021
Page range: -
Received: 04 Dec 2020
Accepted: 11 Apr 2021
Journal Details
First Published
01 Jan 2016
Publication timeframe
2 times per year

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 ei is found the expressions of vector projections (ΔA)i and (rotA)i, where A is a counter variant vector. Examples of orthonormal bases are considered and general expressions for (ΔA)i and (rotA)i for the bases are also given. As a demonstration of the working capacity of the common formulas obtained, detailed calculations of (ΔA)i and (rotA)i as an example are made in cases of spherical and cylindrical coordinate systems.



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 [3], nor in monograph [4].

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 (ΔA)i and (rotA)i, where A is a contravariant differentiable vector, on an arbitrary unit basis ei.

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.

Basic concepts of the method of calculation in orthonormal basis, as an alternative approach
Laplace operator

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 [3] dl2=dxidxi=gikdxidxkd{l^2} = d{x_i}d{x^i} = {g_{ik}}d{x^i}d{x^k} where gik – is the covariant metric tensor, defined via transformations xi = xi(xk) in the usual way [3, 4]such as gik=xnxixnxk{g_{ik}} = {{\partial {x_n}} \over {\partial {x^i}}}{{\partial {x_n}} \over {\partial {x^k}}} where (here and below) summation is implied by repeated indexes.

The relation of the contravariant metric tensor to the covariant (2) is determined by the inverse ratio gisgsk=δik{g_{is}}{g^{sk}} = \delta_i^k where δik\delta_i^k – is the Kronecker symbol representing the unit matrix.

We present Eq. (1) as follows: dl2=igii(dxi)2+ikgikdxidxk.d{l^2} = \sum\limits_i {g_{ii}}{(d{x^i})^2} + \sum\limits_{i \ne k} {g_{ik}}d{x^i}d{x^k}.

By introducing intermediate differentials x˜i=gii(dxi)2\partial {\tilde x^i} = \sqrt {{g_{ii}}{{(d{x^i})}^2}} where i – is fixed, we get from Eq. (4) the following: dl2=i(x˜i)2+ikgikgiigkkx˜ix˜kd{l^2} = \sum\limits_i {(\partial {\tilde x^i})^2} + \sum\limits_{i \ne k} {{{g_{ik}}} \over {\sqrt {{g_{ii}}} \sqrt {{g_{kk}}}}}\partial {\tilde x^i}\partial {\tilde x^k}

Clearly, the square of an element of length Eq. (6) is written in the orthogonal basis. By omitting the summation sign, we have dl2=(x˜i)d{l^2} = (\partial {\tilde x^i})

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 gik=gik=δik=δik=δik{g_{ik}} = {g^{ik}} = {\delta_{ik}} = \delta_i^k = {\delta^{ik}}

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. [4] and also papers [5, 6]); their determination is in the form of the standard formula (see Refs [3, 4]) Γikl=12(gikxl+gilxk-gklxi){\Gamma_{ikl}} = {1 \over 2}({{\partial {g_{ik}}} \over {\partial {x^l}}} + {{\partial {g_{il}}} \over {\partial {x^k}}} - {{\partial {g_{kl}}} \over {\partial {x^i}}}) not applicable, since according to Eq. (8) all. Γikl=Γkli=0{\Gamma_{ikl}} = \Gamma_{kl}^i = 0 .

This simply means that we cannot use Eq. (9), but we must proceed from a direct determination, which is written as (see Ref. [3]) Γikl=xnxi2xnxkxl{\Gamma_{ikl}} = {{\partial {x_n}} \over {\partial {x^i}}}{{{\partial^2}{x_n}} \over {\partial {x^k}\partial {x^l}}}

Since we know development xi = xi(xk), then according to Eq. (5), calculation of the Christoffel symbols will not be difficult. In fact, we have for them: Γikl=Γkli=xnx˜i2xnx˜kx˜l{\Gamma_{ikl}} = \Gamma_{kl}^i = {{\partial {x_n}} \over {\partial {{\tilde x}^i}}}{{{\partial^2}{x_n}} \over {\partial {{\tilde x}^k}\partial {{\tilde x}^l}}}

It should be noted that calculation of the Christoffel symbols by Eq. (11) automatically results in their correct dimension 1L{1 \over L} , where L – is the unit of length. Specific examples of how Eq. (11) ‘works’ can be found, for example, in the articles [5, 6].

Let there be some arbitrary continuous contravariant vector function A, and we are interested in the result of the action of the Laplace operator on it, that is, vector BA.

Then its projections on the orts ei will be as follows: Bi=gik(ΔA)k{B_i} = {g_{ik}}{(\Delta {\bf{A}})^k}

To calculate this expression, we first find the first and second covariant derivatives of vector A = (Ai).

For the first covariant derivative, we have A,ki=Bki=Aixk+ΓkliAlA_{,k}^i = B_k^i = {{\partial {A^i}} \over {\partial {x^k}}} + \Gamma_{kl}^i{A^l} where BkiB_k^i – is the index-mixed tensor of second rank.

The second covariant derivative will be as follows: Bk,li=Bkixl+ΓlsiBks-ΓklsBsiB_{k,l}^i = {{\partial B_k^i} \over {\partial {x^l}}} + \Gamma_{ls}^iB_k^s - \Gamma_{kl}^sB_s^i which is easy to find, if we represent tensor BkiB_k^i as a product of contravariant M and covariant N vectors, that is, as Bki=MiNkB_k^i = {M^i}{N_k} , and if we use the known rules for their differentiation M,ki=Mixk+ΓkliMlM_{,k}^i = {{\partial {M^i}} \over {\partial {x^k}}} + \Gamma_{kl}^i{M^l} , Ni,k=Nixk-ΓiksNs{N_{i,k}} = {{\partial {N_i}} \over {\partial {x^k}}} - \Gamma_{ik}^s{N_s} .

By applying determination Eqs (13) and (14) for tensor BkiB_k^i , we immediately receive Bk,li=Bkixl+ΓlsiBks-ΓklsBsi=xl(Aixk+ΓksiAs)+Γlni(Anxk+ΓksnAs)-Γkln(Aixn+ΓnsiAs)=2Aixkxl+ΓksiAsxl+ΓlsiAsxk-ΓklnAixn+(Γksixl+ΓlniΓksn-ΓnsiΓkln)As=2Aixkxl+ΓksiAsxl+(ΓlsiAs)xk-ΓklnAixn+RklsiAs\matrix{{B_{k,l}^i =} \hfill & {{{\partial B_k^i} \over {\partial {x^l}}} + \Gamma_{ls}^iB_k^s - \Gamma_{kl}^sB_s^i = {\partial \over {\partial {x^l}}}({{\partial {A^i}} \over {\partial {x^k}}} + \Gamma_{ks}^i{A^s}) + \Gamma_{ln}^i({{\partial {A^n}} \over {\partial {x^k}}} + \Gamma_{ks}^n{A^s}) - \Gamma_{kl}^n({{\partial {A^i}} \over {\partial {x^n}}} + \Gamma_{ns}^i{A^s}) =} \hfill \cr {} \hfill & {{{{\partial^2}{A^i}} \over {\partial {x^k}\partial {x^l}}} + \Gamma_{ks}^i{{\partial {A^s}} \over {\partial {x^l}}} + \Gamma_{ls}^i{{\partial {A^s}} \over {\partial {x^k}}} - \Gamma_{kl}^n{{\partial {A^i}} \over {\partial {x^n}}} + ({{\partial \Gamma_{ks}^i} \over {\partial {x^l}}} + \Gamma_{ln}^i\Gamma_{ks}^n - \Gamma_{ns}^i\Gamma_{kl}^n){A^s} =} \hfill \cr {} \hfill & {{{{\partial^2}{A^i}} \over {\partial {x^k}\partial {x^l}}} + \Gamma_{ks}^i{{\partial {A^s}} \over {\partial {x^l}}} + {{\partial (\Gamma_{ls}^i{A^s})} \over {\partial {x^k}}} - \Gamma_{kl}^n{{\partial {A^i}} \over {\partial {x^n}}} + R_{kls}^i{A^s}} \hfill \cr} where the Riemann curvature tensor is defined in a conventional manner [4], as Rklsi=Γksixl-Γlsixk+ΓlniΓksn-ΓnsiΓklnR_{kls}^i = {{\partial \Gamma_{ks}^i} \over {\partial {x^l}}} - {{\partial \Gamma_{ls}^i} \over {\partial {x^k}}} + \Gamma_{ln}^i\Gamma_{ks}^n - \Gamma_{ns}^i\Gamma_{kl}^n

By collapsing expression (15) by indexes k,l, taking into account the explicit expression (16), we come to the following result: (ΔA)i=2Ai(xk)2+2ΓksiAsxk-ΓkknAixn+(Γksixk+ΓkniΓksn-ΓnsiΓkkn)As{(\Delta {\bf{A}})^i} = {{{\partial^2}{A^i}} \over {{{\left({\partial {x^k}} \right)}^2}}} + 2\Gamma_{ks}^i{{\partial {A^s}} \over {\partial {x^k}}} - \Gamma_{kk}^n{{\partial {A^i}} \over {\partial {x^n}}} + \left({{{\partial \Gamma_{ks}^i} \over {\partial {x^k}}} + \Gamma_{kn}^i\Gamma_{ks}^n - \Gamma_{ns}^i\Gamma_{kk}^n} \right){A^s}

Highlighting here the Laplace operator in an explicit form, that is, as Δ=2(xi)2-Γkknxn\Delta = {{{\partial^2}} \over {{{\left({\partial {x^i}} \right)}^2}}} - \Gamma_{kk}^n{\partial \over {\partial {x^n}}} we finally get the following expression for vector Bi as Bi=(ΔA)i=ΔAi+2ΓksiAsxk+(Γksixk+ΓkniΓksn-ΓnsiΓkkn)As{B^i} = (\Delta {\bf{A}}{)^i} = \Delta {A^i} + 2\Gamma_{ks}^i{{\partial {A^s}} \over {\partial {x^k}}} + \left({{{\partial \Gamma_{ks}^i} \over {\partial {x^k}}} + \Gamma_{kn}^i\Gamma_{ks}^n - \Gamma_{ns}^i\Gamma_{kk}^n} \right){A^s}

Thus, the covariant projection according to Eq. (12) will be Bi=gik(ΔA)k=gik[ΔAk+2ΓlskAsxl+(Γlskxl+ΓlnkΓlsn-ΓnskΓlln)As]{B_i} = {g_{ik}}{(\Delta {\bf{A}})^k} = {g_{ik}}\left[ {\Delta {A^k} + 2\Gamma_{ls}^k{{\partial {A^s}} \over {\partial {x^l}}} + \left({{{\partial \Gamma_{ls}^k} \over {\partial {x^l}}} + \Gamma_{ln}^k\Gamma_{ls}^n - \Gamma_{ns}^k\Gamma_{ll}^n} \right){A^s}} \right]

Let us now consider another problem. Let there be a covariant vector

B = Δ A

where A = (Ai) and we are interested in the result Ci=(ΔA)i{C_i} = (\Delta {\bf{A}}{)_i}

As above, in the case of a contravariant vector, we find the first two covariant derivatives of vector A = (Ai). For the first derivative, we have Ai,k=Bik=Aixk-ΓiksAs{A_{i,k}} = {B_{ik}} = {{\partial {A_i}} \over {\partial {x^k}}} - \Gamma_{ik}^s{A_s} where Bik – is the second rank covariant tensor.

For the second covariant derivative (representing this tensor in factorised form, as Bik = MiNk), we get Bik,l=Bikxl-ΓilsBsk-ΓklsBis{B_{ik,l}} = {{\partial {B_{ik}}} \over {\partial {x^l}}} - \Gamma_{il}^s{B_{sk}} - \Gamma_{kl}^s{B_{is}}

By applying determination Eqs (22) and (23) for tensor Bik, we immediately find here that Bik,l=Ai,k,l=2Aixkxl-ΓiksAsxl-ΓilsAsxk-ΓklsAixs+(ΓilnΓnks+ΓklnΓins-Γiksxl)As{B_{ik,l}} = {A_{i,k,l}} = {{{\partial^2}{A_i}} \over {\partial {x^k}\partial {x^l}}} - \Gamma_{ik}^s{{\partial {A_s}} \over {\partial {x^l}}} - \Gamma_{il}^s{{\partial {A_s}} \over {\partial {x^k}}} - \Gamma_{kl}^s{{\partial {A_i}} \over {\partial {x^s}}} + \left({\Gamma_{il}^n\Gamma_{nk}^s + \Gamma_{kl}^n\Gamma_{in}^s - {{\partial \Gamma_{ik}^s} \over {\partial {x^l}}}} \right){A_s}

By collapsing this expression by the indices k,l, we get Ci=2Ai(xk)2-2ΓiksAsxk-ΓkksAixs+(ΓiknΓnks+ΓkknΓins-Γiksxk)As==ΔAi-2ΓiksAsxk+(ΓiknΓnks+ΓkknΓins-Γiksxk)As\matrix{{{C_i} = {{{\partial^2}{A_i}} \over {{{\left({\partial {x^k}} \right)}^2}}} - 2\Gamma_{ik}^s{{\partial {A_s}} \over {\partial {x^k}}} - \Gamma_{kk}^s{{\partial {A_i}} \over {\partial {x^s}}} + \left({\Gamma_{ik}^n\Gamma_{nk}^s + \Gamma_{kk}^n\Gamma_{in}^s - {{\partial \Gamma_{ik}^s} \over {\partial {x^k}}}} \right){A_s} =} \hfill \cr {= \Delta {A_i} - 2\Gamma_{ik}^s{{\partial {A_s}} \over {\partial {x^k}}} + \left({\Gamma_{ik}^n\Gamma_{nk}^s + \Gamma_{kk}^n\Gamma_{in}^s - {{\partial \Gamma_{ik}^s} \over {\partial {x^k}}}} \right){A_s}} \hfill \cr}

Comparing Eq. (25) with Eq. (20), we can see that, in general, vectors Bi and Ci, as it should be, do not coincide. To bring them in compliance, let us now consider the expression Ai = gisAs and find a second-rank covariant derivative of it.

Due to Ricci's lemma, according to which the covariant derivative of the metric tensor is zero, that is, we have Ai,k,l=gis[2Asxkxl+ΓknsAnxl+ΓlnsAnxk-ΓklnAsxn+Ap(Γkpsxl+ΓlnsΓkpn-ΓklnΓnps)]==gis2Asxkxl+ΓiksAsxl+ΓilsAsxk-gisΓklnAsxn+As(Γiksxl-Γksnginxl+ΓilnΓksn-ΓklnΓins)\matrix{{{A_{i,k,l}} = {g_{is}}\left[ {{{{\partial^2}{A^s}} \over {\partial {x^k}\partial {x^l}}} + \Gamma_{kn}^s{{\partial {A^n}} \over {\partial {x^l}}} + \Gamma_{ln}^s{{\partial {A^n}} \over {\partial {x^k}}} - \Gamma_{kl}^n{{\partial {A^s}} \over {\partial {x^n}}} + {A^p}\left({{{\partial \Gamma_{kp}^s} \over {\partial {x^l}}} + \Gamma_{ln}^s\Gamma_{kp}^n - \Gamma_{kl}^n\Gamma_{np}^s} \right)} \right] =} \hfill \cr {= {g_{is}}{{{\partial^2}{A^s}} \over {\partial {x^k}\partial {x^l}}} + {\Gamma_{iks}}{{\partial {A^s}} \over {\partial {x^l}}} + {\Gamma_{ils}}{{\partial {A^s}} \over {\partial {x^k}}} - {g_{is}}\Gamma_{kl}^n{{\partial {A^s}} \over {\partial {x^n}}} + {A^s}\left({{{\partial {\Gamma_{iks}}} \over {\partial {x^l}}} - \Gamma_{ks}^n{{\partial {g_{in}}} \over {\partial {x^l}}} + {\Gamma_{iln}}\Gamma_{ks}^n - \Gamma_{kl}^n{\Gamma_{ins}}} \right)} \hfill \cr}

But since ginxl=Γinl+Γnil{{\partial {g_{in}}} \over {\partial {x^l}}} = {\Gamma_{inl}} + {\Gamma_{nil}} , it follows that Ai,k,l=gis2Asxkxl+ΓiksAsxl+ΓilsAsxk-gisΓklnAsxn+As(Γiksxl-Γksn(Γinl+Γnil)+ΓilnΓksn-ΓklnΓins){A_{i,k,l}} = {g_{is}}{{{\partial^2}{A^s}} \over {\partial {x^k}\partial {x^l}}} + {\Gamma_{iks}}{{\partial {A^s}} \over {\partial {x^l}}} + {\Gamma_{ils}}{{\partial {A^s}} \over {\partial {x^k}}} - {g_{is}}\Gamma_{kl}^n{{\partial {A^s}} \over {\partial {x^n}}} + {A^s}\left({{{\partial {\Gamma_{iks}}} \over {\partial {x^l}}} - \Gamma_{ks}^n\left({{\Gamma_{inl}} + {\Gamma_{nil}}} \right) + {\Gamma_{iln}}\Gamma_{ks}^n - \Gamma_{kl}^n{\Gamma_{ins}}} \right)

By collapsing expression (26) by the indices k,l, we get Ai,k,k=gisΔAs+2ΓiksAsxk+As(Γiksxk-ΓksnΓnik-ΓkknΓins){A_{i,k,k}} = {g_{is}}\Delta {A^s} + 2{\Gamma_{iks}}{{\partial {A^s}} \over {\partial {x^k}}} + {A^s}\left({{{\partial {\Gamma_{iks}}} \over {\partial {x^k}}} - \Gamma_{ks}^n{\Gamma_{nik}} - \Gamma_{kk}^n{\Gamma_{ins}}} \right) where again, an abbreviated notation is introduced for the usual Laplace operator according to Eq. (18). Therefore, Ci=gisΔAs+2ΓiksAsxk+As(Γiksxk-ΓksnΓnik-ΓkknΓins){C_i} = {g_{is}}\Delta {A^s} + 2{\Gamma_{iks}}{{\partial {A^s}} \over {\partial {x^k}}} + {A^s}\left({{{\partial {\Gamma_{iks}}} \over {\partial {x^k}}} - \Gamma_{ks}^n{\Gamma_{nik}} - \Gamma_{kk}^n{\Gamma_{ins}}} \right)

It is clear that in order to obtain the correct formula, we need Eqs (20) and (25) to bring in compliance with each other, but not Eqs (20) and (28).

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 (ΔA)i=gisΔ˜As+Δ˜Ai2+ΓiskAsxk-ΓiksAsxk-Γkts2(gipApxs+Aixs)++As2(Γiskxk-ΓkknΓisn-ΓiknΓnsk)+As2(ΓiknΓnks+ΓkknΓins-Γiksxk)\matrix{{{{(\Delta {\bf{A}})}_i} = {{{g_{is}}\tilde \Delta {A^s} + \tilde \Delta {A_i}} \over 2} + {\Gamma_{isk}}{{\partial {A^s}} \over {\partial {x^k}}} - \Gamma_{ik}^s{{\partial {A_s}} \over {\partial {x^k}}} - {{\Gamma_{kt}^s} \over 2}\left({{g_{ip}}{{\partial {A^p}} \over {\partial {x^s}}} + {{\partial {A_i}} \over {\partial {x^s}}}} \right) +} \hfill \cr {+ {{{A^s}} \over 2}\left({{{\partial {\Gamma_{isk}}} \over {\partial {x^k}}} - \Gamma_{kk}^n{\Gamma_{isn}} - \Gamma_{ik}^n{\Gamma_{nsk}}} \right) + {{{A_s}} \over 2}\left({\Gamma_{ik}^n\Gamma_{nk}^s + \Gamma_{kk}^n\Gamma_{in}^s - {{\partial \Gamma_{ik}^s} \over {\partial {x^k}}}} \right)} \hfill \cr}

In the case of an orthonormal basis, where, as we know, the formulas are satisfied Γkli=Γikl,  Ai=Ai,  gik=δik\Gamma_{kl}^i = {\Gamma_{ikl}}, {A_i} = {A^i}, {g_{ik}} = {\delta_{ik}}

It immediately follows that (ΔA)i=ΔAi+(Γisk-Γsik)Asxk+As2[xk(Γisk-Γsik)+Γnkk(Γsin-Γin)+Γnkk(Γsnk-Γnsk)]{(\Delta {\bf{A}})_i} = \Delta {A_i} + \left({{\Gamma_{isk}} - {\Gamma_{sik}}} \right){{\partial {A_s}} \over {\partial {x^k}}} + {{{A_s}} \over 2}\left[ {{\partial \over {\partial {x^k}}}\left({{\Gamma_{isk}} - {\Gamma_{sik}}} \right) + {\Gamma_{nkk}}\left({{\Gamma_{sin}} - {\Gamma_{in}}} \right) + {\Gamma_{nkk}}\left({{\Gamma_{snk}} - {\Gamma_{nsk}}} \right)} \right]

Eq. (31) is the final formula that allows us to calculate the projections of vector ΔA in any orthonormal coordinate system.

Vector function curl

We now calculate the projections of the curl of vector A on the covariant orts ei. As Ci = (rotA)i then its obvious generalisation to an arbitrary curved basis can be written as (rotA)i=geiklgln(Akxn+ΓsnkAs){(rot{\bf{A}})_i} = \sqrt g {e_{ikl}}{g^{\ln}}\left({{{\partial {A^k}} \over {\partial {x^n}}} + \Gamma_{sn}^k{A^s}} \right)

On the other hand, the same expression can be represented as gik(rotA)k=gikeklsg(Asxl-ΓslnAn){g_{ik}}{(rot{\bf{A}})^k} = {g_{ik}}{{{e^{kls}}} \over {\sqrt g}}\left({{{\partial {A_s}} \over {\partial {x^l}}} - \Gamma_{sl}^n{A_n}} \right)

It is clear that they should both lead to the same result. Therefore, taking their half-sum, we find (RotA)i=12[geiklgln(Akxn+ΓsnkAs)+gikeklsg(Asxl-ΓslnAn)]{(Rot{\bf{A}})_i} = {1 \over 2}\left[ {\sqrt g {e_{ikl}}{g^{\ln}}\left({{{\partial {A^k}} \over {\partial {x^n}}} + \Gamma_{sn}^k{A^s}} \right) + {g_{ik}}{{{e^{kls}}} \over {\sqrt g}}\left({{{\partial {A_s}} \over {\partial {x^l}}} - \Gamma_{sl}^n{A_n}} \right)} \right]

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 (RotA)i=eikl2gln[(g+1g)Akxn+(gΓsnk+gkpggspxn)As]{(Rot{\bf{A}})_i} = {{{e_{ikl}}} \over 2}{g^{ln}}\left[ {\left({\sqrt g + {1 \over {\sqrt g}}} \right){{\partial {A^k}} \over {\partial {x^n}}} + \left({\sqrt g \Gamma_{sn}^k + {{{g^{kp}}} \over {\sqrt g}}{{\partial {g_{sp}}} \over {\partial {x^n}}}} \right){A^s}} \right]

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) gik=gik=δik,  Γkli=Γikl,  Ai=Ai,  g=l{g_{ik}} = {g^{ik}} = {\delta_{ik}}, \Gamma_{kl}^i = {\Gamma_{ikl}}, {A_i} = {A^i}, g = l we have (RotA)i=eikl(Akxl+12ΓkslAs){(Rot{\bf{A}})_i} = {e_{ikl}}\left({{{\partial {A_k}} \over {\partial {x^l}}} + {1 \over 2}{\Gamma_{ksl}}{A_s}} \right)

Special cases of general Eqs (31) and (36)

Let us now consider the application of Eqs (31) and (36) in specific special cases. Let us start with Eq. (31).

Spherical coordinates

Since the length element in this case is dl2=dr2+r2dθ2+r2sin2θdφ2d{l^2} = d{r^2} + {r^2}d{\theta^2} + {r^2}\mathop {\sin}\nolimits^2 \theta d{\varphi^2} and there are intermediate differentials x1=dr,x2=rdθ,x3=rsinθdφ\partial {x^1} = dr,\partial {x^2} = rd\theta, \partial {x^3} = r\sin \theta d\varphi then, leaving only the non-zero Christoffel characters with the help of the table given, let us say, in Ref. [5], according to which Γrθθ=Γθθr=-1r,Γrφφ=Γφφr=-1r,Γθφφ=Γφφθ=-ctgθr,Γθrθ=Γθθr=Γrθθ=Γθrθ=1r,Γφrφ=Γφφr=Γφrφ=Γrφφ=1r,Γφθφ=Γφφθ=Γφθφ=Γθφφ=ctgθr\matrix{{{\Gamma_{r\theta \theta}} = \Gamma_{\theta \theta}^r = - {1 \over r},{\Gamma_{r\varphi \varphi}} = \Gamma_{\varphi \varphi}^r = - {1 \over r},{\Gamma_{\theta \varphi \varphi}} = \Gamma_{\varphi \varphi}^\theta = - {{{\rm ctg}\theta} \over r},{\Gamma_{\theta r\theta}} = {\Gamma_{\theta \theta r}} = \Gamma_{r\theta}^\theta = \Gamma_{\theta r}^\theta = {1 \over r},} \hfill \cr {{\Gamma_{\varphi r\varphi}} = {\Gamma_{\varphi \varphi r}} = \Gamma_{\varphi r}^\varphi = \Gamma_{r\varphi}^\varphi = {1 \over r},{\Gamma_{\varphi \theta \varphi}} = {\Gamma_{\varphi \varphi \theta}} = \Gamma_{\varphi \theta}^\varphi = \Gamma_{\theta \varphi}^\varphi = {{{\rm ctg}\theta} \over r}} \hfill \cr} in accordance with Eq. (31), for the projection on the ort er of a spherical unit basis (er, eθ, eφ), we have (ΔA)r=ΔAr+(Γrsk-Γsrk)Asxk+As2[xk(Γrsk-Γsrk)+Γnkk(Γsm-Γrsn)+Γnrk(Γsnk-Γnsk)]==ΔAr+(Γrθθ-Γθrθ)Aθxθ+(Γrφφ-Γφrφ)Aφxφ++Ar2[Γθrθ(Γrθθ-Γθrθ)+Γφrφ(Γrφφ-Γφrφ)]++Aθ2[xθ(Γrθθ-Γθrθ)+Γθφφ(Γθrθ-Γrθθ)+Γφrφ(Γθφφ-Γφθφ)]++Aφ2[xφ(Γrφφ-Γφrφ)+Γφθθ(Γφrφ-Γrφφ)+Γθrθ(Γφθθ-Γθφθ)]\matrix{{{{(\Delta {\bf{A}})}_r} = \Delta {A_r} + \left({{\Gamma_{rsk}} - {\Gamma_{srk}}} \right){{\partial {A_s}} \over {\partial {x^k}}} + {{{A_s}} \over 2}\left[ {{\partial \over {\partial {x^k}}}\left({{\Gamma_{rsk}} - {\Gamma_{srk}}} \right) + {\Gamma_{nkk}}\left({{\Gamma_{sm}} - {\Gamma_{rsn}}} \right) + {\Gamma_{nrk}}\left({{\Gamma_{snk}} - {\Gamma_{nsk}}} \right)} \right] =} \hfill \cr {= \Delta {A_r} + \left({{\Gamma_{r\theta \theta}} - {\Gamma_{\theta r\theta}}} \right){{\partial {A_\theta}} \over {\partial {x^\theta}}} + \left({{\Gamma_{r\varphi \varphi}} - {\Gamma_{\varphi r\varphi}}} \right){{\partial {A_\varphi}} \over {\partial {x^\varphi}}} +} \hfill \cr {+ {{{A_r}} \over 2}\left[ {{\Gamma_{\theta r\theta}}\left({{\Gamma_{r\theta \theta}} - {\Gamma_{\theta r\theta}}} \right) + {\Gamma_{\varphi r\varphi}}\left({{\Gamma_{r\varphi \varphi}} - {\Gamma_{\varphi r\varphi}}} \right)} \right] +} \hfill \cr {+ {{{A_\theta}} \over 2}\left[ {{\partial \over {\partial {x^\theta}}}\left({{\Gamma_{r\theta \theta}} - {\Gamma_{\theta r\theta}}} \right) + {\Gamma_{\theta \varphi \varphi}}\left({{\Gamma_{\theta r\theta}} - {\Gamma_{r\theta \theta}}} \right) + {\Gamma_{\varphi r\varphi}}\left({{\Gamma_{\theta \varphi \varphi}} - {\Gamma_{\varphi \theta \varphi}}} \right)} \right] +} \hfill \cr {+ {{{A_\varphi}} \over 2}\left[ {{\partial \over {\partial {x^\varphi}}}\left({{\Gamma_{r\varphi \varphi}} - {\Gamma_{\varphi r\varphi}}} \right) + {\Gamma_{\varphi \theta \theta}}\left({{\Gamma_{\varphi r\varphi}} - {\Gamma_{r\varphi \varphi}}} \right) + {\Gamma_{\theta r\theta}}\left({{\Gamma_{\varphi \theta \theta}} - {\Gamma_{\theta \varphi \theta}}} \right)} \right]} \hfill \cr}

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 (ΔA)r=ΔAr-2r2Aθθ-2r2sinθAφφ-2Arr2-2ctgθAθr2==ΔAr-2r2(Ar+1sinθ(sinθAθ)θ+1sinθAφφ)\matrix{{{{(\Delta {\bf{A}})}_r} = \Delta {A_r} - {2 \over {{r^2}}}{{\partial {A_\theta}} \over {\partial \theta}} - {2 \over {{r^2}\sin \theta}}{{\partial {A_\varphi}} \over {\partial \varphi}} - {{2{A_r}} \over {{r^2}}} - {{2{\rm ctg}\theta {A_\theta}} \over {{r^2}}} =} \hfill \cr {= \Delta {A_r} - {2 \over {{r^2}}}\left({{A_r} + {1 \over {\sin \theta}}{{\partial \left({\sin \theta {A_\theta}} \right)} \over {\partial \theta}} + {1 \over {\sin \theta}}{{\partial {A_\varphi}} \over {\partial \varphi}}} \right)} \hfill \cr}

For the projection on the ort eθ, we get (ΔA)θ=ΔAθ+(Γθsk-Γsθk)Asxk+As2[xk(Γθsk-Γsθk)+Γnkk(Γsθn-Γθsn)+Γnθk(Γsnk-Γnsk)]==ΔAθ+(Γθrθ-Γrθθ)Arxθ+(Γθφφ-Γφθφ)Aφxφ++Ar2[xθ(Γθrθ-Γrθθ)+Γθφφ(Γrθθ-Γθrθ)+Γφθφ(Γrφφ-Γφrφ)]++Aθ2[Γrθθ(Γθrθ-Γrθθ)+Γφθθ(Γθφθ-Γφθθ)+Γφθp(Γθφφ-Γφθφ)]++Aφ2[xθ(Γθpθ-Γφθθ)+xφ(Γθφφ-Γφθφ)+Γθφφ(Γφθθ-Γθφθ)+Γφθθ(Γφθφ-Γθφφ)+Γθθφ(Γφθφ-Γθφφ)]\matrix{{{{(\Delta {\bf{A}})}_\theta} = \Delta {A_\theta} + \left({{\Gamma_{\theta sk}} - {\Gamma_{s\theta k}}} \right){{\partial {A_s}} \over {\partial {x^k}}} + {{{A_s}} \over 2}\left[ {{\partial \over {\partial {x^k}}}\left({{\Gamma_{\theta sk}} - {\Gamma_{s\theta k}}} \right) + {\Gamma_{nkk}}\left({{\Gamma_{s\theta n}} - {\Gamma_{\theta sn}}} \right) + {\Gamma_{n\theta k}}\left({{\Gamma_{snk}} - {\Gamma_{nsk}}} \right)} \right] =} \hfill \cr {= \Delta {A_\theta} + \left({{\Gamma_{\theta r\theta}} - {\Gamma_{r\theta \theta}}} \right){{\partial {A_r}} \over {\partial {x^\theta}}} + \left({{\Gamma_{\theta \varphi \varphi}} - {\Gamma_{\varphi \theta \varphi}}} \right){{\partial {A_\varphi}} \over {\partial {x^\varphi}}} +} \hfill \cr {+ {{{A_r}} \over 2}\left[ {{\partial \over {\partial {x^\theta}}}\left({{\Gamma_{\theta r\theta}} - {\Gamma_{r\theta \theta}}} \right) + {\Gamma_{\theta \varphi \varphi}}\left({{\Gamma_{r\theta \theta}} - {\Gamma_{\theta r\theta}}} \right) + {\Gamma_{\varphi \theta \varphi}}\left({{\Gamma_{r\varphi \varphi}} - {\Gamma_{\varphi r\varphi}}} \right)} \right] +} \hfill \cr {+ {{{A_\theta}} \over 2}\left[ {{\Gamma_{r\theta \theta}}\left({{\Gamma_{\theta r\theta}} - {\Gamma_{r\theta \theta}}} \right) + {\Gamma_{\varphi \theta \theta}}\left({{\Gamma_{\theta \varphi \theta}} - {\Gamma_{\varphi \theta \theta}}} \right) + {\Gamma_{\varphi \theta p}}\left({{\Gamma_{\theta \varphi \varphi}} - {\Gamma_{\varphi \theta \varphi}}} \right)} \right] +} \hfill \cr {+ {{{A_\varphi}} \over 2}\left[ {{\partial \over {\partial {x^\theta}}}\left({{\Gamma_{\theta p\theta}} - {\Gamma_{\varphi \theta \theta}}} \right) + {\partial \over {\partial {x^\varphi}}}\left({{\Gamma_{\theta \varphi \varphi}} - {\Gamma_{\varphi \theta \varphi}}} \right) + {\Gamma_{\theta \varphi \varphi}}\left({{\Gamma_{\varphi \theta \theta}} - {\Gamma_{\theta \varphi \theta}}} \right) + {\Gamma_{\varphi \theta \theta}}\left({{\Gamma_{\varphi \theta \varphi}} - {\Gamma_{\theta \varphi \varphi}}} \right) + {\Gamma_{\theta \theta \varphi}}\left({{\Gamma_{\varphi \theta \varphi}} - {\Gamma_{\theta \varphi \varphi}}} \right)} \right]} \hfill \cr}

By applying here the non-zero values of the Christoffel symbols according to Eq. (39) and the intermediate differentials from Eq. (38), we get (ΔA)θ=ΔAθ+2r2Arθ-2ctgθr2sinθAφφ-Aθr2sin2θ{(\Delta {\bf{A}})_\theta} = \Delta {A_\theta} + {2 \over {{r^2}}}{{\partial {A_r}} \over {\partial \theta}} - {{2{\rm ctg}\theta} \over {{r^2}\sin \theta}}{{\partial {A_\varphi}} \over {\partial \varphi}} - {{{A_\theta}} \over {{r^2}\mathop {\sin}\nolimits^2 \theta}}

Finally, the projection on the ort eφ gives

Using Eqs (38) and (39), we have from here (ΔA)φ=ΔAφ+2r2sinθArφ+2ctgθr2sinθAθφ-Aφr2sin2θ{(\Delta {\bf{A}})_\varphi} = \Delta {A_\varphi} + {2 \over {{r^2}\sin \theta}}{{\partial {A_r}} \over {\partial \varphi}} + {{2{\rm ctg}\theta} \over {{r^2}\sin \theta}}{{\partial {A_\theta}} \over {\partial \varphi}} - {{{A_\varphi}} \over {{r^2}\mathop {\sin}\nolimits^2 \theta}} where the usual Laplace operator in a spherical coordinate system is Δ=1r2r(r2r)+1r2sinθθ(sinθθ)+1r2sin2θ2φ2\Delta = {1 \over {{r^2}}}{\partial \over {\partial r}}\left({{r^2}{\partial \over {\partial r}}} \right) + {1 \over {{r^2}\sin \theta}}{\partial \over {\partial \theta}}\left({\sin \theta {\partial \over {\partial \theta}}} \right) + {1 \over {{r^2}\mathop {\sin}\nolimits^2 \theta}}{{{\partial^2}} \over {\partial {\varphi^2}}}

Putting Eqs (40)(42) together, we find {(ΔA)r=ΔAr-2r2(Ar+1sinθ(sinθAθ)θ+1sinθAφφ),(ΔA)θ=ΔAθ+2r2(Arθ-ctgθsinθAφφ-Aθ2sin2θ)(ΔA)φ=ΔAφ+2r2sinθ(Arφ+ctgθAθφ-Aφ2sinθ)\left\{{\matrix{{{{(\Delta {\bf{A}})}_r} = \Delta {A_r} - {2 \over {{r^2}}}\left({{A_r} + {1 \over {\sin \theta}}{{\partial \left({\sin \theta {A_\theta}} \right)} \over {\partial \theta}} + {1 \over {\sin \theta}}{{\partial {A_\varphi}} \over {\partial \varphi}}} \right),} \hfill \cr {{{(\Delta {\bf{A}})}_\theta} = \Delta {A_\theta} + {2 \over {{r^2}}}\left({{{\partial {A_r}} \over {\partial \theta}} - {{{\rm ctg}\theta} \over {\sin \theta}}{{\partial {A_\varphi}} \over {\partial \varphi}} - {{{A_\theta}} \over {2\mathop {\sin}\nolimits^2 \theta}}} \right)} \hfill \cr {{{(\Delta {\bf{A}})}_\varphi} = \Delta {A_\varphi} + {2 \over {{r^2}\sin \theta}}\left({{{\partial {A_r}} \over {\partial \varphi}} + {\rm ctg}\theta {{\partial {A_\theta}} \over {\partial \varphi}} - {{{A_\varphi}} \over {2\sin \theta}}} \right)} \hfill \cr}} \right.

Comparing projections (44) with the formulas given, for example, in monograph [1] on page 615 (see also Ref. [2]), 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.

Cylindrical coordinate system

this, there are non-zero Christoffel symbols Γrφφ=-1r,Γφrφ=Γφφr=1r{\Gamma_{r\varphi \varphi}} = - {1 \over r},{\Gamma_{\varphi r\varphi}} = {\Gamma_{\varphi \varphi r}} = {1 \over r} and intermediate differentials xr=dr,xφ=rdφ,xz=dz\partial {x^r} = dr,\partial {x^\varphi} = rd\varphi, \partial {x^z} = dz

Using the general Eq. (31) for projections of vector A on an orthonormal unit basis er, eφ, ez, we find for projection on er: (ΔA)r=ΔAr+(Γrsk-Γsrk)Asxk+As2[xk(Γrsk-Γsrk)+Γnkk(Γsm-Γrsn)+Γnrk(Γsnk-Γnsk)]==ΔAr+(Γrφφ-Γφrφ)Aφxφ+Ar2[Γφrφ(Γrφφ-Γφrφ)]+Aφ2xφ(Γrφφ-Γφrφ)\matrix{{{{(\Delta {\bf{A}})}_r} = \Delta {A_r} + \left({{\Gamma_{rsk}} - {\Gamma_{srk}}} \right){{\partial {A_s}} \over {\partial {x^k}}} + {{{A_s}} \over 2}\left[ {{\partial \over {\partial {x^k}}}\left({{\Gamma_{rsk}} - {\Gamma_{srk}}} \right) + {\Gamma_{nkk}}\left({{\Gamma_{sm}} - {\Gamma_{rsn}}} \right) + {\Gamma_{nrk}}\left({{\Gamma_{snk}} - {\Gamma_{nsk}}} \right)} \right] =} \hfill \cr {= \Delta {A_r} + \left({{\Gamma_{r\varphi \varphi}} - {\Gamma_{\varphi r\varphi}}} \right){{\partial {A_\varphi}} \over {\partial {x^\varphi}}} + {{{A_r}} \over 2}\left[ {{\Gamma_{\varphi r\varphi}}\left({{\Gamma_{r\varphi \varphi}} - {\Gamma_{\varphi r\varphi}}} \right)} \right] + {{{A_\varphi}} \over 2}{\partial \over {\partial {x^\varphi}}}\left({{\Gamma_{r\varphi \varphi}} - {\Gamma_{\varphi r\varphi}}} \right)} \hfill \cr}

By applying Eqs (45) and (46) here, we get (ΔA)r=ΔAr-2r2Aφφ-Arr2{(\Delta {\bf{A}})_r} = \Delta {A_r} - {2 \over {{r^2}}}{{\partial {A_\varphi}} \over {\partial \varphi}} - {{{A_r}} \over {{r^2}}}

For the projection on the ort eφ: (ΔA)φ=ΔAφ+(Γφsk-Γsφk)Asxk+As2[xk(Γφsk-Γsφk)+Γnkk(Γsφn-Γφsn)+Γnφk(Γsnk-Γnsk)]==ΔAφ+(Γφrφ-Γrφφ)Arxφ+Ar2xφ(Γφrφ-Γrφφ)+Aφ2Γrφφ(Γφrφ-Γrφφ)\matrix{{{{(\Delta {\bf{A}})}_\varphi} = \Delta {A_\varphi} + \left({{\Gamma_{\varphi sk}} - {\Gamma_{s\varphi k}}} \right){{\partial {A_s}} \over {\partial {x^k}}} + {{{A_s}} \over 2}\left[ {{\partial \over {\partial {x^k}}}\left({{\Gamma_{\varphi sk}} - {\Gamma_{s\varphi k}}} \right) + {\Gamma_{nkk}}\left({{\Gamma_{s\varphi n}} - {\Gamma_{\varphi sn}}} \right) + {\Gamma_{n\varphi k}}\left({{\Gamma_{snk}} - {\Gamma_{nsk}}} \right)} \right] =} \hfill \cr {= \Delta {A_\varphi} + \left({{\Gamma_{\varphi r\varphi}} - {\Gamma_{r\varphi \varphi}}} \right){{\partial {A_r}} \over {\partial {x^\varphi}}} + {{{A_r}} \over 2}{\partial \over {\partial {x^\varphi}}}\left({{\Gamma_{\varphi r\varphi}} - {\Gamma_{r\varphi \varphi}}} \right) + {{{A_\varphi}} \over 2}{\Gamma_{r\varphi \varphi}}\left({{\Gamma_{\varphi r\varphi}} - {\Gamma_{r\varphi \varphi}}} \right)} \hfill \cr}

By applying Eqs (45) and (46) here, we will finally have (ΔA)φ=ΔAφ+2r2Arφ-Aφr2{(\Delta {\bf{A}})_\varphi} = \Delta {A_\varphi} + {2 \over {{r^2}}}{{\partial {A_r}} \over {\partial \varphi}} - {{{A_\varphi}} \over {{r^2}}}

For the projection on the ort ez: (ΔA)z=ΔAz{(\Delta {\bf{A}})_z} = \Delta {A_z}

Putting Eqs (47)(49) together, we have {(ΔA)r=ΔAr-2r2Aφφ-Arr2(ΔA)φ=ΔAφ+2r2Arφ-Aφr2(ΔA)z=ΔAz.\left\{{\matrix{{{{(\Delta {\bf{A}})}_r} = \Delta {A_r} - {2 \over {{r^2}}}{{\partial {A_\varphi}} \over {\partial \varphi}} - {{{A_r}} \over {{r^2}}}} \hfill \cr {{{(\Delta {\bf{A}})}_\varphi} = \Delta {A_\varphi} + {2 \over {{r^2}}}{{\partial {A_r}} \over {\partial \varphi}} - {{{A_\varphi}} \over {{r^2}}}} \hfill \cr {{{(\Delta {\bf{A}})}_z} = \Delta {A_z}.} \hfill \cr}} \right.

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 ∂ x1 = dr, ∂x2 = rdθ, ∂ x3 = r sinθ dφ.

Then, for the projections of the curl on the unit orts er, eθ, eφ, from Eq. (36) we find as follows: {(rotA)r=Aθx3-Aφx2+12(Γθφs-Γφθs)As,(rotA)θ=Arx3-Aφx1+12(Γrφs-Γφrs)As(rotA)φ=Arx2-Aθx1+12(Γrθs-Γθrs)As.\left\{{\matrix{{{{(rot{\bf{A}})}_r} = {{\partial {A_\theta}} \over {\partial {x^3}}} - {{\partial {A_\varphi}} \over {\partial {x^2}}} + {1 \over 2}\left({{\Gamma_{\theta \varphi s}} - {\Gamma_{\varphi \theta s}}} \right){A_s},} \hfill \cr {{{(rot{\bf{A}})}_\theta} = {{\partial {A_r}} \over {\partial {x^3}}} - {{\partial {A_\varphi}} \over {\partial {x^1}}} + {1 \over 2}\left({{\Gamma_{r\varphi s}} - {\Gamma_{\varphi rs}}} \right){A_s}} \hfill \cr {{{(rot{\bf{A}})}_\varphi} = {{\partial {A_r}} \over {\partial {x^2}}} - {{\partial {A_\theta}} \over {\partial {x^1}}} + {1 \over 2}\left({{\Gamma_{r\theta s}} - {\Gamma_{\theta rs}}} \right){A_s}.} \hfill \cr}} \right.

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 {(rotA)r=1rsinθ[(sinθAφ)θ-Aθφ],(rotA)θ=1rr(rAφ)-1rsinθArφ(rotA)φ=1rr(rAθ)-1rArθ.\left\{{\matrix{{{{(rot{\bf{A}})}_r} = {1 \over {r\sin \theta}}\left[ {{{\partial \left({\sin \theta {A_\varphi}} \right)} \over {\partial \theta}} - {{\partial {A_\theta}} \over {\partial \varphi}}} \right],} \hfill \cr {{{(rot{\bf{A}})}_\theta} = {1 \over r}{\partial \over {\partial r}}\left({r{A_\varphi}} \right) - {1 \over {r\sin \theta}}{{\partial {A_r}} \over {\partial \varphi}}} \hfill \cr {{{(rot{\bf{A}})}_\varphi} = {1 \over r}{\partial \over {\partial r}}\left({r{A_\theta}} \right) - {1 \over r}{{\partial {A_r}} \over {\partial \theta}}.} \hfill \cr}} \right.

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 [1]).

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 (ΔA)i and (rotA)i in an arbitrary curvilinear basis is proposed;

General expressions for (ΔA)i and (rotA)i in an arbitrary orthonormal basis are obtained;

Special cases of calculation (ΔA)i and (rotA)i in spherical and cylindrical coordinate systems have been considered.

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