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

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] (1) dl2=dxidxi=gikdxidxk d{l^2} = d{x_i}d{x^i} = {g_{ik}}d{x^i}d{x^k} where g_ik – is the covariant metric tensor, defined via transformations x_i = x_i(x^k) in the usual way [3, 4]such as (2) gik=∂xn∂xi∂xn∂xk {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 (3) 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: (4) dl2=∑igii(dxi)2+∑i≠kgikdxidxk. 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 (5) ∂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: (6) dl2=∑i(∂x˜i)2+∑i≠kgikgiigkk∂x˜i∂x˜k d{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 (7) 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 (8) 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]) (9) Γikl=12(∂gik∂xl+∂gil∂xk-∂gkl∂xi) {\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]) (10) Γikl=∂xn∂xi∂2xn∂xk∂xl {\Gamma_{ikl}} = {{\partial {x_n}} \over {\partial {x^i}}}{{{\partial^2}{x_n}} \over {\partial {x^k}\partial {x^l}}}

Since we know development x_i = x_i(x^k), then according to Eq. (5), calculation of the Christoffel symbols will not be difficult. In fact, we have for them: (11) Γikl=Γkli=∂xn∂x˜i∂2xn∂x˜k∂x˜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 B=ΔA.

Then its projections on the orts e_i will be as follows: (12) 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 (13) A,ki=Bki=∂Ai∂xk+ΓkliAl A_{,k}^i = B_k^i = {{\partial {A^i}} \over {\partial {x^k}}} + \Gamma_{kl}^i{A^l} where Bki B_k^i – is the index-mixed tensor of second rank.

The second covariant derivative will be as follows: (14) Bk,li=∂Bki∂xl+ΓlsiBks-ΓklsBsi B_{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 Bki B_k^i as a product of contravariant M and covariant N vectors, that is, as Bki=MiNk B_k^i = {M^i}{N_k} , and if we use the known rules for their differentiation M,ki=∂Mi∂xk+ΓkliMl M_{,k}^i = {{\partial {M^i}} \over {\partial {x^k}}} + \Gamma_{kl}^i{M^l} , Ni,k=∂Ni∂xk-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 Bki B_k^i , we immediately receive (15) Bk,li=∂Bki∂xl+ΓlsiBks-ΓklsBsi=∂∂xl(∂Ai∂xk+ΓksiAs)+Γlni(∂An∂xk+ΓksnAs)-Γkln(∂Ai∂xn+ΓnsiAs)=∂2Ai∂xk∂xl+Γksi∂As∂xl+Γlsi∂As∂xk-Γkln∂Ai∂xn+(∂Γksi∂xl+ΓlniΓksn-ΓnsiΓkln)As=∂2Ai∂xk∂xl+Γksi∂As∂xl+∂(ΓlsiAs)∂xk-Γkln∂Ai∂xn+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 (16) Rklsi=∂Γksi∂xl-∂Γlsi∂xk+ΓlniΓksn-ΓnsiΓkln R_{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: (17) (ΔA)i=∂2Ai(∂xk)2+2Γksi∂As∂xk-Γkkn∂Ai∂xn+(∂Γksi∂xk+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 (18) Δ=∂2(∂xi)2-Γkkn∂∂xn \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 (19) Bi=(ΔA)i=ΔAi+2Γksi∂As∂xk+(∂Γksi∂xk+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 (20) Bi=gik(ΔA)k=gik[ΔAk+2Γlsk∂As∂xl+(∂Γlsk∂xl+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 = (A_i) and we are interested in the result (21) 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 = (A_i). For the first derivative, we have (22) Ai,k=Bik=∂Ai∂xk-ΓiksAs {A_{i,k}} = {B_{ik}} = {{\partial {A_i}} \over {\partial {x^k}}} - \Gamma_{ik}^s{A_s} where B_ik – is the second rank covariant tensor.

For the second covariant derivative (representing this tensor in factorised form, as B_ik = M_iN_k), we get (23) Bik,l=∂Bik∂xl-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 B_ik, we immediately find here that (24) Bik,l=Ai,k,l=∂2Ai∂xk∂xl-Γiks∂As∂xl-Γils∂As∂xk-Γkls∂Ai∂xs+(ΓilnΓnks+ΓklnΓins-∂Γiks∂xl)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 (25) Ci=∂2Ai(∂xk)2-2Γiks∂As∂xk-Γkks∂Ai∂xs+(ΓiknΓnks+ΓkknΓins-∂Γiks∂xk)As==ΔAi-2Γiks∂As∂xk+(ΓiknΓnks+ΓkknΓins-∂Γiks∂xk)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 B_i and C_i, as it should be, do not coincide. To bring them in compliance, let us now consider the expression A_i = g_isA^s 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[∂2As∂xk∂xl+Γkns∂An∂xl+Γlns∂An∂xk-Γkln∂As∂xn+Ap(∂Γkps∂xl+ΓlnsΓkpn-ΓklnΓnps)]==gis∂2As∂xk∂xl+Γiks∂As∂xl+Γils∂As∂xk-gisΓkln∂As∂xn+As(∂Γiks∂xl-Γksn∂gin∂xl+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 ∂gin∂xl=Γinl+Γnil {{\partial {g_{in}}} \over {\partial {x^l}}} = {\Gamma_{inl}} + {\Gamma_{nil}} , it follows that (26) Ai,k,l=gis∂2As∂xk∂xl+Γiks∂As∂xl+Γils∂As∂xk-gisΓkln∂As∂xn+As(∂Γiks∂xl-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 (27) Ai,k,k=gisΔAs+2Γiks∂As∂xk+As(∂Γiks∂xk-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, (28) Ci=gisΔAs+2Γiks∂As∂xk+As(∂Γiks∂xk-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 (29) (ΔA)i=gisΔ˜As+Δ˜Ai2+Γisk∂As∂xk-Γiks∂As∂xk-Γkts2(gip∂Ap∂xs+∂Ai∂xs)++As2(∂Γisk∂xk-ΓkknΓisn-ΓiknΓnsk)+As2(ΓiknΓnks+ΓkknΓins-∂Γiks∂xk) \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 (30) Γkli=Γikl, Ai=Ai, gik=δik \Gamma_{kl}^i = {\Gamma_{ikl}}, {A_i} = {A^i}, {g_{ik}} = {\delta_{ik}}

It immediately follows that (31) (ΔA)i=ΔAi+(Γisk-Γsik)∂As∂xk+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.

We now calculate the projections of the curl of vector A on the covariant orts e_i. As C_i = (rotA)_i then its obvious generalisation to an arbitrary curved basis can be written as (32) (rotA)i=geiklgln(∂Ak∂xn+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 (33) gik(rotA)k=gikeklsg(∂As∂xl-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 (34) (RotA)i=12[geiklgln(∂Ak∂xn+ΓsnkAs)+gikeklsg(∂As∂xl-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 (35) (RotA)i=eikl2gln[(g+1g)∂Ak∂xn+(gΓsnk+gkpg∂gsp∂xn)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 (36) (RotA)i=eikl(∂Ak∂xl+12ΓkslAs) {(Rot{\bf{A}})_i} = {e_{ikl}}\left({{{\partial {A_k}} \over {\partial {x^l}}} + {1 \over 2}{\Gamma_{ksl}}{A_s}} \right)

Since the length element in this case is (37) dl2=dr2+r2dθ2+r2sin2θdφ2 d{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 (38) ∂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 (39) 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 e_r of a spherical unit basis (e_r, e_θ, e_φ), we have (ΔA)r=ΔAr+(Γrsk-Γsrk)∂As∂xk+As2[∂∂xk(Γrsk-Γsrk)+Γnkk(Γsm-Γrsn)+Γnrk(Γsnk-Γnsk)]==ΔAr+(Γrθθ-Γθrθ)∂Aθ∂xθ+(Γrφφ-Γφrφ)∂Aφ∂xφ++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-2r2∂Aθ∂θ-2r2sinθ∂Aφ∂φ-2Arr2-2ctgθAθr2==ΔAr-2r2(Ar+1sinθ∂(sinθAθ)∂θ+1sinθ∂Aφ∂φ) \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 e_θ, we get (ΔA)θ=ΔAθ+(Γθsk-Γsθk)∂As∂xk+As2[∂∂xk(Γθsk-Γsθk)+Γnkk(Γsθn-Γθsn)+Γnθk(Γsnk-Γnsk)]==ΔAθ+(Γθrθ-Γrθθ)∂Ar∂xθ+(Γθφφ-Γφθφ)∂Aφ∂xφ++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 (40) (ΔA)θ=ΔAθ+2r2∂Ar∂θ-2ctgθr2sinθ∂Aφ∂φ-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 e_φ gives

Using Eqs (38) and (39), we have from here (41) (ΔA)φ=ΔAφ+2r2sinθ∂Ar∂φ+2ctgθr2sinθ∂Aθ∂φ-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 (42) Δ=1r2∂∂r(r2∂∂r)+1r2sinθ∂∂θ(sinθ∂∂θ)+1r2sin2θ∂2∂φ2 \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 (43) {(ΔA)r=ΔAr-2r2(Ar+1sinθ∂(sinθAθ)∂θ+1sinθ∂Aφ∂φ),(ΔA)θ=ΔAθ+2r2(∂Ar∂θ-ctgθsinθ∂Aφ∂φ-Aθ2sin2θ)(ΔA)φ=ΔAφ+2r2sinθ(∂Ar∂φ+ctgθ∂Aθ∂φ-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.

this, there are non-zero Christoffel symbols (44) 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 (45) ∂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 e_r, e_φ, e_z, we find for projection on e_r: (ΔA)r=ΔAr+(Γrsk-Γsrk)∂As∂xk+As2[∂∂xk(Γrsk-Γsrk)+Γnkk(Γsm-Γrsn)+Γnrk(Γsnk-Γnsk)]==ΔAr+(Γrφφ-Γφrφ)∂Aφ∂xφ+Ar2[Γφrφ(Γrφφ-Γφrφ)]+Aφ2∂∂xφ(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 (46) (ΔA)r=ΔAr-2r2∂Aφ∂φ-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 e_φ: (ΔA)φ=ΔAφ+(Γφsk-Γsφk)∂As∂xk+As2[∂∂xk(Γφsk-Γsφk)+Γnkk(Γsφn-Γφsn)+Γnφk(Γsnk-Γnsk)]==ΔAφ+(Γφrφ-Γrφφ)∂Ar∂xφ+Ar2∂∂xφ(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 (47) (ΔA)φ=ΔAφ+2r2∂Ar∂φ-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 e_z: (48) (ΔA)z=ΔAz {(\Delta {\bf{A}})_z} = \Delta {A_z}

Putting Eqs (47)–(49) together, we have (49) {(ΔA)r=ΔAr-2r2∂Aφ∂φ-Arr2(ΔA)φ=ΔAφ+2r2∂Ar∂φ-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 e_r, e_θ, e_φ, from Eq. (36) we find as follows: (50) {(rotA)r=∂Aθ∂x3-∂Aφ∂x2+12(Γθφs-Γφθs)As,(rotA)θ=∂Ar∂x3-∂Aφ∂x1+12(Γrφs-Γφrs)As(rotA)φ=∂Ar∂x2-∂Aθ∂x1+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 (51) {(rotA)r=1rsinθ[∂(sinθAφ)∂θ-∂Aθ∂φ],(rotA)θ=1r∂∂r(rAφ)-1rsinθ∂Ar∂φ(rotA)φ=1r∂∂r(rAθ)-1r∂Ar∂θ. \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.