The Study on the Differential and Integral Calculus in Pseudoeuclidean Space

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1 Journal of Appled Mathematcs and Physcs, 217, 5, ISSN Onlne: ISSN Prnt: The Study on the Dfferental and Integral Calculus n Pseudoeucldean Space Gennady Tarabrn Department of Appled Mathematcs, Volgograd State Techncal Unversty, Volgograd, Russa How to cte ths paper: Tarabrn, G. (217) The Study on the Dfferental and Integral Calculus n Pseudoeucldean Space. Journal of Appled Mathematcs and Physcs, 5, Receved: August 19, 217 Accepted: September 16, 217 Publshed: September 19, 217 Copyrght 217 by author and Scentfc Research Publshng Inc. Ths work s lcensed under the Creatve Commons Attrbuton Internatonal Lcense (CC BY 4.). Open Access Abstract In the vector space of real vectors, comparson was executed of the multlnear forms, covarant dervatves, the total dfferentals and dervatves n the drecton whch are calculated wth dfferent metrcs wth Eucldean metrc and wth pseudoeucldean metrc of a zero ndex. Comparson was executed of the Taylor s formulas to dfferent metrcs. What s establshed by us s that multlnear forms of dfferent metrcs have dfferent values; covarant dervatves have dentcal values; the total dfferentals and dervatves n the drecton have dfferent values. In Eucldean space, Taylor s formula wth any order of accuracy assgned n advance s equal, but n pseudoeucldean space Taylor s formula s not equal wth any order of accuracy. It s concluded that n space wth a pseudoeucldean metrc, the computng sense of the dfferental and ntegral calculus created n Eucldean space s lost and the possblty of mathematcal model operaton of real physcal processes n vector space wth pseudoeucldean metrc s called nto queston. Keywords The Vector Spaces, Taylor s Formula, Pseudoeucldean Metrc 1. Introducton Ths work s a statement n Englsh of the results publshed by us n Russan n journal artcles [1] [2] can be desgnated as the begnnng of ths work when we were forced to begn profoundly studyng the tensor calculus and ts applcatons. Snce then we stopped never addressng tensors. One of results of ths close communcaton wth tensors was wrtng of the book Appled Tensor Calculus [3]. In well-known fundamental lterature [4]-[1], propertes of pseudoeucldean space are studed and mathematcal models of physcal processes n t are under DOI: /jamp Sep. 19, Journal of Appled Mathematcs and Physcs

2 constructon. However, the queston of applcablty of dfferental and ntegral calculus of Eucldean space n pseudoeucldean space s not consdered at all, or s consdered n specal cases of studed problems, but generally remans unexplored. It, owng to the bg authorty of the called lterature authors, can generate confdence that such problem does not exst. The executed research by us gves the grounds to assert that the problem of correct applcaton of mathematcs of Eucldean space n the vector space wth pseudoeucldean metrcs exsts and that attempts to gnore ths problem can lead to the mathematcal models whch are not adequately descrbng processes and the phenomena of the materal world. 2. Comparatve Defntons of Eucldean and Pseudoeucldean Spaces Ths part of artcle contans elements of algebra and math analyss whch wth suffcent completeness are explaned n textbook [3]. Requrement of ncluson of elementary detals of ths materal n artcle s dctated by need to reasonably show the common propertes of Eucldean and pseudoeucldean spaces, to show ntal dstncton of these spaces, to clearly recognze the stage of creaton of the lnearly vector space when ths dstncton s entered, and as t nfluences further creaton of space, to study how behave algebrac forms and dfferental calculus at the same tme. Even the nsgnfcant ambguty from all ths can generate doubt n relablty of results and therefore nothng from wrtten below t can be excluded as the partal fact, for reducton of volume of artcle. We wll defne the vector space L of dmenson n + 1 of real vectors x, y, and ts bass tradtonally how we may read about t n the textbooks for students. For vsual demonstraton we wll determne the vectors as the rankorder sets of real numbers, representng them ether matrxes columns, or matrxes lnes. Vectors of affne bass we wll desgnate a, a1,, an. Let s agree that the Greek ndexes accept values from to n, and Latn ndexes from 1 to n. Therefore vector decomposton x on bass a wth use of the rule of totng on twce repeatng ndex t s possble to wrte down doubly: x = a = a + a. ξ ξ ξ As bass vectors a we wll accept matrxes lnes a = 1, a = 1,, a = 1. [ ] [ ] [ ] 1 In such bass elements ξ of any vector x L wll be components of decomposton of ths vector on bass a,.e. wll be x = a ξ. The concept of a pont of the vector space and communcaton of ponts wth vectors of ths vector space s establshed axomatcally: to each couple of ponts of the vector space, the gven n a partcular order, one s put n complance and only one vector of ths space. Let s construct a radus vector of the vector space. For ths purpose we wll n DOI: /jamp Journal of Appled Mathematcs and Physcs

3 take a zero vector (at t all elements are equal to zero) and we wll call t an orgn pont of coordnates that we wll agree to wrte down n a look O(,,,). Let M the current pont of the vector space. Accordng to an axom of ponts of the vector space, each couple of ponts OM the vector of ths space unambguously answers. Let t wll be a ξ vector,.e. OM = a ξ. We wll call a set of numbers ξ, ξ,, ξ affne coordnates of a pont M that we wll agree to wrte down M ( ξ, ξ,, ξ ). We wll call a vector OM = a ξ a poston vector of a pont M, we wll desgnate r a poston vector and we wll wrte r = a ξ. Let s remnd that a s affne bass. The poston vector of the current pont of the vector space s a varable vector. Therefore t s vector functon of scalar arguments ξ, ξ,, ξ. Let s take a set of functons ξ = ξ ( x, x,, x ), where =,1,, n, ndependent from each other varables x, x,, x. Let s beleve that these func- tons one-to-one establsh connecton between affne coordnates ξ and varables x. When replacng by ad of these functons of varables ξ by varables x the poston vector becomes functon of varables x,.e. (, 1,, n r = r x x x ). We wll call varables x, x,, x the curvlnear coordnates of the current pont of space L, and a set of functons 1 (,,, n ξ = ξ x x x ), where =,1,, n, we wll call a transformaton of coordnates. Further everywhere n the vector space L we wll use curvlnear coordnates X : x, x,, x whch have coordnate lnes x (lnes on whch 1 n x = const,, x = const ) are parallel straght lnes wth the drectng vector a, and all other coordnate lnes x are, generally speakng, curves. In such coordnates the poston vector s functon of a look r = a x + f ( x, x,, x ) and 1 2 n 1 2 n ξ = x, ξ = ξ ( x, x,, x ), = 1, 2,, n. In each pont M L we wll enter the bass e called by local bass on a formula r, f e. = e = a e = (1) x x It s necessary attenton that formulas (1) defne vectors of local bass e n form of decomposton on affne bass a. Consderng that f s the composte functon wth the ntermedate arguments ξ and noncontguous varables x, we wll wrte down these decompostons: j f ξ f ξ ξ e = = a, e = = a. (2) j ξ x ξ x x Let s assume that curvlnear coordnates X are locally the lnear coordnates,.e. such that n any pont of space L the local bass e can be used as affne bass of rather small vcnty of ths pont. Both n Eucldean and n pseudoeucldean spaces the vector norm s formulated equally: the vector norm s equal to a square root from a scalar square of a vector: x = xx. However scalar squares of vectors n Eucldean and n pseudoeucldean spaces are dfferent wth each other as scalar products of vec- DOI: /jamp Journal of Appled Mathematcs and Physcs

4 tors are defned n dfferent ways. Therefore vector norms metrcs of spaces are dfferent. Dfferently njected scalar multplcaton of vectors s a boundary at whch there s a branchng of the vector space on Eucldean and on pseudoeucldean. Let s defne a scalar product of vectors of the vector space wth Eucldean metrcs. We wll enter the defnton of vectors multplcaton,.e. we wll formulate the law under whch to each couple of vectors x, y L ths law sets the partcular value of real number whch we wll desgnate xy. At the same tme operaton of a scalar multplcaton of vectors has to satsfy to axoms of commutaton, assocaton, dstrbuton and a postve sgn of a scalar square of vectors. In Eucldean space t s axomatcally clamed that the scalar square of a vector s postve defnte: xx > when x >, xx = when x =. The scalar multplcaton of vectors becomes the gven f to defne to set the law of a scalar multplcaton of vectors of affne bass. For the vector space L wth affne bass a and wth Eucldean metrc we wll determne scalar multplcaton by the followng rule: the scalar product of any couple of vectors of affne bass a s equal to the sum of products of ther correspondng components. Ths rule s expressed by formulas: aa = δ aa = 1, aa =, aa = δ, (3) j j where δ unt matrx (Kronecker delta). Havng multpled vectors x = aξ, y = a η wth use of formulas (3) and wth applcaton of axoms of commutaton, assocaton and dstrbuton, we wll receve a formula of a scalar multplcaton n Eucldean space of the vectors set by coordnates n affne bass a, j xy = aa ξ η = ξ η + δjξη. (4) On ths formula we receve a scalar square of a vector ( ξ ) ( ξ ) ( ξ ) n 2 x = xx = By ths formula we conclude that the rule of a scalar multplcaton of vectors of affne bass (3) s guarantee of realzaton of an axom of a postve determnaton of a scalar square of a vector n Eucldean space. Follows from ths formula also that n Eucldean space the norm of the vector x = a ξ preset by coordnates n affne bass a s defned by a formula ( ξ ) ( ξ ) ( ξ ) x = xx = Vectors e of all set determned by formulas (1) are lnearly ndependent and therefore they form bass at any choce of curvlnear coordnates (locally the lnear coordnates). Vectors e, apparently from (2), are defned n affne bass. Therefore ther scalar products at each other and on themselves can be calculated on formulas (4). The set of these scalar products forms the square matrx g of order n + 1 called by the covarant Gram matrx representng a DOI: /jamp Journal of Appled Mathematcs and Physcs

5 covarant tensor of the second rank. ξ ξ g = ee g = 1, g =, gj = δ. (5) j x x j Let s make a scalar multplcaton of vectors x = eξ + e ξ, y = eη + e jη (the vectors defned so agreed to call contravarant) and we use for ths purpose a covarant Gram matrx g at multplcaton. As a result we wll receve a formula of a scalar multplcaton of contravarant vectors n Eucldean space j xy = = +. (6) gξ η ξ η g j ξη Let s defne other local basc system. Vectors of ths basc system agreed to number the top ndexes e and to call vectors e manual vectors to vectors e. We wll accept that scalar product of vectors of basc system e on vectors of basc system e have to satsfy equalty ee = δ, (7) where δ unt matrx (Kronecker delta). Ths equaton agreed to call the mutualty equaton of basc systems e, e, and basc systems e, e agreed to call the mutual basc systems. The vectors y = e η preset by decomposton on bass e agreed to call covarant. Let s make scalar multplcaton of a contravarant vector x = eξ + e ξ on a j covarant y = e η + e η j and wrte down scalar product wth use of the mutualty Equaton (7). In result we wll receve n Eucldean space a formula of a scalar multplcaton of the vectors reset by components n the mutual basc systems e and e xy = ξ η = ξ η + ξη. (8) Scalar products of bass vectors e at each other and on themselves form a contravarant Gram matrx g = ee whch represents a contravarant tensor of the second rank. Usng ths matrx at a scalar multplcaton of vectors j x = e ξ + e ξ, y = e η + e η j, we wll receve a formula of a scalar multplcaton of covarant vectors n Eucldean space j xy = ξη + g ξη. (9) Formulas (1) gve us e = a, formulas (3) gve us aa = 1, and from the equaton (7) we receve ee =. These three equaltes gve us 1 j a e e. (1) = = Let s pay attenton (t s extremely mportant for a comprehenson of the further text) that vectors e of the mutual basc system do not depend on the ac- cepted rule of a scalar multplcaton of vectors n Eucldean space. It can seem not so as the mutualty Equaton (7) represents a scalar multplcaton of vectors n Eucldean space. Let s prove valdty of the made statement. In any pont of the vector space there s the nfnte set of local bases. We choose from ths nfnte set (we do not calculate, we do not use the rule of a scalar multplcaton, we DOI: /jamp Journal of Appled Mathematcs and Physcs

6 choose) such local bass whch satsfes to the mutualty Equaton (7). The chosen bass n the nfnte set cannot absent. Therefore the statement s proved. Now we wll construct the vector space wth a pseudoeucldean metrc. At the same tme lnear space L, ts affne bass a, curvlnear coordnates X : x, x,, x and the mutual basc systems e, e we wll keep former by what they were accepted n Eucldean space. The bass for such decson s that all called elements of the vector space are taken wthout use of a scalar multplcaton of vectors n Eucldean space and therefore the scalar multplcaton of vectors n Eucldean space studed above wll not be bound wth the rule of a scalar multplcaton of vectors whch we wll construct below n pseudoeucldean space. Let s defne a scalar multplcaton of vectors of affne bass a of the vector space L wth a pseudoeucldean metrc the followng rule: aa = 1, aa =, aa =δ. (11) j j The vector space wth such scalar multplcaton of bass vectors s called pseudoeucldean space of a zero ndex. Havng multpled vectors x = a ξ, y = a η wth use of formulas (11) and wth applcaton of axoms of a commutaton, assocaton and dstrbuton, we wll receve a formula of a scalar multplcaton n pseudoeucldean space of the vectors preset by coordnates n affne bass a, j j j xy= aaξ η + aaξη + aaξ η + aa ξη = ξ η + δ ξη. j j j On ths formula we receve a scalar square of a vector n pseudoeucldean space ( ξ ) ( ξ ) ( ξ ) n 2 x = xx = Ths formula allows us to conclude that the rule of a scalar multplcaton of vectors n pseudoeucldean space (11) does not lead to a postve determnaton of a scalar square of a vector. From ths formula we conclude that n pseudoeucldean space the scalar square of a nonzero vector can be postve, negatve and equal to zero. In pseudoeucldean space of zero ndex the formula ( ξ ) ( ξ ) ( ξ ) x = xx = defnes the norm of the vector x = a ξ preset by decomposton n affne bass a. Follows from ths formula that n the vector space of real vectors wth a pseudoeucldean metrc there s a set of nonzero vectors for whch the norm s not defned. From formulas (11) we have aa = 1. Consderng t and usng equaltes (1) we conclude that n pseudoeucldean space of a zero ndex ee = ee = ee =. 1 In pseudoeucldean space of a zero ndex the scalar products of other bass vectors at each other and on themselves reman the same by what they were receved n Eucldean space DOI: /jamp Journal of Appled Mathematcs and Physcs

7 e e ee e e ee j j =, j = gj; =, = g ; j j = =, = δ,, j = 1, 2,, n. e e ee ee Therefore to receve formulas of a scalar multplcaton n pseudoeucldean space of a zero ndex of the vectors preset by coordnates n local basc systems t s enough the summands ξη, ξη, ξη n formulas ((6), (8) and (9)) to take wth a mnus-sgn. Let s agree further n the presence of a double sgn ± we wll take a plus-sgn n a Eucldean space, and we wll take a mnus-sgn n pseudoeucldean space. Wth use of ths agreement of a formula of a scalar multplcaton of the vectors x = eξ = e ξ, y = eη = e η preset by coordnates n local basc systems n spaces wth Eucldean and pseudoeucldean of a zero ndex metrcs can be wrtten down unformly j j xy =± ξ η + g ξη =± ξη + g ξη =± ξη + ξη. (12) 3. Multlnear Form j j Lemma 1. In each pont of the vector space of real vectors the multlnear form of any order changes the numercal value when replacng Eucldean metrc n ths space to a pseudoeucldean metrc of a zero ndex. Proof. In arbtrary pont M of the vector space L of real vectors we wll take tensors of frst rank vectors v, ξ, η and tensor of second rank w and consder the lnear vξ and blnear wξ η forms. On the formula (12) we derve vξ =± vξ + v ξ. (13) Exstence here summand wth dfferent sgns for dfferent metrcs (plus for Eucldean and mnus for pseudoeucldean) s ndcaton of valdty of the lemma 1 relatve to lnear form. If at a tensor of the second rank w mentally to reject one of ndexes, then, accordng to the quotent rule of tensor calculus, we wll derve the covarant tensor of the frst rank a vector. Let s desgnate ths vector w. On a formula (12) we wll make convoluton transform a vector w wth a vector ξ. Let s wrte down result of convoluton and we wll return the rejected ndex w ξ =± w ξ + w ξ, w ξ =± w ξ + w ξ = v. We derve vector v. Let s make compresson of ths vector wth vector We wll derve j v =± v + v j. η η η η. If to substtute here v =± w ξ + w ξ, then we wll receve the developed look of a blnear form j j wξ η =± w ξη + w ξ η + w ξ η + w ξη. (14) ( ) j j Exstence here summands wth dfferent sgns for dfferent metrcs (plus for Eucldean and mnus for pseudoeucldean) s ndcaton of valdty of the DOI: /jamp Journal of Appled Mathematcs and Physcs

8 lemma 1 relatve to blnear form. Ths proof can seem to someone not qute convncng as s under constructon on the bass of the quotent rule whch proof out of ths text. Let s consder other proof. The covarant tensor of the second rank w by defnton s equal to an external product of two covarant tensors of the frst rank p, q. The external product of two vectors p and q represents the set of elements of a square matrx of an order n + 1 receved by multplcaton of each component p to each component q,.e. w = pq. The blnear form wξ η s a polynomal. Therefore t can be derved as the product of a polynomal pξ to a polynomal qη,.e. The lnear forms on formula (12) w ξη = p ξq η. (15) j p ξ =± p ξ + pξ, q η =± qη + qη. Havng multpled these lnear forms these polynomals, we wll have ( ) p ξ q η =± p ξ qη + pξ qη + p ξ qη + pξ qη. j j j j Havng substtuted (15) here, we wll derve a formula whch n accuracy repeats a formula (14). It s easy to see that the proofs executed for the lnear (13) and blnear (14) forms can be contnued wth use of the same concepts and methods for multlnear forms of any hgher order. # 4. Dervatves, Dfferentals and Taylor s Formula Dervatves, dfferentals and Taylor s formula whch were defned n the vector space wth Eucldean metrc wll be the objects of our researches. We wll observe change of numercal values of the consdered objects upon transton to space wth a pseudoeucldean metrcs. Chrstoffel symbols of the 2 nd sort Γ γ are components of decomposton of the second partal dervatves of a poston vector 2 r x x n local bass e,.e. γ 2 r x x =Γ γ e γ. From ths t follows that Chrstoffel symbols of the 2 nd sort are not bound to a type of a metrcs of the vector space. They depend only on the choce of a type of a curvlnear coordnates X : x, x,, x. Let F( X ) s real functon of ponts X ( x, x,, x ) L. Let s accept that t s contnuous and has the contnuous partal dervatves on all varables to the necessary order nclusve. Its covarant dervatves we wll agree to desgnate an nferor ndex after an astersk. Covarant dervatves of the ncreasng orders are defned by the followng formulas: j DOI: /jamp Journal of Appled Mathematcs and Physcs

9 F F F * γ * δ δ F* =, F* = Γ F* γ, F* γ = ΓγF* δ ΓγF* δ, γ x x x Partal dervatves F x, 2 F x 3 γ x, F x x x, and Chrstoffel symbols of the 2 nd sort Γ γ do not depend on a type of a metrc. Therefore numercal values of covarant dervatves of any order n any pont of the vector space reman nvarable at change of Eucldean metrc on pseudoeucldean. Let C( c, c,, c ) the fxed pont of the vector space L and ( d, 1 d 1,, n d n X c + x c + x c + x ) the current pont of ths space n rather small vcnty of pont C (n the vcnty n whch the local bass e n pont C can be consdered as affne bass of ths vcnty). Then t s possble to accept that the ncrement of a poston vector r of pont C upon transton to pont X s approxmately equal to poston vector dfferental dr,.e. ( C) r r( X) r( C) dr = dx = e dx. x From ths t follows that the set of dfferentals dx represents a contravarant vector. Lemma 2. In each pont of the vector space the total dfferentals of any order and dervatves n the drecton of any order of scalar functon change the numercal values when replacng Eucldean metrc n ths space to a pseudoeucldean metrc of a zero ndex. Proof. As covarant dervatves F *, F * are covarant tensors of the frst and second ranks and dx s a contravarant vector, the total dfferentals of the frst and second orders df = F d x, d F = F dx dx 2 * * are the lnear and square forms and, accordng to the lemma 1, for them the lemma 2, that ther numercal values change at change of metrcs, s the exact. The norm of a vector dx s defned by a formula 1 ( ) ( ) ( ) dl = ± dx + dx + + dx n. Let s remnd that we take a plus-sgn for Eucldean metrc, and we take a mnus-sgn for pseudoeucldean. We wll wrte down dervatves n a pont C towards a pont X : 2 d F d x d F d x d x = F* = F 2 *,. dl dl dl dl dl Exstence of dfferent sgns n a formula dl says to us about what dervatves n the drecton n spaces wth dfferent metrcs have dfferent numercal values. Follows from a formula dl also that n the drectons satsfyng to nequalty dervatves are not defned. 1 n ( x ) ( x ) ( x ) d + d + + d DOI: /jamp Journal of Appled Mathematcs and Physcs

10 Here proofs for dfferentals and dervatves frst and second orders were made. It s not dffcult to see that these proofs can be repeated for any hgher order. # Theorem 1. In the vector space wth Eucldean metrc Taylor s formula s equalty wth beforehand gven accuracy. In the same space wth pseudoeucldean metrc of a zero ndex Taylor s formula s not equalty wth any order of accuracy. Proof. Taylor s formula 1 2 F( X) F( C) = df( C) + d F( C) (16) 2 defnes the dfference of values of scalar functon F of real varables x, x,, x n rather close located ponts C, X of the vector space L wth Eucldean metrc wth second order of accuracy. The left-hand member of ths equalty s a dfference of values of functon F( X ) n two mentally the fxed ponts XC, of the vector space L. Operaton of change of a metrcs does not change space L, does not change the provson of the chosen ponts XC, n ths space and does not change functon F( X ) of ponts X L n any way. Therefore the left-hand member of Taylor s formula remans nvarable at change of a metrc of the vector space L from Eucldean metrc on pseudoeucldean. The rght-hand member of Taylor s formula changes the value at change of a metrcs as represents the lnear combnaton of dfferentals whch values, accordng to the lemma 2, change upon transton from Eucldean metrc to pseudoeucldean metrc of a zero ndex. Comparson of behavor of the left-hand and rght-hand members of Taylor s formula at change of a metrcs leads to the concluson that n pseudoeucldean space of a zero ndex Taylor s formula s not equalty, as was to be shown. Here the proof s executed for the second order of accuracy. It s clear, that t can be made for somehow hgh order of accuracy. # Theorem 2. The operatons of dfferental and ntegral calculatons developed for Eucldean space n pseudoeucldean space do not make computng sense. Proof. Follows from the theorem 1 that n pseudoeucldean space the dfference of values of functon cannot be calculated by means of the devce (16) of the dfferental calculus created for Eucldean space. It follows from ths that n pseudoeucldean space the theory of dfference schemes and, n general, all methods of fnte dfferences created for Eucldean space cannot be used. The ntegral n the vector space wth Eucldean metrc of any fnte-dmensonal of a measure of ntegraton doman s a lmt of the ntegral sum,.e. sum of dfferentals of antdervatve of ntegrand. Dfferentals, accordng to the lemma 2, change the numercal value at change of a metrcs. Therefore values of ntegrals n pseudoeucldean space wll dffer from ther values n Eucldean space. It follows from ths that the ntegral calculus created for Eucldean space s not sutable for numercal methods n pseudoeucldean space. # DOI: /jamp Journal of Appled Mathematcs and Physcs

11 5. Conclusons It s establshed that values of multlnear forms, dervatves of a scalar functon and ts dfferentals n space wth pseudoeucldean metrc dffer from ther values n the same space wth Eucldean metrc. Taylor s formula whch Eucldean space s the equalty expressng an ncrement of scalar functon through dfferentals of ths functon n pseudoeucldean space equalty s not. The executed researches lead to the concluson that the dfferental and ntegral calculus developed for space wth Eucldean metrc n space wth pseudoeucldean metrc are not sutable. As dfferental and ntegral calculus of real functons of real varables s a consttuent of appled mathematcs, t s possble to draw a concluson on loss of a possblty of adequate mathematcal model operaton of actual natural phenomena and processes of human actvty by methods of appled mathematcs n pseudoeucldean space. Results of work can be consdered n theoretcal and expermental studes where conclusons of the specal theory of relatvty are used, n partcular, at creaton of mathematcal models of the phenomena of gravtaton, cosmology and physcs of elementary partcles and propagaton of electromagnetc waves. References [1] Tarabrn, G.T. (216) Taylor s Formula n Pseudoeucldean Space. Strotelnaya Mechanka Raschet Sooruzheny, 1, (In Russan) [2] Tarabrn, G.T. (216) About Dfferental Calculus n Pseudoeucldean Space. Strotelnaya Mechanka Raschet Sooruzheny, 6, (In Russan) [3] Tarabrn, G. (214) Appled Tensor Calculus. Palmarum Academc Publshng, Saarbrucken. (In Russan) [4] Fok, V.A. (1955) Theory of Space, Tme and Gravtaton. Gostehteorzdat, Moscow. (In Russan) [5] Landau, L.D. and Lfshc, E.M. (1988) Theoretcal Physcs. Vol. 2, Feld Theory, Nauka, Moscow. (In Russan) [6] Paul, W. (1981) Theory of Relatvty. Dover Publcaton. [7] Rashevsky, P.K. (1953) Remannan Geometry and Tensor Analyss. Gostehteorzdat, Moscow. (In Russan) [8] Schouten, J.A. (1951) Tensor Analyss for Physcs. Oxford at the Clarendon Press. [9] Tolman, R. (1969) Relatvty, Thermodynamcs and Cosmology. Oxford at the Clarendon Press. [1] Wenberg, S. (1972) Gravtaton and Cosmology. John Wley and Sons, Inc., New York, London, Sydney, Toronto. DOI: /jamp Journal of Appled Mathematcs and Physcs

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