The Study on the Differential and Integral Calculus in Pseudoeuclidean Space

In the vector space of real vectors, comparison was executed of the multilinear forms, covariant derivatives, the total differentials and derivatives in the direction which are calculated with different metrics—with Euclidean metric and with pseudoeuclidean metric of a zero index. Comparison was executed of the Taylor’s formulas to different metrics. What is established by us is that multilinear forms of different metrics have different values; covariant derivatives have identical values; the total differentials and derivatives in the direction have different values. In Euclidean space, Taylor’s formula with any order of accuracy assigned in advance is equal, but in pseudoeuclidean space Taylor’s formula is not equal with any order of accuracy. It is concluded that in space with a pseudoeuclidean metric, the computing sense of the differential and integral calculus created in Euclidean space is lost and the possibility of mathematical model operation of real physical processes in vector space with pseudoeuclidean metric is called into question.


Comparative Definitions of Euclidean and Pseudoeuclidean Spaces
This part of article contains elements of algebra and math analysis which with sufficient completeness are explained in textbook [3]. Requirement of inclusion of elementary details of this material in article is dictated by need to reasonably show the common properties of Euclidean and pseudoeuclidean spaces, to show initial distinction of these spaces, to clearly recognize the stage of creation of the linearly vector space when this distinction is entered, and as it influences further creation of space, to study how behave algebraic forms and differential calculus at the same time. Even the insignificant ambiguity from all this can generate doubt in reliability of results and therefore nothing from written below it can be excluded as the partial fact, for reduction of volume of article.
We will define the vector space L of dimension 1 n + of real vectors , , x y and its basis traditionally how we may read about it in the textbooks for students. For visual demonstration we will determine the vectors as the rankorder sets of real numbers, representing them either matrixes columns, or matrixes lines.
Vectors of affine basis we will designate 0 1 , , , n a a a . Let's agree that the Greek indexes accept values from 0 to n, and Latin indexes-from 1 to n.
Therefore vector decomposition x on basis α a with use of the rule of toting on twice repeating index it is possible to write down doubly: As basis vectors α a we will accept matrixes lines In such basis elements α ξ of any vector ∈ x L will be components of decomposition of this vector on basis α a , i.e. will be Let's construct a radius vector of the vector space. For this purpose we will , , , n M ξ ξ ξ . We will call a vector α α ξ = OM a a position vector of a point M , we will designate r a position vector and we will write α α ξ = r a . Let's remind that α a is affine basis.
The position vector of the current point of the vector space is a variable vector. Therefore it is vector function of scalar arguments 0 1 , , , n ξ ξ ξ .
Let's take a set of functions Further everywhere in the vector space L we will use curvilinear coordinates , , , In each point M ∈ L we will enter the basis α e called by local basis on a formula 0 0 , .
It is necessary attention that formulas (1)  x α , we will write down these decompositions: Let's assume that curvilinear coordinates X are locally the linear coordinates, i.e. such that in any point of space L the local basis α e can be used as affine basis of rather small vicinity of this point. Let's define a scalar product of vectors of the vector space with Euclidean metrics. We will enter the definition of vectors multiplication, i.e. we will formulate the law under which to each couple of vectors , ∈ x y L this law sets the particular value of real number which we will designate xy . At the same time operation of a scalar multiplication of vectors has to satisfy to axioms of commutation, association, distribution and a positive sign of a scalar square of vectors.
In Euclidean space it is axiomatically claimed that the scalar square of a vector is positive definite: The scalar multiplication of vectors becomes the given if to define-to set the law of a scalar multiplication of vectors of affine basis.
For the vector space L with affine basis α a and with Euclidean metric we will determine scalar multiplication by the following rule: the scalar product of any couple of vectors of affine basis α a is equal to the sum of products of their corresponding components. This rule is expressed by formulas: where αβ δ -unit matrix (Kronecker delta).
Having multiplied vectors , x a y a with use of formulas (3) and with application of axioms of commutation, association and distribution, we will receive a formula of a scalar multiplication in Euclidean space of the vectors set by coordinates in affine basis α a , On this formula we receive a scalar square of a vector By this formula we conclude that the rule of a scalar multiplication of vectors of affine basis (3) is guarantee of realization of an axiom of a positive determination of a scalar square of a vector in Euclidean space.
Follows from this formula also that in Euclidean space the norm of the vector α α ξ = x a preset by coordinates in affine basis α a is defined by a formula Vectors α e of all set determined by formulas (1) are linearly independent and therefore they form basis at any choice of curvilinear coordinates (locally the linear coordinates). Vectors α e , apparently from (2), are defined in affine basis. Therefore their scalar products at each other and on themselves can be calculated on formulas (4). The set of these scalar products forms the square matrix g αβ of order 1 n + called by the covariant Gram matrix representing a Journal of Applied Mathematics and Physics covariant tensor of the second rank.
Let's make a scalar multiplication of vectors (the vectors defined so agreed to call contravariant) and we use for this purpose a covariant Gram matrix g αβ at multiplication. As a result we will receive a formula of a scalar multiplication of contravariant vectors in Euclidean space Let's define other local basic system. Vectors of this basic system agreed to number the top indexes β e and to call vectors β e manual vectors to vectors α e . We will accept that scalar product of vectors of basic system α e on vectors of basic system β e have to satisfy equality where β Formulas (1) give us 0 0 = e a , formulas (3) give us 0 0 1 = a a , and from the equation (7) we receive . These three equalities give us Let's pay attention (it is extremely important for a comprehension of the further text) that vectors β e of the mutual basic system do not depend on the accepted rule of a scalar multiplication of vectors in Euclidean space. It can seem not so as the mutuality Equation (7) represents a scalar multiplication of vectors in Euclidean space. Let's prove validity of the made statement. In any point of the vector space there is the infinite set of local bases. We choose from this infinite set (we do not calculate, we do not use the rule of a scalar multiplication, we Journal of Applied Mathematics and Physics choose) such local basis which satisfies to the mutuality Equation (7). The chosen basis in the infinite set cannot absent. Therefore the statement is proved. Now we will construct the vector space with a pseudoeuclidean metric. At the same time linear space L , its affine basis α a , curvilinear coordinates 0 1 : , , , n X x x x and the mutual basic systems , β α e e we will keep former by what they were accepted in Euclidean space. The basis for such decision is that all called elements of the vector space are taken without use of a scalar multiplication of vectors in Euclidean space and therefore the scalar multiplication of vectors in Euclidean space studied above will not be bound with the rule of a scalar multiplication of vectors which we will construct below in pseudoeuclidean space.
Let's define a scalar multiplication of vectors of affine basis α a of the vector space L with a pseudoeuclidean metric the following rule: The vector space with such scalar multiplication of basis vectors is called pseudoeuclidean space of a zero index.

Having multiplied vectors
with use of formulas (11) and with application of axioms of a commutation, association and distribution, we will receive a formula of a scalar multiplication in pseudoeuclidean space of the vectors preset by coordinates in affine basis α a ,  6), (8) and (9)) to take with a minus-sign.
Let's agree further in the presence of a double sign ± we will take a plus-sign in a Euclidean space, and we will take a minus-sign in pseudoeuclidean space.
With use of this agreement of a formula of a scalar multiplication of the vectors If at a tensor of the second rank w αβ mentally to reject one of indexes, then, according to the quotient rule of tensor calculus, we will derive the covariant tensor of the first rank-a vector. Let's designate this vector w α • . On a formula (12) we will make convolution transform a vector w α • with a vector α ξ . Let's write down result of convolution and we will return the rejected index We derive vector v β . Let's make compression of this vector with vector β η .
We will derive If to substitute here The bilinear form w α β αβ ξ η is a polynomial. Therefore it can be derived as the product of a polynomial p α α ξ to a polynomial q β β η , i.e.
The linear forms on formula (12) Having multiplied these linear forms-these polynomials, we will have ( ) Having substituted (15) here, we will derive a formula which in accuracy repeats a formula (14).
It is easy to see that the proofs executed for the linear (13) and bilinear (14) forms can be continued with use of the same concepts and methods for multilinear forms of any higher order. #

Derivatives, Differentials and Taylor's Formula
Derivatives, differentials and Taylor's formula which were defined in the vector space with Euclidean metric will be the objects of our researches. We will observe change of numerical values of the considered objects upon transition to space with a pseudoeuclidean metrics. , , , n X x x x ∈ L . Let's accept that it is continuous and has the continuous partial derivatives on all variables to the necessary order inclusive. Its covariant derivatives we will agree to designate an inferior index after an asterisk.
Covariant derivatives of the increasing orders are defined by the following formulas: From this it follows that the set of differentials dx α represents a contravariant vector.
Lemma 2. In each point of the vector space the total differentials of any order and derivatives in the direction of any order of scalar function change the numerical values when replacing Euclidean metric in this space to a pseudoeuclidean metric of a zero index. The norm of a vector dx α is defined by a formula ( ) ( ) ( ) Let's remind that we take a plus-sign for Euclidean metric, and we take a minus-sign for pseudoeuclidean.
We will write down derivatives in a point C towards a point X : Here proofs for differentials and derivatives first and second orders were made. It is not difficult to see that these proofs can be repeated for any higher order. # Theorem 1. In the vector space with Euclidean metric Taylor's formula is equality with beforehand given accuracy. In the same space with pseudoeuclidean metric of a zero index Taylor's formula is not equality with any order of accuracy.
Proof. Taylor's formula Comparison of behavior of the left-hand and right-hand members of Taylor's formula at change of a metrics leads to the conclusion that in pseudoeuclidean space of a zero index Taylor's formula is not equality, as was to be shown.
Here the proof is executed for the second order of accuracy. It is clear, that it can be made for somehow high order of accuracy. # Theorem 2. The operations of differential and integral calculations developed for Euclidean space in pseudoeuclidean space do not make computing sense.
Proof. Follows from the theorem 1 that in pseudoeuclidean space the difference of values of function cannot be calculated by means of the device (16) of the differential calculus created for Euclidean space. It follows from this that in pseudoeuclidean space the theory of difference schemes and, in general, all methods of finite differences created for Euclidean space cannot be used.
The integral in the vector space with Euclidean metric of any finite-dimensional of a measure of integration domain is a limit of the integral sum, i.e. sum of differentials of antiderivative of integrand. Differentials, according to the lemma 2, change the numerical value at change of a metrics. Therefore values of integrals in pseudoeuclidean space will differ from their values in Euclidean space. It follows from this that the integral calculus created for Euclidean space is not suitable for numerical methods in pseudoeuclidean space. #

Conclusions
It is established that values of multilinear forms, derivatives of a scalar function and its differentials in space with pseudoeuclidean metric differ from their values in the same space with Euclidean metric. Taylor's formula which Euclidean space is the equality expressing an increment of scalar function through differentials of this function in pseudoeuclidean space equality is not.
The executed researches lead to the conclusion that the differential and integral calculus developed for space with Euclidean metric in space with pseudoeuclidean metric are not suitable.
As differential and integral calculus of real functions of real variables is a con-