Translation:On the spacetime lines of a Minkowski world/Paragraph 1

Integral forms. Transformation of a $$p$$-fold integral into a $$p+1$$-fold integral. Exact differentials.
Let $$\left(\begin{array}{c} n\\ p \end{array}\right)$$ be the functions $$A_{\alpha_{1}\alpha_{2}\dots\alpha_{p}}$$ of the variables $$x^{(1)}x^{(2)}\dots x^{(n)}$$; a permutation of the indices $$\alpha_{1}\alpha_{2}\dots\alpha_{p}$$ of $$A_{\alpha_{1}\alpha_{2}\dots\alpha_{p}}$$ shall give $$\pm A_{\alpha_{1}\alpha_{2}\dots\alpha_{p}}$$ here, depending on whether it is straight or not. The expression

$\sum_{\alpha_{1}=1}^{n}\sum_{\alpha_{2}=1}^{n}\cdots\sum_{\alpha_{p}=1}^{n}A_{\alpha_{1}\alpha_{2}\dots\alpha_{p}}dx^{\left(\alpha_{1}\right)}dx^{\left(\alpha_{2}\right)}\dots dx^{\left(\alpha_{p}\right)}$|undefined

is then denoted as an integral form. If we imagine $$n$$ variables $$x^{(\alpha)}$$ as functions of $$p\leqq n$$ parameters $$u^{(1)}\dots u^{(p)}$$, and if $$A_{\alpha_{1}\dots\alpha_{p}}$$ are well defined and integrable within this value-area of $$x$$, then $$M_{p}$$ is defined in $$S_{n}\left(x^{(1)}\dots x^{(n)}\right)$$, on which the $$p$$-fold integral

$\int du^{(1)}du^{(2)}\dots du^{(p)}\sum_{\left(\alpha_{1}\dots\alpha_{p}\right)}A_{\alpha_{1}\alpha_{2}\dots\alpha_{p}}\frac{\partial\left(x^{\left(\alpha_{1}\right)}x^{\left(\alpha_{2}\right)}\dots x^{\left(\alpha_{p}\right)}\right)}{\partial\left(u^{\left(1\right)}u^{\left(2\right)}\dots u^{\left(p\right)}\right)}$|undefined

shall be extended; the sum is herein extended over all combinations without repetition of $$n$$ numbers $$1,2\dots n$$ to $$p$$ each.

If $$M_{p}$$ is “closed”, and if furthermore $$A_{\alpha_{1}\alpha_{2}\dots\alpha_{p}}$$ together with their first partial derivatives are well-defined within $$M_{p+1}$$:

$x^{(\alpha)}=x^{(\alpha)}\left(v^{(1)}v^{(2)}\dots v^{(p)}v^{(p+1)}\right)\quad(\alpha=1,2\dots n)$

and if it satisfies certain regulatory conditions, then it is

where the $$p+1$$-fold integral is to be extended over these $$M_{p+1}$$, and the $$p$$-fold integral over its “boundary”, the “closed” $$M_{p}$$. The $$A_{\alpha_{1}\alpha_{2}\dots\alpha_{p+1}}$$ have the meaning:

$A_{\alpha_{1}\alpha_{2}\dots\alpha_{p+1}}=\frac{\partial A_{\alpha_{1}\alpha_{2}\dots\alpha_{p}}}{\partial x^{\left(\alpha_{p+1}\right)}}-\frac{\partial A_{\alpha_{1}\alpha_{2}\dots\alpha_{p-1}\alpha_{p+1}}}{\partial x^{\left(\alpha_{p}\right)}}+\dots+(-)^{p}\frac{\partial A_{\alpha_{2}\dots\alpha_{p}\alpha_{p+1}}}{\partial x^{\left(\alpha_{1}\right)}}$|undefined

If a $$p$$-fold integral extended over an arbitrary “open” $$M_{p}$$ shall only depend on their (closed) boundary-$$M_{p-1}$$, then to that end it is necessary and sufficient, that


 * for all formations $$A_{\alpha_{1}\alpha_{2}\dots\alpha_{p+1}}\equiv0\qquad\left(\alpha_{1}\alpha_{2}\dots\alpha_{p+1}\right)$$;

the related integral form of $$p$$-th order is then called an exact differential and their coefficients $$A_{\alpha_{1}\alpha_{2}\dots\alpha_{p}}$$ allow the representation as $$\mathrm{A}_{\alpha_{1}\alpha_{2}\dots\alpha_{p}}$$, by which the transformation into a $$p-1$$-fold integral extended over the closed boundary-$$M_{p-1}$$, appears to be given.

Invariance of the integral forms.
It is required, that the integral forms goes over into itself at the passage to the new coordinates $$\bar{x}$$ or

$\sum_{\left(\alpha_{1}\dots\alpha_{p}\right)}A_{\alpha_{1}\alpha_{2}\dots\alpha_{p}}\frac{\partial\left(x^{\left(\alpha_{1}\right)}x^{\left(\alpha_{2}\right)}\dots x^{\left(\alpha_{p}\right)}\right)}{\partial\left(u^{\left(1\right)}u^{\left(2\right)}\dots u^{\left(p\right)}\right)}=\sum_{\left(\alpha_{1}\dots\alpha_{p}\right)}\bar{A}_{\alpha_{1}\alpha_{2}\dots\alpha_{p}}\frac{\partial\left(\bar{x}^{\left(\alpha_{1}\right)}\bar{x}^{\left(\alpha_{2}\right)}\dots\bar{x}^{\left(\alpha_{p}\right)}\right)}{\partial\left(u^{\left(1\right)}u^{\left(2\right)}\dots u^{\left(p\right)}\right)},$|undefined

from which

$\bar{A}_{\alpha_{1}\alpha_{2}\dots\alpha_{p}}=\sum_{\beta_{1}=1}^{n}\sum_{\beta_{2}=1}^{n}\cdots\sum_{\beta_{p}=1}^{n}A_{\beta_{1}\beta_{2}\dots\beta_{p}}\frac{\partial x^{\left(\beta_{1}\right)}}{\partial\bar{x}^{\left(\alpha_{1}\right)}}\frac{\partial x^{\left(\beta_{2}\right)}}{\partial\bar{x}^{\left(\alpha_{2}\right)}}\dots\frac{\partial x^{\left(\beta_{p}\right)}}{\partial\bar{x}^{\left(\alpha_{p}\right)}}$|undefined

The line element is unchanged with respect to this transformation, corresponding to the passage to new coordinates.

The covariance of $$A_{\alpha_{1}\alpha_{2}\dots\alpha_{p}}$$.
Following the theory of differential invariants of a quadratic differential form given by and, we have to formulate the previous result as follows:

$$\mathrm{A}_{\alpha_{1}\alpha_{2}\dots\alpha_{p}}$$ considered as functions of $$x$$ form a covariant system of $$p$$-th order.

This system is a special one, following from the circumstance that it doesn't contain $$np$$, but only $$\left(\begin{array}{c} n\\ p \end{array}\right)$$ linear independent parts. The ordinary vectors of $$p$$-th kind of mathematical physics are identical with such special covariant or also contravariant systems of $$p$$-th order.

As the most simple example of a covariant system of different kind, we refer to the system of coefficients of the line element

$ds^{2}=\sum_{\alpha}\sum_{\beta}c_{\alpha\beta}dx^{(\alpha)}dx^{(\beta)},$,

where $$c_{\alpha\beta}=c_{\beta\alpha}$$.

For the passage to other coordinates it is known that

$\bar{c}_{\alpha\beta}=\sum_{\gamma}\sum_{\delta}c_{\gamma\delta}\frac{\partial x^{\left(\gamma\right)}}{\partial\bar{x}^{\left(\alpha\right)}}\frac{\partial x^{\left(\delta\right)}}{\partial\bar{x}^{\left(\beta\right)}}$|undefined

If space $$S_{n}$$ is Euclidean – and only with such one we will concern ourselves in the following – and if $$x$$ are Cartesian orthogonal coordinates in its interior, then and only then it is given:

$\begin{array}{rlrlr} c_{\gamma\delta} & =[\gamma\delta] & =0, & \text{if} & \gamma\ne\delta\\ & & =1, & \text{if} & \gamma=\delta \end{array}$

For the passage to curvilinear coordinates, the previous statement of covariance of $$c_{\gamma\delta}$$ gives:

the well-known formula from the theory of surfaces.

If $$c$$ denotes the determinant

$c=\left

where we now may again use the arbitrary coordinates $$x$$, furthermore $$C_{\alpha\beta}$$ as the adjuncts of $$c_{\alpha\beta}$$ in $$c$$ (taken together with the affiliated signs), then we have

$c^{(\alpha\beta)}=\frac{C_{\alpha\beta}}{c},\qquad c^{(\alpha\beta)}=c^{(\beta\alpha)}$|undefined

as a simple example for a (symmetric) contravariant system of second order; i.e. it is given for the passage to new coordinates $$\bar{x}$$:

In order to confirm this by computation, we only need (except the theorem of the multiplication of matrices) the following known formulas:

where

$\frac{\partial(\bar{x})}{\partial(x)}=\frac{\partial\left(\bar{x}^{(1)}\dots\bar{x}^{(n)}\right)}{\partial\left(x^{(1)}\dots x^{(n)}\right)}$

and:

where $$\alpha_{1}\alpha_{2}\dots\alpha_{n-1}\alpha_{n}$$ and $$\beta_{1}\beta_{2}\dots\beta_{n-1}\beta_{n}$$ each denote a positive permutation of $$1,2\dots n$$.

As a simple contravariant of first order we additionally may denote the differential $$dx^{(\alpha)}$$ itself; because

$d\bar{x}^{(\alpha)}=\sum_{\beta}\frac{\partial\bar{x}^{\left(\alpha\right)}}{\partial x^{\left(\beta\right)}}dx^{\left(\beta\right)}$. |undefined

The covariance of $$A_{\alpha_{1}\alpha_{2}\dots\alpha_{p}\alpha_{p+1}}$$
The covariance of $$A_{\alpha_{1}\alpha_{2}\dots\alpha_{p}\alpha_{p+1}}$$ follows from their property as coefficients of an integral form of $$p+1$$-th order. From that it follows:

$\bar{A}_{\alpha_{1}\alpha_{2}\dots\alpha_{p}\alpha_{p+1}}=\frac{\partial}{\partial\bar{x}^{\left(\alpha_{p+1}\right)}}\bar{A}_{\alpha_{1}\alpha_{2}\dots\alpha_{p}}-\frac{\partial}{\partial\bar{x}^{\left(\alpha_{p}\right)}}\bar{A}_{\alpha_{1}\dots\alpha_{p-1}\alpha_{p+1}}+\dots+(-)^{p}\frac{\partial}{\partial\bar{x}^{\left(\alpha_{1}\right)}}\bar{A}_{\alpha_{2}\dots\alpha_{p}\alpha_{p+1}}$,|undefined

if

$\bar{A}_{\alpha_{1}\alpha_{2}\dots\alpha_{p+1}}=\sum_{(\beta)}A_{\beta_{1}\dots\beta_{p+1}}\frac{\partial x^{\left(\beta_{1}\right)}}{\partial\bar{x}^{\left(\alpha_{1}\right)}}\dots\frac{\partial x^{\left(\beta_{p+1}\right)}}{\partial\bar{x}^{\left(\alpha_{p+1}\right)}}$|undefined

and

$\bar{A}_{\alpha_{1}\alpha_{2}\dots\alpha_{p}}=\sum_{(\beta)}A_{\beta_{1}\dots\beta_{p}}\frac{\partial x^{\left(\beta_{1}\right)}}{\partial\bar{x}^{\left(\alpha_{1}\right)}}\dots\frac{\partial x^{\left(\beta_{p}\right)}}{\partial\bar{x}^{\left(\alpha_{p}\right)}}$|undefined

are given.

For computational confirmation we need the equations for the second derivatives given by $$\frac{\partial^{2}x^{\left(\alpha\right)}}{\partial\bar{x}^{\left(\beta\right)}\partial\bar{x}^{\left(\gamma\right)}}$$.

When the previous formulas are applied to the special case of the passage from Cartesian orthogonal coordinates $$x$$ of an Euclidean $$S_{n}$$ to orthogonal curvilinear coordinates, the known Jacobian integral transformation follows. Our formulas of course apply to an arbitrary $$S_{n}$$ and arbitrary oblique curvilinear coordinates. The doctrine of the integral forms, in particular the formations $$A_{\alpha_{1}\alpha_{2}\dots\alpha_{p+1}}$$, in connection with 's and 's theories contains everything that is necessary for a generalized vector analysis from a unified standpoint, for arbitrary many dimensions, arbitrary metric and arbitrary coordinates.

In an Euclidean $$S_{n}$$ for orthogonal Cartesian coordinates $$x$$ as well as $$\bar{x}$$, covariants and contravariants coincide due to the properties of the orthogonal matrix $$\frac{\partial(x)}{\partial(\bar{x})}$$ of determinant $$+1$$. In addition, for instance, the homogeneous coordinates of the plane of $$S_{3}$$ are comparable to the covariants of first order as long as the transformation of the homogeneous $$x$$ to the homogeneous $$\bar{x}$$ is projective (i.e. linear), and the point coordinates of $$S_{3}$$ are comparable to the contravariants of first order. The polar vectors of $$S_{3}$$ are contravariants of first order, the axial vectors are special contravariants of second order, from which the so-called supplement (a covariant system of first order) has to be derived.

Supplement.
We define the following covariant system $$n-p$$-th order as the supplement of a contravariant system of $$p$$-th order:

Let $$\alpha_{1}\dots\alpha_{p}\alpha_{p+1}\dots\alpha_{n}$$ be a positive permutation of $$1,2,\dots u$$, then for the supplement $$B_{\alpha_{p+1}\dots\alpha_{n}}$$ of the system $$A^{\left(\alpha_{1}\dots\alpha_{p}\right)}$$(which is assumed to be special), we have

$\sqrt{c}A^{\left(\alpha_{1}\dots\alpha_{p}\right)}=B_{\alpha_{p+1}\dots\alpha_{n}}$|undefined

with the remark that in the case of motions changing the direction, i.e. for the case

$\frac{\partial(x)}{\partial(\bar{x})}<0$,

the root $$\sqrt{c}$$ has to be changed to $$-\sqrt{\bar{c}}$$. It can be easily confirmed by the aid of formulas (4) and (5), that $$B_{\alpha_{p+1}\dots\alpha_{n}}$$is really a covariant system of $$n-p$$-th order.

Similarly, for the supplement $$B^{\left(\alpha_{p+1}\dots\alpha_{n}\right)}$$ of a covariant system $$A_{\alpha_{1}\dots\alpha_{p}}$$ it is given:

$\sqrt{\frac{1}{c}}A_{\alpha_{1}\dots\alpha_{p}}=B^{\left(\alpha_{p+1}\dots\alpha_{n}\right)}$|undefined

with the same remark regarding the change of sign of the root. During the application of the Euclidean $$S_{3}$$ and Cartesian orthogonal coordinates $$x$$ we have $$c\equiv1,$$ and the equations

$A_{23}=B^{(1)},\quad A_{31}=B^{(2)},\quad A_{12}=B^{(3)}$,

or

$A^{(23)}=B_{1},\quad A^{(31)}=B_{2},\quad A^{(12)}=B_{3}$

provide the known classification of the supplement $$B$$ to the vector of second kind $$A$$, where $$A$$ retains its sign when there is a change of the motion's direction (i.e. of all three coordinate axes), as it can be seen from the formulas

$A_{\alpha\beta}=\sum_{\gamma,\delta}A_{\gamma\delta}\frac{\partial x^{(\gamma)}}{\partial\bar{x}^{(\alpha)}}\frac{\partial x^{(\delta)}}{\partial\bar{x}^{(\beta)}},\qquad\left(A_{\gamma\delta}=-A_{\delta\gamma}\right)$,|undefined

and therefore $$B$$ is doing the same, contrary to ordinary polar vectors of first kind (contra- or contravariants of first order), which is just the result of the rule of sign change from $$\sqrt{c}\equiv1$$ to $$-\sqrt{\bar{c}}\equiv-1$$ (axial vectors).

The supplement in an Euclidean $$S_{n}$$ is of course nothing other than the duality of polar correlation in the infinitely distant $$S_{n-1}$$ with respect to the absolute $$M_{n-1}^{2}$$ of Euclidean metric, therefore the name dual system is used.

As the reciprocal system, on the other hand, we denote the following covariant or contravariant system, which emerges from the contravariant or covariant system of same order be means of the coefficients $$c_{\alpha\beta}$$ of $$ds^{2}$$:

$$A^{\left(\alpha_{1}\dots\alpha_{p}\right)}=\sum_{\beta_{1}=1}^{n}\sum_{\beta_{2}=1}^{n}\cdots\sum_{\beta_{p}=1}^{n}c^{\left(\alpha_{1}\beta_{1}\right)}c^{\left(\alpha_{2}\beta_{2}\right)}\dots c^{\left(\alpha_{p}\beta_{p}\right)}A_{\beta_{1}\beta_{2}\dots\beta_{p}},\qquad\alpha_{1}\alpha_{2}\dots\alpha_{p}=1,2,\dots n$$

with the solution

$$A_{\alpha_{1}\dots\alpha_{p}}=\sum_{\beta}c_{\alpha_{1}\beta_{1}}c_{\alpha_{2}\beta_{2}}\dots c_{\alpha_{p}\beta_{p}}A^{\left(\beta_{1}\dots\beta_{p}\right)},$$,

where the $$A$$ are not forming a special system this time.

Orientation questions at multiple integrals.
In order to discuss them for the general transformation (1), we will shortly remember the proof of these integral theorems. At first we have for $$p=n-1$$:

$\begin{aligned} & \int\dots\int dx^{(1)}\dots dx^{(n)}\left\{ \frac{\partial}{\partial x^{(n)}}A_{12\dots n-1}-\frac{\partial}{\partial x^{(n-1)}}A_{12\dots n-2}+\dots+(-)^{n-1}\frac{\partial}{\partial x^{(1)}}A_{23\dots n}\right\} \\ & \quad=\int\dots\int dx^{(1)}\dots dx^{(n-1)}\left|A_{12\dots n-1}\right|_{\left.x^{(n)}\right._{\mathrm{inf.}}}^{\left.x^{(n)}\right._{\mathrm{sup.}}}+\\ & \qquad+\int\dots\int dx^{(1)}\dots dx^{(n-2)}dx^{(n)}\left|-A_{12\dots n-2n}\right|_{\left.x^{(n-1)}\right._{\mathrm{inf.}}}^{\left.x^{(n-1)}\right._{\mathrm{sup.}}}+\\ & \qquad\dots+\int\dots\int dx^{(2)}\dots dx^{(n)}\left|(-)^{n-1}A_{23\dots n}\right|_{\left.x^{(1)}\right._{\mathrm{inf.}}}^{\left.x^{(1)}\right._{\mathrm{sup.}}} \end{aligned} $|undefined

If the Euclidean $$S_{n}$$ and the Cartesian orthogonal coordinates are presupposed now, then the $$n-1$$-fold integrals transform into hypersurface integrals, if one writes:

$\begin{aligned} & \int\dots\int dx^{(1)}\dots dx^{(n-1)}A_{12\dots n-1}+\\ & \qquad+\int\dots\int dx^{(1)}\dots dx^{(n-2)}dx^{(n)}\left(-A_{12\dots n-2n}\right)+\dots\\ & \qquad+\int\dots\int dx^{(2)}\dots dx^{(n)}\left((-)^{n-1}A_{23\dots n}\right)\\ & \quad=\int\dots\int df_{n-1}\left\{ \cos\left(Nx^{(n)}\right)A_{12\dots n-1}+\dots+\cos\left(Nx^{(1)}\right)\cdot(-)^{n-1}A_{23\dots n}\right\} \end{aligned} $

There we have assumed for simplicities sake, that any of the parallels to the axes intersects the boundary-$$M_{n-1}$$ at only two points (being closed hypersurface, the boundary-$$M_{n-1}$$ has to be intersected by every line at an even number of intersections). At the exit point of the parallel of the axes, $$\left.x^{(n)}\right._{\mathrm{sup.}}$$, we have

$\cos\left(Nx^{(n)}\right)>0$

thus

$df_{n-1}\cos\left(Nx^{(n)}\right)=dx^{(1)}\dots dx^{(n-1)}$

at the entry point of the parallel of the axes, $$\left.x^{(n)}\right._{\mathrm{inf.}}$$, we have

$\cos\left(Nx^{(n)}\right)<0$

thus

$-df_{n-1}\cos\left(Nx^{(n)}\right)=dx^{(1)}\dots dx^{(n-1)}$

thus $$N$$ has to go from the interior to the exterior; the possibility of such an orientation of $$M_{n-1}$$, i.e. their two-sidedness, is always presupposed here.

The integral theorem $$p=0-2$$ can be proven most simply by reduction to $$n=1$$, $$p=n-2$$ according to, by considering $$M_{n-1}$$ at first as “plane” $$S_{n-1}$$, which will be made to $$S_{n-1}x^{(n)}=0$$ by an orthogonal transformation of the coordinates. In which, however, the framing-$$M_{n-2}$$ will be oriented correctly, if one chooses the exterior normal, as shown above. The general case of the integral theorem $$p=n-2$$ can be obtained by decomposition of the now arbitrary curvilinear $$M_{n-1}$$ into infinitesimal “plane” pieces and application of the obtained ones onto them. In this way, one finds the rule for the orientation of $$M_{n-2}$$ which limits the curvilinear $$M_{n-1}$$:

The normal-plane $$[N'N]$$, which definitely belongs to the framing-$$M_{n-2}$$ by virtue of the Euclidean metric, will be oriented correctly, if one predefines a direction $$N$$ within it in an arbitrary way, and in this way determines the necessary second direction $$N'$$: One intersects the normal plane with the curvilinear $$M_{n-1}$$, by which an intersection curve $$M_{1}$$ emerges which forms with $$M_{n-2}$$ an even number of intersections; since $$M_{n-2}$$ is closed, an exit point belongs to any entry point of a curve $$M_{1}$$ drawn upon $$M_{n-1}$$ into the area enclosed by the framing $$M_{n-2}$$. As the second necessary normal $$N$$, one then chooses (in the respective point of our $$M_{n-2}$$) that direction of the tangent of the mentioned intersection-$$M_{1}$$ which goes to the exterior. In accordance with the things now said, this direction must be traceable – again under certain presuppositions regarding the constitution of $$M_{n-2}$$.

In the case $$n$$, $$p=0-3$$: We have to provide two arbitrary directions $$NN$$, then we search for the tangent (which is directed to the exterior) of the intersection-$$M_{1}$$ of the normal plane of the boundary-$$M_{n-3}$$ with the $$M_{n-2}$$, so this direction provides the third necessary normal $$N'$$, and the normal space $$[N'NN]$$ is oriented correctly etc.

If parameters $$u^{(1)}\dots u^{(p)}$$ upon $$M_{p}$$ is provided, which should limit $$M_{p+1}$$ in the general case (1), and if one determines the “directions” (better: locations) of the Normal-$$S_{n-p}$$ with the aid of the values

$\sqrt{\frac{c}{b}}\frac{\partial\left(x^{\left(\alpha_{1}\right)}\dots x^{\left(\alpha_{p}\right)}\right)}{\partial\left(u^{(1)}\dots u^{(p)}\right)}=N_{\alpha_{p+1}\dots\alpha_{n}}$|undefined

where $$b$$ is the discriminant of the arc-element on $$M_{p}$$, then this must be in agreement with the orientations of the normal-$$S_{n-p}$$ mentioned above; i.e., when $$N_{p+1}$$ is the normal determined by the intersection curve of the normal-$$S_{n-p}$$ with $$M_{p+1}$$, and $$N_{p+2}$$ to $$N_{n}$$ are the arbitrarily given $$n-p-1$$ normals (which serve for the orientation of the normal-$$S_{n-p-1}$$ of $$M_{p+1}$$), then the directions

$\frac{\partial x}{\partial u^{(1)}},\ \frac{\partial x}{\partial u^{(2)}}\dots\frac{\partial x}{\partial u^{(p)}},\ N_{p+1},\ N_{p+2}\dots N_{n}$|undefined

must exhibit the same order of succession as the coordinate axes $$x^{(1)}x^{(2)}\dots x^{(n)}$$. Otherwise the sign of the left-hand side of (1) would have to be changed.

Difference of the components of a vector determined by the differential invariant theory compared to the ordinary representation.
The latter one operates by the method of vectorial splitting with respect to the directions of the $$n$$-gon, which form in any spacepoint the $$n$$ passing parameter lines. So, let $$x$$ be Cartesian orthogonal coordinates in $$S_{n}$$ and $$\bar{x}$$ be any generalized coordinates, then by (2) it is given:

$ds^{2}=\sum_{\alpha=1}^{n}\left(dx^{(\alpha)}\right)^{2}=\sum_{\alpha,\beta}\sum_{\lambda}\frac{\partial x^{(\lambda)}}{\partial\bar{x}^{(\alpha)}}\frac{\partial x^{(\lambda)}}{\partial\bar{x}^{(\beta)}}d\bar{x}^{(\alpha)}d\bar{x}^{(\beta)}=\sum\bar{c}_{\alpha\beta}d\bar{x}^{(\alpha)}d\bar{x}^{(\beta)}$|undefined

and in consequence of (3):

$\bar{c}_{\alpha\beta}=\sum_{\lambda}\frac{\partial\bar{x}^{(\alpha)}}{\partial x^{(\lambda)}}\frac{\partial\bar{x}^{(\beta)}}{\partial x^{(\lambda)}}$|undefined

So if one has a covariant system of $$p$$-th order in Cartesian coordinates $$A_{\alpha_{1}\dots\alpha_{p}}$$, then it is given

$\bar{A}_{\alpha_{1}\alpha_{2}\dots\alpha_{p}}=\sum_{\beta_{1}=1}^{n}\sum_{\beta_{2}=1}^{n}\cdots\sum_{\beta_{p}=1}^{n}A_{\beta_{1}\beta_{2}\dots\beta_{p}}\frac{\partial x^{\left(\beta_{1}\right)}}{\partial\bar{x}^{\left(\alpha\right)}}\frac{\partial x^{\left(\beta_{2}\right)}}{\partial\bar{x}^{\left(\alpha\right)}}\dots\frac{\partial x^{\left(\beta_{p}\right)}}{\partial\bar{x}^{\left(\alpha_{p}\right)}}$|undefined

Instead of this, the method of vectorial splitting with respect to all directions of the generalized $$n$$-gon gives:

$\sum_{\beta_{1}=1}^{n}\sum_{\beta_{2}=1}^{n}\cdots\sum_{\beta_{p}=1}^{n}A_{\beta_{1}\beta_{2}\dots\beta_{p}}\frac{\frac{\partial x^{\left(\beta_{1}\right)}}{\partial\bar{x}^{\left(\alpha_{1}\right)}}}{\sqrt{\bar{c}_{\alpha_{1}\alpha_{1}}}}\frac{\frac{\partial x^{\left(\beta_{2}\right)}}{\partial\bar{x}^{\left(\alpha_{2}\right)}}}{\sqrt{\bar{c}_{\alpha_{2}\alpha_{2}}}}\dots\frac{\frac{\partial x^{\left(\beta_{p}\right)}}{\partial\bar{x}^{\left(\alpha_{p}\right)}}}{\sqrt{\bar{c}_{\alpha_{p}\alpha_{p}}}}$|undefined

and for the contravariant system (vector $$p$$-th order), instead of

$\bar{A}^{\left(\alpha_{1}\alpha_{2}\dots\alpha_{p}\right)}=\sum_{\beta_{1}=1}^{n}\sum_{\beta_{2}=1}^{n}\cdots\sum_{\beta_{p}=1}^{n}A^{\left(\beta_{1}\beta_{2}\dots\beta_{p}\right)}\frac{\partial\bar{x}^{\left(\alpha_{1}\right)}}{\partial x^{\left(\beta_{1}\right)}}\frac{\partial x^{\left(\alpha_{2}\right)}}{\partial\bar{x}^{\left(\beta_{2}\right)}}\dots\frac{\partial x^{\left(\alpha_{p}\right)}}{\partial\bar{x}^{\left(\beta_{p}\right)}}$|undefined

we have the form

$\sum_{\beta_{1}=1}^{n}\sum_{\beta_{2}=1}^{n}\cdots\sum_{\beta_{p}=1}^{n}A^{\left(\beta_{1}\beta_{2}\dots\beta_{p}\right)}\frac{\frac{\partial\bar{x}^{\left(\alpha_{1}\right)}}{\partial x^{\left(\beta_{1}\right)}}}{\sqrt{\bar{c}^{\left(\alpha_{1}\alpha_{1}\right)}}}\frac{\frac{\partial x^{\left(\alpha_{2}\right)}}{\partial\bar{x}^{\left(\beta_{2}\right)}}}{\sqrt{\bar{c}^{\left(\alpha_{2}\alpha_{2}\right)}}}\dots\frac{\frac{\partial\bar{x}^{\left(\alpha_{p}\right)}}{\partial x^{\left(\beta_{p}\right)}}}{\sqrt{\bar{c}^{\left(\alpha_{p}\alpha_{p}\right)}}}$ |undefined

The components formed by means of the differential invariants, when for instance the angle arises as generalized coordinate, can provide a physically incorrect dimension, as this indeed also happens in other areas such as e.g. the Lagrangian generalized forces of mechanics; as shown above, this can be easily improved after finishing the computation. However, to carry them out in the representation of the vectorial splitting, would mean to give away the advantages of the differential invariants.

For the sake of explanation and at the same time to obtain formulas which are important for the following, the cases $$n=3$$ and $$n=4$$ shall be discussed now.