In one of the letters written to the Infeld group in Warsaw Einstein wrote:
"A new manner of thinking is essential if humankind is to survive."

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Vectors
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We shall try to define here the concept of vector and that of its variance 
encompassing two types, contravariance and covariance.

The reader is supposed to know elements of calculus, namely the differential 
and the partial derivative.

In Appendix 2 we see the simplest possible example of contravariant and 
covariant coordinates in an oblique coordinate system.
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DEFINITION
We shall call vector A(i) in the nD #space a set of numbers A(i), i=1,2,...,n 
called its components, such that the properties listed in Appendix 1 are 
satisfied.

A vector whose components are functions of n variables x(i) i=1,2,...,n defining 
a point of the nD #space will be called 
"vector function of point" A(x(1),x(2),...,x(n)), or a "vector field".
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NOTATIONS AND CONVENTIONS.
Unless they are elementary displacements noted dx,dy..., vectors are usually 
noted with upper case letters (A,B...) and their components with lower case 
letters designating indexes (i,j,k...) written as upper or lower, following 
vector's name:

A
.i

.j
B

In ASCII context we shall write them with help of brackets and slash, as 
follows:

A(i/), B(/j)

Upper indexes designate contravariant components and lower indexes covariant 
ones (definitions below).

Thus A(i/) designates the i-th covariant component of vector A and B(/j) the 
j-th contravariant component of vector B.

In cases when the variance is not yet defined we shall skip the slash and C(k) 
will mean the k-th component of C of unknown variance.

We shall write derivative of y with respect to x as: d(y)/d(x) and partial 
derivative of u with respect to v: p(u)/p(v).

We shall abbreviate contravariance and covariance respectively with CNV, COV.

We shall call vectors of the same and different variance respectively homo- 
and hetero-variant.

Let's further introduce Einstein's indexing notation implying summation over 
each index repeated within a monome as upper and lower one. Thus, for 3D:

A(i/)B(/i)=A(1/)B(/1)+A(2/)B(/2)+A(3/)B(/3)
                                          
This convention applies also to partial derivatives as follows:

(px(/i)/py(/j))dy(/j) = 
(px(/i)/py(/1))dy(/1)+ (px(/i)/py(/2))dy(/2)+(px(/i)/py(/3))dy(/3)
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ELEMENTARY DISPLACEMENT and CNV variance.
 
For two systems of curvilinear coordinates x,y x(i) is function of all y(j) 
and reciprocally:

x(i)=x(i)(y(1),y(2),...y(n)) or x(i)(y(j)),j=1,n              (1.1)

y(j)=y(j)(x(1),x(2),...x(n)) or y(j)(x(i)),i=1,n              (1.2)
 
Total differentials of (1.1),(1.2) are:
 
dx(i)=(px(i)/py(j))dy(j)         (2.1)
dy(j)=(py(j)/px(i))dx(i)         (2.2)
 
(2.1),(2.2) define a type of transformation between coordinate systems x,y 
which we shall call by definition CNV (or CONTRAVARIANT). 
Components of vectors transforming accordingly will be called CNV components
and indicated by upper, or CNV indexes

Thus, finally (2.1),(2.2) become:
 
dx(/i)=(px(/i)/py(/j))dy(/j)     (3.1)
dy(/j)=(py(/j)/px(/i))dx(/i)     (3.2)
 
Conclusion:
Elementary displacement and its homo-variant vectors such as speed will be 
taken as CNV.

NOTE: Vector is identical with its set of components and "as such", else than 
this set does not exist. 
However, for convenience sake we may talk about "vector X" or say that 
"X is COV", 
ALWAYS implying a set of COV components X(i/). Thus we may say in metalanguage 
that speed is CNV by being homo-variant with displacement.
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GRADIENT and COV variance.
 
For a scalar S(x(i)) we have: 
dS=(pS/py(/j))dy(/j)= ((pS/px(/i))(px(/i)/py(/j)))dy(/j)
Thus:
Gradient Gy(j)=pS/py(j) transforms into Gx(i):
 
Gy(j)=(px(/i)/py(/j))Gx(i)   (4)
 
(4) defines a new type of transformation, hetero-variant with CNV which we 
shall call by definition COV (COVARIANT). 
Vector components transforming accordingly will be called COV components and 
indicated by lower, or COV indexes Thus, finally (4) becomes:
 
Gy(j/)=(px(/i)/py(/j))Gx(i/)   (5)
 
Conclusion:
Gradient and vectors homo-variant with it such as force, will be taken as COV.            
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APPENDIX 1 Vector Properties.

A=B when A(i)=B(i)
C=A+B --> C(i)=A(i)+B(i) (C is "sum" of a and B)
Given scalar l, P product of A by l: 
P = lA --> P(i) = lA(i)
A+B = B+A
A+(B+C) = (A+B)+C
There is a vector 0 (zero) such that A+0=A
Each vector A involves -A such that A+(-A)=0
Given scalars a,b:
a(bA)=(ab)A
(a+b)A = aA + bA
a(A+B) = aA +aB
Vector multiplied by scalar 1: 1A = A
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APPENDIX 2, EXAMPLE:
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1.Orthogonal system X (X1,X2) and X-coordinates of P

.......X2
.......|
.......|
.......|
.......|
.......|
.......|
.......|
x(2/)=.+--------------------#P
x(/2)..|....................|
.......|....................|
.......|....................|
.......|--------------------+---X1
............................x(1/)=
............................x(/1)
>
> In an orthogonal system COV and CNV coordinates
> are confused: x(1/)=x(/1) and x(2/)=x(/2)
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2.Oblique system Y (Y1,Y2) and Y-coordinates of P

....................Y2
.................../
................../
................./
................+y(2/)
.............../...*
............../.......*
............./...........*
............+ y(/2) --------#P
.........../.............../|
........../.............../.|
........./.............../..|
........+---------------+---+------Y1
....................y(/1)...y(1/)

y(1/),y(2/) are COV coordinates. They are orthogonal projections of P on 
respective axes.

y(/1),y(/2) are CNV coordinates. They are hetero-parallel projections:
P-y(/2) is parallel to Y1
P-y(/1) is parallel to Y2
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3.Both systems together.

We consider transformations between orthogonal system X and oblique Y:
We shall abbreviate by "sin", "cos"; etc. the trygonometric functions
of the angle Y2-Y1.
 
.......X2...........Y2
.......|.........../
.......|........../
.......|........./
.......|........+y(2/).orthogonal projection
.......|......./...*...of P on Y2
.......|....../.......*
.......|...../...........*
x(2/)=.+----+ y(/2) --------#P
x(/2)..|.../.............../|
.......|../.............../.|
.......|./.............../..|
.......|----------------+---+---X1/Y1
....................y(/1)...y(1/)=
............................x(1/)=
............................x(/1)

Coordinates y(/1), y(/2) related to
x(/1),x(/2) with relation
 
x(/1) = y(/1) + y(/2)cos   
x(/2) = y(/2)sin
 
y(/1) = x(/1)-x(/2)/tg     
y(/2) = x(/2)/sin
 
are CNV: 

indeed:
(p(x(/m))/p(y(/n)))y(/n) involves:
 
(p(x(/1))/p(y(/1)))y(/1)+(p(x(/1))/p(y(/2)))y(/2) = y(/1) + y(/2)cos = x(/1)
and
(p(x(/2))/p(y(/1)))y(/1)+(p(x(/2))/p(y(/2)))y(/2) = y(/2)sin = x(/2)

COV coordinates are: y(1/), y(2/)

which can be shown with a similar simple procedure, which we shall leave to 
the reader as an exercise.

Let's note that COV and CNV coordinates are respectively orthogonal and 
hetero-parallel projections on system's axes.

NOTE: In an orthogonal system COV and CNV components are obviously identical.
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