What is behind the physical phenomenon of the velocity; of the force; there is the mathematical concept of the vector. This is a new concept, since force has direction, sense, and magnitude, and we accept the physical principle that the forces exerted on a body can be added to the rule of the parallelogram. This is the first axiom of Newton. Newton essentially requires that the power is a " vectorial " size , without writing clearly , and Galileo that applies the principle of the independence of forces .
Physics Serway Jewett 6th edition for Scientists and Engineers
The mathematical and philosophical concept of vector
1. The mathematical and philosophical concept of vector 1
The mathematical and philosophical concept of vector
by George Mpantes www.mpantes .gr
Historical summary
The vectors
The transformation equations of coordinates
the transformation equations of vectors
Euclidean Geometry and Newtonian physics .
Philosophical comments, Aristotle
Historical summary
“There is un unspoken hypothesis which underlies all the physical theories so far
created, namely that behind physical phenomena lies a unique mathematical structure
which is the purpose of theory to reveal. According to this hypothesis , the
mathematical formulae of physics are discovered not invented, the Lorentz
transformation , for example ,being as much a part of physical reality as a table or a
chair”. ( RELATIVITY: THE SPECIAL THEORY J.L.Synge p.163)
Indeed in our example the physical phenomenon is the force, and the
underlying mathematical structure is vector analysis. But looking the historical
process, mathematics create their truths independently, discover new entities,
and their tendency for generalization goes ahead exceeding the initial physical
presuppositions. The new discoveries of the mathematical process return to the
physical theory where they create new unifications and generalizations, now for
the phenomena of the actual world. Can we trust them? Can mathematics lead
2. The mathematical and philosophical concept of vector 2
the physical theory? The answer seems to be positive, if the measurements
agree with the mathematical conjectures (electromagnetic waves!). that is that
mathematical structure extends the physical theory. So, for example, with the
support of the use of vector methods , we had a development of theoretical
physics and by the beginning of the twentieth century, vector analysis had
become firmly entrenched as a tool for the development of geometry and
theoretical physics.
As we look back on the nineteenth century it is apparent that a
mathematical theory in terms of which physical laws could be described and
their universality checked was needed. Figuratively speaking two men stepped
forward in this direction, Hamilton and Grassman. Hamilton was trying to find
the appropriate mathematical tools with which he could apply Newtonian
mechanics to various aspects of astronomy and physics. Grassman tried to
develop an algebraic structure on which geometry of any number of dimension
could be based. The quaternions of Hamilton and Grassmann’s calculus of
extension proved to be too complicated for quick mastery and easy application ,
but from them emerged the much more easily learned and more easily applied
subject of vector analysis. This work was due principally to the American
physicist John Willard Gibbs (1839-1903) and is encountered by every student
of elementary physics.
The vectors .
What is behind the physical phenomenon of the velocity; of the force;
there is the mathematical concept of the vector. This is a new concept, since
force has direction, sense, and magnitude, and we accept the physical principle
that the forces exerted on a body can be added to the rule of the
parallelogram. This is the first axiom of Newton. Newton essentially requires
that the power is a " vectorial " size , without writing clearly , and Galileo that
applies the principle of the independence of forces .
3. The mathematical and philosophical concept of vector 3
These are the basic physical indications for the mathematical treatment,
for “vector1 geometry”, where
the term vector denotes a translation or a displacement a in the space.2 The
statement that the displacement a transfers the point P to the point Q (“transforms” P
into Q ) may also be expressed by saying that Q is the end-point of the vector a whose
starting point is at P. if P and Q are ant two points then there is one and only one
displacement a which transforms P to Q. We shall cal it the vector defined by P and Q
.
and indicate it by PQ
There are two fundamental operations, which are subject to a system of
laws, viz. addition of two vectors (the translation which arises through two
successive translations (law of parallelogram), and multiplication of a vector by a
number (is defined through the addition). These laws are
A.Addition: a+b=c
with the properties
a+b=b+a
(a+b)+c=a+(b+c)
If a and c are any two vectors , then there is one and only one value of x
for which the equation a+x=c holds
B. Multiplication b=λ.a
with the properties
(λ+μ)a=(λa)+(μa)
λ(μa)=(λμ)a
1.a=a
λ(a+b)=(λa)+(λb)
In elementary physics , a vector is graphically regarded as a directed line
segment , or arrow. This is the translation or the displacement described by
Weyl. So in elementary physics , vector was something apparent, something
concrete and intuitively simple. It was geometrical. In theoretical physics it
became an idea, something cerebral, connected with algebra. The first was a
1 The term vector was introduced from Hamilton
2 This definition is from Weyl, (Space, time, matter)
4. The mathematical and philosophical concept of vector 4
sketch of the second. This is the course of mathematics. The formula for
algebraic vector was the old bold Cartesian binding of geometry with algebra viz
this of a picture with the abstract and compact truth of numbers, a good
combination between intuition and rigor, through concepts, in the center of
which was the well-known coordinate system, one of the more significant
generalization of mathematics.
By means of a coordinate system, a set of ordered triples of real
numbers can be put into one-to-one correspondence with the points of a three
dimensional Euclidean space. However many aspects of modern-day science
cannot be adequately described in terms of a three-dimensional Euclidean model.
The ideas of vector analysis when expressed in a notational fashion are
immediately extendable to n-dimensional space and their physical usage is amply
demonstrated in the development of special and general relativity theory.
With the change of the figurativeness of the points, change also the
description of the vector.
The set {A1, A2, A3} of all triples (A1, A2, A3), (A1΄, A2΄, A3΄) etc.,
determined by orthogonal projections of a common arrow representation on the
axes of the associated rectangular Cartesian coordinate system is said to be a
Cartesian vector. Many triples means many systems, but all these represent
the same Cartesian vector, which has a family of arrows as its geometrical
representative. The binding of orthogonal projections with the law of
parallelogram is the base of all the formalism of vector analysis.3
A Cartesian vector (A1, A2, A3), (3-tuple), can be represented
graphically by an arrow, with it’s initial point at the origin and it’s terminal
point at the position with coordinates (A1, A2, A3), but it is not the only possible
arrow representation. An arrow with initial and terminal points (a,b,c) and
(A,B,C) such that A1=A-a, A2=B-b, A3=C-c can be considered a representative
of a 3-tuple.
3 See my article “the mathema tical forms of nature, the tensors”
5. The mathematical and philosophical concept of vector 5
A Cartesian vector with respect to a coordinate system, is
characterized by a magnitude , a direction and a sense , and its components in
any coordinate system satisfy the algebraic laws of the triples, viz the laws 1
and 2 for the vectors, expressed algebraically , if we define a=(a1,a2,…..an)
b=(b1,b2,….bn)
i.e (a1,a2,…..an)+(b1,b2,….bn)=(a1+b1, a2+b2+……an,bn).
λ.a=λ(a1,a2,…..an)= (λa1,λa2,…..λan) .
Now an analytical treatment of vector geometry is possible, in which
every vector is represented by it’s components and every point by its
coordinates.
How all these triples, (A1, A2, A3), (A1΄, A2΄, A3΄) etc., are related?
The transformation equations of coordinates
A fundamental problem of theoretical physics is that formulating
universally valid laws relating natural phenomena. Because the transformation
idea is of such importance, the development of vector geometry and later of
vector analysis is build around this.
A rectangular Cartesian coordinate system4 imposes a one-to-one
correspondence between the points of Euclidean three-space and the set of all
ordered triples of real numbers. A second rectangular Cartesian system brings
about another correspondence of the same point . What is the nature of those
transformations that relate such coordinate representations of the three-space?
The specific transformations of coordinates for our example in the
development of vector analysis, are called translations and rotations. They are
linear transformations and they connect orthogonal Cartesian systems. All linear
transformations have the characteristic that the fundamental relations (A) and
(B) are not disturbed by the transformation viz they hold for the transformed
points and vectors :
4 We examine this particular case in our example.
6. The mathematical and philosophical concept of vector 6
α΄+b΄=c΄ b΄=λ .a΄………
DEFINITION 1. The transformation equations that relate the
coordinates ( , , ) and (x , , ) 1 2 3 1 2 3 x x x x x in rectangular coordinate systems , the
axes of which are parallels are
j j j x x x
.......... .......... ....(1) 0
where ( , , ) 3
1
0 x x x represent the unbarred coordinates of the origin of the
0
2
0
barred system O’. These are called equations of translation. The Cartesian
vector concept is employed in obtaining them.
DEFINITION 2.
The transformation equations that relate the coordinates
( , , ) and (x , , ) 1 2 3 1 2 3 x x x x x in rectangular coordinate systems, having a common
origin and such that there is no change of unit distance along coordinate axes,
are related by the transformation equations
j x c x
.......... ..(2) j k
k
where the coefficients of
transformations j
k a are direction
cosines satisfying the conditions
p
3
c j
c j
j
k p
1 k
7. The mathematical and philosophical concept of vector 7
These are called equations of rotation.
The transformations of the coordinates (2) are a subset of the linear or
affine transformations, with the general form
.........( 3)
1
2
3
c c c
3
3
c c c
3
2
3
1
2
3
2
2
2
1
1
3
1
2
1
1
1
2
3
x
x
x
c c c
x
x
x
where apply the conditions of orthogonality, they are the orthogonal
transformations that connect orthogonal Cartesian systems with common origin
and are produced from the vectorial behavior of the vectorial units (bases) in
the axes of the two systems
. Physically they describe, as we have mention, the rotation of an
orthogonal Cartesian system. The orthogonal transformations fulfill the first
unification of geometry (the Euclidean metrical geometry in every orthogonal
system) and as the geometry is a fundamental branch of physics, this unification
will be the model of the unification of physical laws in all the
systems.(universality)
But what about the vectors? What is their deepest behavior in the scene
of coordinate systems?
the transformation equations of vectors
We have seen that a Cartesian vector (A1, A2, A3) can be represented
graphically by an arrow, but
The components of this arrow, transform under rotation, as the coordinates .
Proof: If the transformation (2) is applied to the coordinates of P0 and P1 , the
coordinate differences { j j x x1 0 } satisfy
j j x x c x c x c x x
j k k
k
j k
k
j k
k
( )......... .......... ..(4) 1 0 1 0 1 0
that is the transformation (2).
A corresponding verification of the statement holds for translations, where the
vector components remain unaltered.
8. The mathematical and philosophical concept of vector 8
So we have the definition of the Cartesian vector under the light of both
transformations:
A Cartesian vector (A1, A2, A3), is a collection of ordered triples , each
associated with a rectangular Cartesian coordinate system and such that any
two satisfy the transformation law
.......... .......... .......... ......( 5) k
j
x
j A
k
x
A
where the partial derivatives are the coefficients j
i c of the linear
transformation (3), of coordinates.
We must notice that every component of the vector in the new system is
a linear combination of the components in the initial variables. So if all
components of the vector are zero in the initial system they will be also zero in
the new variables. This is the more important property of vectors: a vectorial
equation holds in every rectangular Cartesian system (for our paradigm), if it
holds in one! This is the root of the universality of the physical or geometrical
laws, as we see in the end of the article. Newton’s law is universal because it is
written in vectorial form. It’s invariance in translation is the mathematical
acceptance of the Newtonian principle of relativity.
The scalars .
A second concept which has evolved in the development of vector analysis is
that of the scalar. The definition of scalar states that it is a quantity
possessing magnitude but no direction. Such entities as mass, time, density and
temperature are given as examples. But for mathematics, the prize example is
the real number, as it does not have to be associated with magnitude. From a
historical point of view scalar is a quantity invariant under all transformations
of coordinates (Felix Klein). Whether a given algebraic form is invariant depends
on the group of transformations under consideration. Again the scalars , as
vectors, are associated with coordinate systems and transformations.
9. The mathematical and philosophical concept of vector 9
Euclidean Geometry and Newtonian physics .
The mathematical investigation showed that our Known geometrical
vector (arrow) has hidden qualities which are raised by their correlation with
coordinate systems: The laws of it’s transformation. The vector concept
received much of its impetus from this fact, so it plays a fundamental role in
many aspects of geometry and physics. This mathematical result underlies the
principles of relativity of Newton and Einstein, that would be ungrounded
without the mathematical discovery of the transformation theory of the vectors
and (later) of tensors.
Magnitude and angle are fundamental to the metric structure of
Euclidean space. They are scalar invariants under the transformations of the
orthogonal Cartesian set.
The inner product transforms
k
P j Q j
c j P r
c j Q s
c j
c j P r Q s
P r Q s
P k
Q
r
s
r
s
r
j
j
j
k
3
1
s
3
1
3
1
3
1
( )( )
1 0 , X j X (through 4)
and the distance of the points j
1
1 (x x ) (x x ) (x x ) (x x ) (x x ) (x x )
3 2
0
3
1
2 2
0
2
1
1 2
0
1
1
3 2
0
3
1
2
0
2
1
1 2
0
These formulas carry out the first unification of metrical Euclidean geometry.
An observer who measures a distance and an angle in a orthogonal Cartesian
system uses the same formulas and finds the same results with somebody else
who measures the same magnitudes in another orthogonal Cartesian system,
which subsists a translation or a rotation of the first. The question of finding
those entities (as distance and inner product) that have an absolute meaning
transcending the coordinate system, is of prime significance. This gives us a
direction as to which of the concepts considered in the framework of
rectangular Cartesian systems should be generalized as well as how to bring
about the generalizations. This is the criminal point of the universality of the
physical laws. Moreover, the availability of the Cartesian systems of reference
10. The mathematical and philosophical concept of vector 10
will be valuable when considering the special theory of relativity, in a later
article.
In vector formalism, we will now show the covariance of Newton's
law in linear systems with given origin ( rotation) .
In the system K we have
d
F
( ) k k m
Multiply by
k
r
dt
x
x
and summing with respect to k (from 5) we have
x
k
m
d
F
m
( ) ( ) k r r
x
r
F
k
k
x
x
r
dt
d
dt
So the form of the equation remains the same in the new system (
covariant ) , and the mathematical formalism demonstrates that the laws of
Newton have a universal application in Euclidean space, where we can adjust
orthogonal Cartesian systems. The physical laws are invariant in form in all the
orthogonal Cartesians systems (my article “covariance and invariance in physics”)
Philosophical comments, Aristotle .
The quotation of Synge about the discovery of mathematical structure
of vector analysis , where the vector is as much a part of physical reality as a
table, is very poetic, having construct a reality of visible and invisible objects,
as the vector and the table. Vector exists as the table but it is invisible!
Aristotle in his ideas of the theory of knowledge (Analytica posterioria)
says that “the knowledge of a fact differs from the knowledge of a reasoned fact” .
The theoretical foundations of the systems of this deductive reasoning,
account of first principles where are the bases of every science. …. The
scientific knowledge through reasoning is impossible if we do not know the first
principles. ….it is clear that in science of nature as elsewhere we should try first to
determine questions about the first principles …..αληθείς και πρωταρχικές και άμεσες και
πρωθύστερες και αιτίες του συμπεράσματος,the first basis from which a thing is
11. The mathematical and philosophical concept of vector 11
known……as regards their existence must be assumed for the principle ,(what a straight
line is , what a triangle is ..)but proved for the rest of the system, by logical reasoning.
How are these first principles to be established? ….they are arrived by
the repeated visual sensations, which leave their marks in the memory. Then we reflect
on these memories and arrive by a process of intuition (νους) at the first principles ….if
there in not something intelligible behind the phenomena, there is not science for
anything, science is not created from senses….
So vector is an intelligible creation, a first principle, that is a conviction,
a support of the deductive reasoning. This reasoning constructs a mental logical
reality in our brains, which is the human’s way of comprehension. Mathematics
are neither discoveries nor inventions. Mathematics are creations , as poems,
but logical creations, based on the rules of deductive reasoning.
But in fact, their first principles are founded in nature.
Sources
Herman Weyl (space,time, matter,Dover)
H.Eves (foundations and fundamental concepts of mathematics,Dover)
J.L Synge (Relativity: the special theory, Noth Holland publising Company
Amsterdam New York Oxford)
Robert C.Wrede (introduction to vector and tensor analysis, Dover)
Aristotle (Analytica posterioria, internet)
George Mpantes mathematics teacher www.mpantes.gr