BASICS FOR PHYSICS
MATHEMATICAL PRELIMINARIES
NOTEBANK ID:
ffenztbivaq2002
, 1
MATHEMATICAL PRELIMINARIES
1.1 Invariants
It is a remarkable fact that very few fundamental laws are required to describe the
enormous range of physical phenomena that take place throughout the universe. The
study of these fundamental laws is at the heart of Physics. The laws are found to have a
mathematical structure; the interplay between Physics and Mathematics is therefore
emphasized throughout this book. For example, Galileo found by observation, and
Newton developed within a mathematical framework, the Principle of Relativity:
the laws governing the motions of objects have the same mathematical
form in all inertial frames of reference.
Inertial frames move at constant speed in straight lines with respect to each other – they
are non-accelerating. We say that Newton’s laws of motion are invariant under the
Galilean transformation (see later discussion). The discovery of key invariants of Nature
has been essential for the development of the subject.
Einstein extended the Newtonian Principle of Relativity to include the motions of
beams of light and of objects that move at speeds close to the speed of light. This
extended principle forms the basis of Special Relativity. Later, Einstein generalized the
principle to include accelerating frames of reference. The general principle is known as
the Principle of Covariance; it forms the basis of the General Theory of Relativity ( a theory
of Gravitation).
,2 MATHEMATICAL PRELIMINARIES
A review of the elementary properties of geometrical invariants, generalized
coordinates, linear vector spaces, and matrix operators, is given at a level suitable for a
sound treatment of Classical and Special Relativity. Other mathematical methods,
including contra- and covariant 4-vectors, variational principles, orthogonal functions, and
ordinary differential equations are introduced, as required.
1.2 Some geometrical invariants
In his book The Ascent of Man, Bronowski discusses the lasting importance of the
discoveries of the Greek geometers. He gives a proof of the most famous theorem of
Euclidean Geometry, namely Pythagoras’ theorem, that is based on the invariance of
length and angle ( and therefore of area) under translations and rotations in space. Let a
right-angled triangle with sides a, b, and c, be translated and rotated into the following
four positions to form a square of side c:
c
1
c
2 4
c
b
a 3
c
|← (b – a) →|
The total area of the square = c2 = area of four triangles + area of shaded square.
If the right-angled triangle is translated and rotated to form the rectangle:
, MATHEMATICAL PRELIMINARIES 3
a a
1 4
b b
2 3
then the area of four triangles = 2ab.
The area of the shaded square area is (b – a)2 = b2 – 2ab + a2
We have postulated the invariance of length and angle under translations and rotations and
therefore
c2 = 2ab + (b – a)2
= a 2 + b2 . (1.1)
We shall see that this key result characterizes the locally flat space in which we live. It is
the only form that is consistent with the invariance of lengths and angles under
translations and rotations .
The scalar product is an important invariant in Mathematics and Physics. Its invariance
properties can best be seen by developing Pythagoras’ theorem in a three-dimensional
coordinate form. Consider the square of the distance between the points P[x1 , y 1 , z 1] and
Q[x2 , y2 , z 2] in Cartesian coordinates:
MATHEMATICAL PRELIMINARIES
NOTEBANK ID:
ffenztbivaq2002
, 1
MATHEMATICAL PRELIMINARIES
1.1 Invariants
It is a remarkable fact that very few fundamental laws are required to describe the
enormous range of physical phenomena that take place throughout the universe. The
study of these fundamental laws is at the heart of Physics. The laws are found to have a
mathematical structure; the interplay between Physics and Mathematics is therefore
emphasized throughout this book. For example, Galileo found by observation, and
Newton developed within a mathematical framework, the Principle of Relativity:
the laws governing the motions of objects have the same mathematical
form in all inertial frames of reference.
Inertial frames move at constant speed in straight lines with respect to each other – they
are non-accelerating. We say that Newton’s laws of motion are invariant under the
Galilean transformation (see later discussion). The discovery of key invariants of Nature
has been essential for the development of the subject.
Einstein extended the Newtonian Principle of Relativity to include the motions of
beams of light and of objects that move at speeds close to the speed of light. This
extended principle forms the basis of Special Relativity. Later, Einstein generalized the
principle to include accelerating frames of reference. The general principle is known as
the Principle of Covariance; it forms the basis of the General Theory of Relativity ( a theory
of Gravitation).
,2 MATHEMATICAL PRELIMINARIES
A review of the elementary properties of geometrical invariants, generalized
coordinates, linear vector spaces, and matrix operators, is given at a level suitable for a
sound treatment of Classical and Special Relativity. Other mathematical methods,
including contra- and covariant 4-vectors, variational principles, orthogonal functions, and
ordinary differential equations are introduced, as required.
1.2 Some geometrical invariants
In his book The Ascent of Man, Bronowski discusses the lasting importance of the
discoveries of the Greek geometers. He gives a proof of the most famous theorem of
Euclidean Geometry, namely Pythagoras’ theorem, that is based on the invariance of
length and angle ( and therefore of area) under translations and rotations in space. Let a
right-angled triangle with sides a, b, and c, be translated and rotated into the following
four positions to form a square of side c:
c
1
c
2 4
c
b
a 3
c
|← (b – a) →|
The total area of the square = c2 = area of four triangles + area of shaded square.
If the right-angled triangle is translated and rotated to form the rectangle:
, MATHEMATICAL PRELIMINARIES 3
a a
1 4
b b
2 3
then the area of four triangles = 2ab.
The area of the shaded square area is (b – a)2 = b2 – 2ab + a2
We have postulated the invariance of length and angle under translations and rotations and
therefore
c2 = 2ab + (b – a)2
= a 2 + b2 . (1.1)
We shall see that this key result characterizes the locally flat space in which we live. It is
the only form that is consistent with the invariance of lengths and angles under
translations and rotations .
The scalar product is an important invariant in Mathematics and Physics. Its invariance
properties can best be seen by developing Pythagoras’ theorem in a three-dimensional
coordinate form. Consider the square of the distance between the points P[x1 , y 1 , z 1] and
Q[x2 , y2 , z 2] in Cartesian coordinates:
