1 Scalars, Vectors and Tensors - physics.gmu.edu

Lecture I: Theoretical Physics ( Phys701)
web page: physics.gmu.edu/ isatija/Phys701/TP.html

1 Scalars, Vectors and Tensors

Examples of familiar tensors: Moment of Inertial, stress-energy tensor, conductivity;
(a) in anisotropic solids, elastic, optical, electrical and magnetic properties may involve
tensors:
J = σE is valid in the isotropic medium where σ is a scalar. If the medium is anisotropic,
such as crystals and plasma in the presence of magnetic field, the current in the x direction
depend upon the electric field in the y direction and so on.
In such a case, Ji = σijEj
Analogous things happen for mechanical and magnetic properties.
(b) We will show later that electric and magnetic fields can be described in terms of a single
quantity, a tensor
Important Note about Vectors:
Cross product is defined only in 3D while dot product can be defined in any dimension
Why do we need to worry about higher dimensions when we live in 3D :
(a) Relativistic theories deal with 4-d
(b) String theories:
(c) Possibility of hidden dimensions

What is wrong with the definition of a vector: a quantity that has magni-
tude and direction??

Even scalars such as work have positive and negative sign

Elastic constants, index of refraction in anisotropic media depend upon direction

Unifying Scalars, Vectors and Tensors

We assume that space is isotropic, that is , there is NO preferred direction and laws of
physics should be independent of where we choose x -axis for example
General definition of scalars, vectors and Tensors involve their transformation w.r.t Rota-
tions

A quantity that does not change under rotations of coordinate systems is called scalar

A quantity whose components transform like r: ”those of a distance from a point” is
called a vector.
Let us write down how r transforms under rotation: simple case, choose 2D rotations with
rectangular coordinates.

1

x¯ = x cos φ + y sin φ
y¯ = −x sin φ + y cos φ
z¯ = z

Now we say that a quantity A is a vector provided,

A¯x = Ax cos φ + Ay sin φ
A¯y = −Ax sin φ + Ay cos φ
A¯z = Az

New Notation:

x¯i = aijxj (1)

aij = ∂x¯i = ∂xj (2)
∂xj ∂x¯i

, where x1 = x, x2 = y and x3 = z.
The aij are called direction cosines.

Axial or Pseudo vectors
Define: C = AXB. Note that,
(a) Ci = ǫijkAjBk ≡ ǫijkAjBk where ǫijk is called Levi-Civita symbol defined as
ǫ123 = ǫ231 = ǫ312 = 1 and ǫ132 = ǫ213 = ǫ321 = −1 and zero otherwise.
Note that, we can avoid writing the summation by using a convention that repeated index
is summed over.
(b) C does not change sign under inversion, assuming A and B do. Such vectors are called
Pseudo or Axial vectors.
Examples are angular momentum, area .

The reason for the new notation is that it can be generalized to define Tensors

A tensor T ij of rank two is defined as,

T¯ij = aikajlT kl ≡ aikajlT kl (3)

kl

2

This can be clearly generalized to define a tensor of rank n.
Scalar is called a tensor of rank Zero
Vector is called a tensor of rank One
Some Properties of Second Rank Tensors
(a) Contraction
(b)Symmetric-Antisymmetric Tensor
Any Tensor can be written as a sum of symmetric and anti-symmetric tensor. (note that
the components of a cross product of two vectors define an anti-symmetric tensor.)
(c) Can be written as a 3X3-matrix
Contravriant and Covariant Tensors
In rectangular coordinates, there is no difference between these two types of tensors.
—————————-
HOME WORK:
—————————–
(I) Verity that the expression for aij ( Eq. (2) ) is correct
(II) prove the orthogonality relation , i aijaik = δjk where δjk is the Kronecker delta.
(III) Using above definition of a vector, show that A.B is a scalar and AXB is a vector.
(IV) Given two vectors A = (3, 1, −2) and B = (1, −1, 1) (a) find the angle between the
vectors and (b) find a unit vector normal to these two vectors.
———————————————-

2 Lorentz Covariance of Maxwell’s Equations

Electric and Magnetic Fields as components of a Second-rank Tensor
No notes supplied, See Arfken.

3

3 Curvilinear Coordinates

In the above discussion, we had restricted to Cartesian coordinates.
Why study Curvilinear Coordinates???
Cartesian coordinate system offers a unique advantage that the three unit vectors, (i, j, k)
are constants in magnitude as well as in direction. However, not all problems are well
adopted to the solution in Cartesian coordinates. Example, central force problem.
In solving physics problems, we are primarily interested in coordinates in which the equa-
tion can be solved easily.

Fields

Practically all of Modern physics ( beyond Newtonian mechanics ) deals with Fields
Mathematically speaking, a field is a set of functions of coordinates of a point in space. It
cannot be analyzed in terms of the positions of finite number of particles.
For example, most of the problems involve the equation for the field ψ,

(∇2 + k2)ψ = 0 (4)

Note, k2 = 0 is Laplace equation
k2 negative is the diffusion equation
k2 equal to a constant times kinetic energy describes Schrodinger equation

It has been shown that ( Phys Rev 45, 427, 1937 by Eisenhart ) this equation is sepa-
rable in 11 different coordinate systems. In spite of the fact that ∇2 is very complicated in
non-Cartesian coordinates, one needs to use coordinate system in which partial differential
equation is separable.

Review familiar Coordinate systems
NO notes: Read from any book

4 Separability in more than one coordinate system

Examples
(1) H-atom can be solved in spherical as well as parabolic coordinate systems
(2) Three-dimensional isotropic harmonic oscillator can be solved in rectangular and spher-
ical coordinates.
We will explore in detail the relationship between symmetry and separability in more than
one coordinate system.

4