mathmatical physics

1. NPTEL – Physics – Mathematical Physics - 1
Lecture 25
Transformation properties of vectors
Suppose the components of a vector in 3D (𝑉3) is represented in two
different coordinate systems as,
(𝑥1, 𝑥2, 𝑥3) and (𝑥̅ 1, 𝑥̅ 2, 𝑥̅ 3).
Since they are components of the same vector, there should exist a relation of
the form,
𝑥̅1 = 𝑎11𝑥1 + 𝑎12𝑥2 + 𝑎13𝑥3
𝑥̅2 = 𝑎21𝑥1 + 𝑎22𝑥2 + 𝑎23𝑥3
𝑥̅3 = 𝑎31𝑥1 + 𝑎32𝑥2 + 𝑎33𝑥3
In a compact notation, this can be written as,
𝑥̅ 𝑖 = 𝑎𝑖1𝑥1 + 𝑎𝑖2𝑥2 + 𝑎𝑖3𝑥3; 𝑖 = 1,2,3
Or, 𝑥̅ 𝑖 = ∑3
Joint initiative of IITs and IISc – Funded by MHRD Page 4 of 20
𝑎𝑖 𝑗 𝑥
𝑗
𝑗
=1
𝑖 = 1,2,3
The components of 𝑎𝑖
𝑗
are already discussed in the context of the
transformation between Cartesian and Spherical polar systems. Using the
summation convention, the above equation is written as,
𝑥̅𝑖 = 𝑎𝑖𝑗 𝑥𝑗
As an application of this notation, let us conveniently express the matrix
multiplication equation, where A, B and C are matrices. According to the above
rule, it can be written as,
𝐶𝑖𝑗 = 𝐴𝑖𝑘𝐵𝑘𝑗
where the repeated index k is assumed to be summed
over. Of special importance are the orthogonal matrices for
which 𝐶𝑖𝑗 = 𝛿𝑗 (Krönecker delta is also written as 𝛿𝑖𝑗)
𝑖

2. NPTEL – Physics – Mathematical Physics - 1
Quotient rules
We shall discuss two useful theorems which will establish the tensor character
for sets of functions.
Theorem 1
Let A(i1,i2, …….ir) be a set of functions of the variable xi and let the
inner product A(, i2 …..ir) B with another vector 𝐵⃗⃑ be a tensor of
the form,
𝐴𝑘1………𝑘𝑝
,
𝑗 1……..𝑗
𝑞 then the set of A(i1 ……ir) represents a tensor of the type
𝐴𝑘1………𝑘𝑝 𝑟
Proof
Let us assume that the inner product A(, j, k) B yields a tensor of the type
𝐴𝑘 (x). Now it is to be proved that A(i,j,k) is a tensor of the type 𝐴𝑖𝑘 .
Now A(,j,k) B transforms as,
𝑗 1……..𝑗
𝑞
𝑗
𝑗
𝐶 (, 𝑗, 𝑘)  = 𝑥 𝑦
𝐴(, , 𝛾)𝐵
𝑟 𝑗
𝑦𝑘 𝑥
Where 𝐵(𝑥) =
𝑥
𝑦  
and  is an arbitrary vector.
Putting this expression for B in the right hand of the above formula
and transporing all terms on one side of the equations yields,
[𝐶(𝑖, 𝑗, 𝑘) − 𝑥  𝑟
𝑦  𝑦
𝑘
𝑥 𝑦𝑗
 𝐴(, , 𝛾)]= 0
Since  is arbitrary,
𝑥 𝑥𝑟 𝑦𝑗
𝐶(𝑖, 𝑗, 𝑘) =
𝑦 𝑦𝑘
 𝐴(,, 𝛾)
Joint initiative of IITs and IISc – Funded by MHRD Page 5 of 20
This is precisely the law of transformation of the tensor of the
type 𝐴𝑖𝑘
𝑗

3. NPTEL – Physics – Mathematical Physics - 1
Theorem 2
Let 𝐴 (𝑖𝑙 … … … 𝑖𝑟) be a set of  𝑟 functions defined in the P-coordinate system,
and let 𝐵 (𝑖𝑙 … … … 𝑖𝑟) be the corresponding sets in the Q- coordinate system. If
for every set of vectors with components 𝑙𝑞 relates to P-coordinates
and
1
relates to the Q-coordinates, one has the equality,
𝐵(𝑙 …… . . 𝑟 ) … … … …  = 𝐴(1 … … … . . 𝑟 )1 … … … … 𝑟
1 𝑟
(that is inner product is a scalar), then the set of functions A (i1 ……..ir)
represents a contravariant tensor of rank r in the P-coordinate system.
Proof
Since  =
𝑖 𝑥 
𝑖
𝑦 
𝑖 𝑖
So,
[B (i……….r) - A (1……….r)
𝑦 
𝑖
𝑥𝑖
…….. ]  … … … … .  = 0
𝑦 
𝑟
𝑥𝑟 𝑖 𝑟
Since  … … . are abitary, the [… … ] = 0
𝑖
B (i……….r) =
𝑦 𝑖 𝑦 
𝑟
𝑥𝑖 𝑥
𝑟
…….. A(1 … … … … 𝑟 )
Joint initiative of IITs and IISc – Funded by MHRD Page 6 of 20
Which goes on to confirm that
A(1 … … … … 𝑟 ) = 𝐴1…………𝑟