Page 192 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Unit 16: Tensor Fields Differentiation of Tensors

3 3 Notes
5. Show that in case of orthonormal coordinates, when g = d , formula    g   for
ij
ij
ij
i
j
i 1 j 1


2 2 2
        
the Laplace operator 4 reduces to the standard formula    1    2    3  .
  x    x    x 
The coordinates of the vector rot X in a skew-angular coordinate system are given by
3 3 3
ri
k
formula (rot X)r   g   j X . Then for vector rot X itself we have the expansion:
ijk
i 1 j 1 k 1



3
r
rot X   (rot X) e .
r
r 1

3 3 3 3
ri
r
k
6. Substitute (rot X)r   g   j X into rot X   (rot X) e and show that in the case
ijk
r
i 1 j 1 k 1 r 1




3
r
of a orthonormal coordinate system the resulting formula rot X   (rot X) e reduces to
r
r 1

e 1 e 2 e 3
rot X  det  1  2  3 .
x  x  x 
X 1 X 2 X 3
Answers: Self Assessment
1. Cartesian coordinate system 2. functional array
3. Partial derivatives
16.9 Further Readings
Books Ahelfors, D.V. : Complex Analysis
Conway, J.B. : Function of one complex variable
Pati, T. : Functions of complex variable
Shanti Narain : Theory of function of a complex Variable
Tichmarsh, E.C. : The theory of functions
H.S. Kasana : Complex Variables theory and applications
P.K. Banerji : Complex Analysis
Serge Lang : Complex Analysis
H. Lass : Vector & Tensor Analysis
Shanti Narayan : Tensor Analysis

C.E. Weatherburn : Differential Geometry
T.J. Wilemore : Introduction to Differential Geometry
Bansi Lal : Differential Geometry.

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