Page 181 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Complex Analysis and Differential Geometry

Notes 1 j,
10. Prove that definition     for i  is invariant under a change of basis, if we interpret
i
j
 0 for i  j
 1 for i  j,
the Kronecker symbol as a tensor. Show that both definitions in   d  
ij
ij
 0 for i  j
are not invariant under a change of basis.
3

11. Lower index i of tensor    1 for i  j, by means of X ...p...  g X ....s.... . What tensorial
i
.........

..........
ps
j
 0 for i  j s 1

i  1 for i  j,
object do you get as a result of this operation? Likewise, raise index J in    .
j
 0 for i  j
3 3 3
12. Prove that the vector a with components a   g  ijk x y coincides with cross
r
j
k
ri
i 1 j 1 k 1



product of vectors x and y, i.e. a = [x, y].
Answers: Self Assessment
1. basis-dependent 2. bilinear form.
3. contravariant index. 4. transformation rules
15.16 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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