Page 189 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Complex Analysis and Differential Geometry

Notes Usually the valencies of all tensors composing the tensor field are the same. Let them all
be of type (r, s). Then if we choose some basis e , e , e , we can represent any tensor of our
1 2 3
tensor field as an array X 1 i ...i r s with r + s indices:
1. j ..j
1 i ...i
X j1 1 i ...i s r  X 1 j ...j r s (P).
. ..j
The left hand side is a tensor since the fraction in right hand side is constructed by means
of tensorial operations. Passing to the limit h  0 does not destroy the tensorial nature of
this fraction since the transition matrices S and T are all time-independent.

Differentiation with respect to external parameters is a tensorial operation producing
new tensors from existing ones.
The tensorial nature of partial derivatives established by theorem is a very useful feature.

We can apply it to extend the scope of classical operations of vector analysis. Let’s consider
the gradient, grad = . Usually the gradient operator is applied to scalar fields, i.e. to
functions  = (P) or  = (x , x , x ) in coordinate form:
1
2
3

a     .
q
q
 x q
The Laplace operator is defined as the divergency applied to a vectorial gradient of

something, it is denoted by the triangle sign:  = div grad. From Laplace operator , we
derive the following formula:
3 3
ij
   g   j .
i
i 1 j 1


Denote by the following differential operator:
1  2
=  . 
2
c  t 2
Above operator is called the d’Alambert operator or wave operator. In general relativity
upon introducing the additional coordinate x = ct one usually rewrites the d’Alambert
0
operator in a form quite similar.
And finally, let’s consider the rotor operator or curl operator (the term “rotor” is derived
from “rotation” so that “rotor” and “curl” have approximately the same meaning). The
rotor operator is usually applied to a vector field and produces another vector field:
Y = rot X. Here is the formula for the r-th coordinate of rot X:

3 3 3
k
ri
(rot X)r   g   j X .
ijk
i 1 j 1 k 1



16.6 Keywords

Cartesian coordinate system in space and hence can represent P by its radius-vector r  OP and
p
by its coordinates x , x , x .
2
1
3
1
2
3
2
3
1

 X 1 i ...i r X 1 i ...i r (x  h,x ,x ) X 1 i ...i r (x ,x ,x )
Partial derivatives 1 j ...j s  lim 1 j ...j s 1 j ...j s , taken as a whole, form an
t  h 0 h
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