Page 188 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Unit 16: Tensor Fields Differentiation of Tensors

Operator (7) is called the d’Alambert operator or wave operator. In general relativity upon Notes
introducing the additional coordinate x = ct one usually rewrites the d’Alambert operator in a
0
form quite similar to (6).
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 = rotX. Here is the
formula for the r-th coordinate of rot X:

3 3 3
ri
k
(rot X)r   g   j X . ...(8)
ijk
i 1 j 1 k 1



Exercise 1.1: Formula (8) can be generalized for the case when X is an arbitrary tensor field with
at least one upper index. By analogy with (5) suggest your version of such a generalization.
Note that formulas (6) and (8) for the Laplace operator and for the rotor are different from those
that are commonly used. Here are standard formulas:
2 2 2
        
   1    2    3  , ...(9)
  x    x    x 
e 1 e 2 e 3

rot X  det    . ...(10)
x  1 x  2 x  3
X 1 X 2 X 3

The truth is that formulas (6) and (8) are written for a general skew-angular coordinate system
with a SAB as a basis. The standard formulas (10) are valid only for orthonormal coordinates
with ONB as a basis.
Exercise 2.1: Show that in case of orthonormal coordinates, when g =  , formula (6) for the
ij
ij
Laplace operator 4 reduces to the standard formula (9).
The coordinates of the vector rot X in a skew-angular coordinate system are given by formula
(8). Then for vector rot X itself we have the expansion:

3
r
rot X   (rot X) e . ...(11)
r
r 1

Exercise 3.1: Substitute (8) into (11) and show that in the case of a orthonormal coordinate system
the resulting formula (11) reduces to (10).
16.5 Summary

Indeed, their components are arrays related to bases, while any basis is a triple of free

vectors (not bound to any point). Hence, the tensors previously considered are also not
bound to any point.
Now suppose we want to bind our tensor to some point in space, then another tensor to
another point and so on. Doing so we can fill our space with tensors, one per each point. In
this case we say that we have a tensor field. In order to mark a point P to which our
particular tensor is bound we shall write P as an argument:
X = X(P)

LOVELY PROFESSIONAL UNIVERSITY 181