Page 186 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Unit 16: Tensor Fields Differentiation of Tensors

Differentiation with respect to external parameters (like t in (1)) is a tensorial operation producing Notes
new tensors from existing ones.
Exercise 1.1: Give a more detailed explanation of why the time derivative (1) represents a tensor
of type (r, s).

Now let’s consider the spacial derivative of tensor field X, i.e. its derivative with respect to a
spacial variable, e.g. with respect to x . Here we also can write
1
1
2
1 i ...i
1 i ...i
3
2
3
1

 X 1 j ...j r s  lim X 1 j ...j r s (x  h,x ,x ) X 1 i ...i r s (x ,x ,x ) , ...(2)
1 j ...j
t  h 0 h
but in numerator of the fraction in the right hand side of (2) we get the difference of two tensors
bound to different points of space: to the point P with coordinates x , x , x and to the point P’
1
2
3
with coordinates x + h, x , x . To which point should be attributed the difference of two such
1
3
2
tensors ? This is not clear. Therefore, we should treat partial derivatives like (2) in a different
way.
Let’s choose some additional symbol, say it can be q, and consider the partial derivative of X 1 i ...i r s
1 j ...j
with respect to the spacial variable x :
q
 X 1 i ...i r
Y qj 1 i ...i r s   x 1 j ...j s . ...(3)
1 ...j
q
Partial derivatives (2), taken as a whole, form an (r + s + 1)-dimensional array with one extra
dimension due to index q. We write it as a lower index in Y qj 1 i ...i r s due to the following theorem
1 ...j
concerning (3).
Theorem 1.1: For any tensor field X of type (r, s) partial derivatives (3) with respect to spacial
variables x , x , x in any Cartesian coordinate system represent another tensor field Y of the
3
1
2
type (r, s + 1).
Thus differentiation with respect to x , x , x produces new tensors from already existing ones.
3
2
1
For the sake of beauty and convenience this operation is denoted by the nabla sign: Y = X. In
index form this looks like
Y qj 1 i ...i r s   q X 1 i ...i r s . ...(4)
1 ...j
1 j ...j
Simplifying the notations we also write:

  . ...(5)
q
 x q
Warning: Theorem 1.1 and the equality (5) are valid only for Cartesian coordinate systems.
In curvilinear coordinates things are different.
Exercise 2.1: Prove theorem 1.1. For this purpose consider another Cartesian coordinate system
x , x , x  related to x , x , x . Then in the new coordinate system consider the partial derivatives
1
2
3
3
2
1



 X 1 i ...i r
Y  qj 1 i ...i r s   x 1 j ...j s ...(6)
1 ...j
q
and derive relationships binding (6) and (3).
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