Page 179 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Complex Analysis and Differential Geometry

Notes A geometric object F in each basis represented by some square matrix F and such that
i
 j
3 3
i
q
p
j 
i
q and
i 
components of its matrix F obey transformation rules F  T S F ,
j
p
j
p 1 q 1


3 3
i
p
i
q 
j 
F  S T F . under a change of basis is called a linear operator.
q
p
j
p 1 q 1


Vectors, covectors, and linear operators are all examples of tensors (though we have no

definition of tensors yet). Now we consider another one class of tensorial objects. For the
sake of clarity, let’s denote by a one of such objects. In each basis e , e , e this object is
1
3
2
represented by some square 3 × 3 matrix a of real numbers. Under a change of basis these
ij
3 3
q
ij 
p
numbers are transformed as follows a  S S a .

i
pq
j
p 1 q 1


The covectors, linear operators, and bilinear forms that we considered above were

artificially constructed tensors. However there are some tensors of natural origin. Let’s
remember that we live in a space with measure. We can measure distance between points
(hence we can measure length of vectors) and we can measure angles between two directions
in our space. Therefore for any two vectors x and y we can define their natural scalar
product (or dot product):
(x, y) = |x| |y| cos()
where  is the angle between vectors x and y.
The tensor product is defined by a more tricky formula. Suppose we have tensor X of type

(r, s) and tensor Y of type (p, q), then we can write:
Z 1 i ...i s p  X 1 i ...i r s Y j  i  r 1 ...i s p   .
r p
r p

s 1 ...j
1 j ....j
1 j ...j

The above formula produces new tensor Z of the type (r + p, s + q). It is called the tensor
product of X and Y and denoted Z = X  Y. Don’t mix the tensor product and the cross
product. They are different.
15.13 Keywords
Linear operator: A geometric object F in each basis represented by some square matrix F and
i
j
3 3
j 
p
i
q
q and
i
i 
such that components of its matrix F obey transformation rules F  T S F ,
p
j
j
p 1 q 1


3 3
i
i
p
j 
q 
F  S T F . under a change of basis is called a linear operator.
p
q
j
p 1 q 1


Bilinear form: A geometric object a in each basis represented by some square matrix aij and such
3 3
ij 
q
p
that components of its matrix aij obey transformation rules a  S S a pq and aij =

i
j
p 1 q 1


3 3
p
q
 T T a  pq under a change of basis is called a bilinear form.
j
i
p 1 q 1


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