Page 172 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Unit 15: Tensors in Cartesian Coordinates

Exercise 15.16: Prove that the sum in the right hand side of formula (4) does not depend on the Notes
basis, i.e. prove the equality
3 3 3 3
q
j
p
i
 a x y   a x y .
  
ij
pq
i 1 j 1 p 1 q 1




This equality means that a(x, y) is a number determined by vectors x and y irrespective of the
choice of basis. Hence we can treat (4) as a scalar function of two vectorial arguments.
Exercise 15.17: Let  be some real number, and let x, y, and z be three vectors. Prove the
following properties of function (4):
(1) a(x + y, z) = a(x, z) + a(y, z);
(2) a( x, y) =  a(x, y);
(3) a(x, y + z) = a(x, y) + a(x, z);
(4) a(x, y) =  a(x, y).
Due to these properties function (4) is called a bilinear function or a bilinear form. It is linear
with respect to each of its two arguments.
Note that scalar product is also a bilinear function of its arguments. The arguments of scalar
product are of a different nature: the first argument is a covector, the second is a vector. Therefore,
we cannot swap them. In bilinear form (4) we can swap arguments. As a result we get another
bilinear function
b(x, y) = a(y, x). ...(5)
The matrices of a and b are related to each other as follows:

b = a , b = a . ...(6)
T
ji
ij
Definition: A bilinear form is called symmetric if a(x, y) = a(y, x).
Exercise 15.18: Prove the following identity for a symmetric bilinear form:
a(x + y, x + y) – a(x, x) – a(y, y)
a(x, y) = . ...(7)
2
Definition: A quadratic form is a scalar function of one vectorial argument f(x) produced from
some bilinear function a(x; y) by substituting y = x:
f(x) = a(x, x). ...(8)
Without a loss of generality a bilinear function a in (8) can be assumed symmetric. Indeed, if a is
not symmetric, we can produce symmetric bilinear function
a(x, y) + a(y, x)
c(x, y) = , ...(9)
2
and then from (8) due to (9) we derive

a(x, x) + a(x, x)
f(x) = a(x, x) = = c(x, x).
2
This equality is the same as (8) with a replaced by c. Thus, each quadratic function f is produced
by some symmetric bilinear function a. And conversely, comparing (8) and (7) we get that a is
produced by f:

f(x + y) – f(x) – f(y)
a(x, y) = . ...(10)
2

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