Page 180 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Unit 15: Tensors in Cartesian Coordinates

15.14 Self Assessment Notes

1. Defining the scalar product a,x by means of sum we used the coordinates of vector x and
of covector a, which are ..................
2. A geometric object a in each basis represented by some square matrix aij and such that
3 3
p
q
ij 
components of its matrix aij obey transformation rules a  S S a pq and a =

i
j
ij
p 1 q 1


3 3
 T T a  pq under a change of basis is called a ..................
p
q
j
i
p 1 q 1


3. The coordinates of a vector are numerated by one upper index, which is called the ..................
4. The number of indices and their positions determine the .................., i.e. the way the
components of each particular tensor behave under a change of basis.
15.15 Review Questions
1. Explain why the scalar product a,x is sometimes called the bilinear function of vectorial
argument x and covectorial argument a. In this capacity, it can be denoted as f(a, x).
Remember our discussion about functions with non-numeric arguments.
y 1 F 1 1 F 2 1 F 3 1 x 1 3
j
2
2. Derive y  F 1 2 F 2 2 F 3 2 x 2 from y =  F x .
i
i
j
y 3 F 1 3 F 2 3 F 3 3 x 3 j 1

3. Let  be some real number and let x and y be two vectors. Prove the following properties
of a linear operator:
(1) F(x + y) = F(x) + F(y),
(2) F( x) = F(x).
4. Find the matrix of composite operator F  H if the matrices for F and H in the basis e , e ,
1
2
e are known.
3
5. Remember the definition of the identity map in mathematics (see on-line Math.
Encyclopedia) and define the identity operator id. Find the matrix of this operator.
6. Remember the definition of the inverse map in mathematics and define inverse operator
F for linear operator F. Find the matrix of this operator if the matrix of F is known.
–1
7. Let a be the matrix of some bilinear form a. Let’s denote by b components of inverse
ij
ij
matrix for a . Prove that matrix b under a change of basis transforms like matrix of twice-
ij
ij
contravariant tensor. Hence it determines tensor b of valency (2, 0). Tensor b is called a
dual bilinear form for a.
8. By analogy with exercise Y 1 i ...i r   X 1 i ...i r prove the consistence of formula
1 j ...j s 1 j ....j s
Z 1 i ...i r p    X 1 j ....j r s Y i  r 1 ...i s p  .
1 i ...i
r p
s 1 ...j
1 j ...j
s p
j 

9. Give an example of two tensors such that X  Y  Y  X.

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