Mathematical Physics Vol 1

Chapter 4. Field theory

212

or

    1 ∂ u p ∂ u 1

1 ∂ u 1 2 ∂ u 1 3 ∂ u 1

1 ∂ u 2 2 ∂ u 2 3 ∂ u 2

1 ∂ u 3 2 ∂ u 3 3 ∂ u 3

1 ∂ u

2 ∂ u

3 ∂ u

C 1 = C

+ C

+ C

,

1 ∂ u

2 ∂ u

3 ∂ u

C 2 = C

(4.232)

+ C

+ C

,

1 ∂ u

2 ∂ u

3 ∂ u

C 3 = C

+ C

+ C

,

or shortly

p ∂ u 2

p ∂ u 3

2 ∂ u

3 ∂ u

C p = C

+ C

+ C

p = 1 , 2 , 3 ,

(4.233)

,

or

3 ∑ q = 1

∂ u p ∂ u q

C q

C p =

p = 1 , 2 , 3 .

(4.234)

,

In the same way we obtain

3 ∑ q = 1

∂ u p ∂ u q

C p

C q

p = 1 , 2 , 3 .

(4.235)

=

,

Definition If the coordinate transformation u i = u i ( u j ) transforms the system C i according to the law (4.235), then this system defines a contravariant tensor of the first order . From a geometrical point of view C i determines contravariant coordinates of vector C in terms of the base of the coordinate system u i , namely C = C i g i . This uswhy C i is often called contravariant vector in literature. Exercise 168 Determine how a covariant tensor (covariant vector coordinates) is transformed within a coordinate transformation. Solution Let us write the covariant components of vector A in systems ( u 1 , u 2 , u 3 ) and ( u 1 , u 2 , u 3 ) A = c 1 ∇ u 1 + c 2 ∇ u 2 + c 3 ∇ u 3 = c 1 ∇ u 1 + c 2 ∇ u 2 + c 3 ∇ u 3 . (4.236) Given that u p = u p ( u 1 , u 2 , u 3 ) , where p = 1 , 2 , 3, it follows that  ∂ u p ∂ x = ∂ u p ∂ u 1 ∂ u 1 ∂ x + ∂ u p ∂ u 2 ∂ u 2 ∂ x + ∂ u p ∂ u 3 ∂ u 3 ∂ x ,

   

∂ u p ∂ y ∂ u p ∂ z

∂ u p ∂ u 1 ∂ u p ∂ u 1

∂ u 1 ∂ y ∂ u 1 ∂ z

∂ u p ∂ u 2 ∂ u p ∂ u 2

∂ u 2 ∂ y ∂ u 2 ∂ z

∂ u p ∂ u 3 ∂ u p ∂ u 3

∂ u 3 ∂ y ∂ u 3 ∂ z

(4.237)

=

+

+

,

=

+

+

.