Mathematical Physics Vol 1

4.6 Examples

213

Also

3 = c

∂ x

∂ u 1 ∂ x

∂ u 2 ∂ x

∂ u 3

1 + c

2 + c

c 1 ∇ u

2 ∇ u

3 ∇ u

i +

+ c 2

+ c 3

1

(4.238)

∂ y

j + c 1

∂ z

+ c 1

∂ u 1 ∂ y

∂ u 2 ∂ y

∂ u 3

∂ u 1 ∂ z

∂ u 2 ∂ z

∂ u 3

k ,

+ c 2

+ c 3

+ c 2

+ c 3

or

∂ x

3 = c

∂ u 1 ∂ x

∂ u 2 ∂ x

∂ u 3

1 + c

2 + c

c 1 ∇ u

2 ∇ u

3 ∇ u

i +

+ c 2

+ c 3

1

(4.239)

+ c 1

∂ y

j + c 1

∂ z

∂ u 1 ∂ y

∂ u 2 ∂ y

∂ u 3

∂ u 1 ∂ z

∂ u 2 ∂ z

∂ u 3

k .

+ c 2

+ c 3

+ c 2

+ c 3

Equalizing coefficients next to i , j , k we obtain  c 1 ∂ u 1 ∂ x + c 2 ∂ u 2 ∂ x + c 3 ∂ u 3 ∂ x = c 1

∂ u 1 ∂ x ∂ u 1 ∂ y ∂ u 1 ∂ z

∂ u 2 ∂ x ∂ u 2 ∂ y ∂ u 2 ∂ z

∂ u 3 ∂ x ∂ u 3 ∂ y ∂ u 3 ∂ z

+ c 2

+ c 3

   

,

∂ u 1 ∂ y ∂ u 1 ∂ z

∂ u 2 ∂ y ∂ u 2 ∂ z

∂ u 3 ∂ y ∂ u 3 ∂ z

(4.240)

c 1

+ c 2

+ c 3

= c 1

+ c 2

+ c 3

,

+ c 2

+ c 3

c 1

= c 1

+ c 2

+ c 3

.

From equations (4.237) and (4.240), by equalizing the coefficients, we obtain     c 1 = c 1 ∂ u 1 ∂ u 1 + c 2 ∂ u 2 ∂ u 1 + c 3 ∂ u 3 ∂ u 1 , c 2 = c 1 ∂ u 1 ∂ u 2 + c 2 ∂ u 2 ∂ u 2 + c 3 ∂ u 3 ∂ u 2 , c 3 = c 1 ∂ u 1 ∂ u 3 + c 2 ∂ u 2 ∂ u 3 + c 3 ∂ u 3 ∂ u 3 , which can be written as

(4.241)

∂ u 1 ∂ u p

∂ u 2 ∂ u p

∂ u 3 ∂ u p

c p = c 1

+ c 2

+ c 3

(4.242)

,

or

3 ∑ q = 1

∂ u q ∂ u p

c p =

c q

p = 1 , 2 , 3 .

(4.243)

And analogously

3 ∑ q = 1

∂ u q ∂ u p

c p =

c q

p = 1 , 2 , 3 .

(4.244)

Definition If the coordinate transformation u i = u i ( u j ) transforms the system c law (4.244), then this system defines a covariant tensor of the first order .

i according to the