Mathematical Physics Vol 1

Chapter 4. Field theory

112

The differential of this vector can be expressed in one of the following ways (depending on whether the coordinates are Cartesian or some other - generalized coordinates)

d r = d x i + d y j + d z k =

y

3 ∑ i = 1 3 ∑ i = 1

3 ∑ i = 1

∂ r ∂ q i

d q i =

i =

g i d q

=

(4.138)

∆ s

i .

h i e i d q

=

∆ r

∆ y

If the Cartesian coordinates are denoted by x 1 , x 2 and x 3 , instead of x , y and z , respectively, for the arc element (via Cartesian coordinates) we obtain

∆ x

r + ∆ r

x

r

3 ∑ i , j = 1

3 ∑ i = 1

d s 2 =

i d x j =

d x i d x i

δ i j d x

(4.139)

O

where δ i j is the Kronecker delta symbol.

Figure 4.20: Arc element.

We can then determine d x i , according to (4.127), as

3 ∑ j = 1

∂ x i ∂ q j

x i = x i ( q j ) ⇒ d x i =

d q j ,

(4.140)

and obtain the arc element

3 ∑ i , j

3 ∑ i , j = 1

3 ∑ k , l = 1

∂ x i ∂ q k ·

∂ x j ∂ q l

ds 2 =

i d x j =

d q k d q l =

δ i j d x

δ i j

∂ x j ∂ q l !

3 ∑ k , l = 1 3 ∑ k , l = 1

3 ∑ i , j = 1

∂ x i ∂ q k ·

d q k d q l =

δ i j

(4.141)

=

k d q l .

g kl d q

=

On the other hand, it follows from (4.138) that

3 ∑ k , l = 1

d s 2 = d r · d r =

k d q l .

g k · g l d q

(4.142)

Comparing (4.142) and (4.141) we can conclude that g k · g l = g kl . Definition The variables g kl defined by the relations

3 ∑ i , j = 1

∂ x i ∂ q k ·

∂ x j ∂ q l

δ i j

g kl = g k · g l =

(4.143)

represent the base metric tensor or base space tensor.