Mathematical Physics Vol 1

4.6 Examples

199

62), the area formed by the two lengths d s 1 andd s 2 is given by d A 1 = ( h 2 d u 2 e 2 ) × ( h 3 d u 3 e 3 ) = = h 2 h 3 d u 2 d u 3 | e 2 × e 3 | = h 2 h 3 d u 2 d u 3 | e 1 | = = h 2 h 3 d u 2 d u 3 .

Similarly, for the remaining two areas we obtain

3 d u 1 , 1 d u 2 ,

d A 2 = h 3 h 1 d u d A 3 = h 1 h 2 d u

or more shortly expressed

3 ∑ j , k = 1

j d u k ,

d A i =

e i jk h j h k d u

where e i jk is the alternation tensor, defined by e i jk =    e 123 =+ 1;

+ 1 , if i jk is an even permutation of indices 1,2,3 , − 1 , if i jk is an odd permutation of indices 1,2,3 , 0 , in all other cases .

Exercise 153 Let u 1 , u 2 and u 3 be the generalized orthogonal coordinates. Prove that the Jacobian of the transformation, symbolically denoted in one of the two following forms

∂ x ∂ u 1 ∂ x ∂ u 2 ∂ x ∂ u 3

∂ y ∂ u 1 ∂ y ∂ u 2 ∂ y ∂ u 3

∂ z ∂ u 1 ∂ z ∂ u 3 ∂ z ∂ u 3

∂ ( x , y , z ) ∂ ( u 1 , u 2 , u 3 )

J =

=

is equal to

∂ ( x , y , z ) ∂ ( u 1 , u 2 , u 3 )

J =

= h 1 h 2 h 3 .

Solution If we introduce new variables u 1 , u 2 and u 3 instead of Cartesian coordinates x , y and z