Mathematical Physics Vol 1

4.6 Examples

201

These vectors can be expressed in terms of Lame’s coefficients h i and unit vectors e i (see p. 110, equation (4.133)), and it follows that the value of the determinant is (4.222) ∂ x ∂ u 1 ∂ y ∂ u 1 ∂ z ∂ u 1 ∂ x ∂ u 2 ∂ y ∂ u 2 ∂ z ∂ u 2 ∂ x ∂ u 3 ∂ y ∂ u 3 ∂ z ∂ u 3 = h 1 e 1 · ( h 2 e 2 × h 3 e 3 )= h 1 h 2 h 3 e 1 · ( e 2 × e 3 ) . As e i are orthonormalized vectors, it follows that e 1 · ( e 2 × e 3 )= 1, and we finally obtain

∂ x ∂ u 1 ∂ x ∂ u 2 ∂ x ∂ u 3

∂ y ∂ u 1 ∂ y ∂ u 2 ∂ y ∂ u 3

∂ z ∂ u 1 ∂ z ∂ u 2 ∂ z ∂ u 3

J =

= h 1 h 2 h 3 .

R Note. if the Jacobian is identically equal to zero then ∂ r ∂ u 1 , ∂ r ∂ u 2 and ∂ r ∂ u 3 are coplanar vectors, namely, they lie in one plane and are linearly dependent. Thus, in that case, x , y and z are not independent, that is, there exists a function in the form F ( x , y , z )= 0. The opposite is also true. Thus, J̸ = 0 is the necessary and sufficient condition for the following coordinate transformation to exist

x = x ( u 1 , u 2 , u 3 ) , y = y ( u 1 , u 2 , u 3 ) , z = z ( u 1 , u 2 , u 3 )

as well as its inverse transformation

u i = u i ( x , y , z ) ,

i = 1 , 2 , 3 .

Exercise 154 Let u 1 , u 2 and u 3 be the generalized curvilinear coordinates. Prove that ∂ r ∂ u 1 , ∂ r ∂ u 2 , ∂ r ∂ u 3 and ∇ u 1 , ∇ u 2 , ∇ u 3 are reciprocal vectors. Solution The necessary and sufficient condition for the vectors to be reciprocal is ∂ r ∂ u p · ∇ u q = 1 , for p = q , 0 , for p̸ = q , p , q = 1 , 2 , 3 . In this case d r = ∂ r ∂ u 1 d u 1 + ∂ r ∂ u 2 d u 2 + ∂ r ∂ u 3 d u 3 ,