Mathematical Physics Vol 1

Chapter 4. Field theory

200

by the following relations

x = x ( u 1 , u 2 , u 3 ) , y = y ( u 1 , u 2 , u 3 ) , z = z ( u 1 , u 2 , u 3 ) , where x ( u i ) , y ( u i ) and z ( u i ) , are continuous functions differentiable by u i , i = 1 , 2 , 3, in a region V , then the total differentials are

∂ x ∂ u 1 ∂ y ∂ u 1 ∂ z ∂ u 1

∂ x ∂ u 2 ∂ y ∂ u 2 ∂ z ∂ u 2 ∂ x ∂ u 2 ∂ y ∂ u 2 ∂ z ∂ u 2

∂ x ∂ u 3 ∂ y ∂ u 3 ∂ z ∂ u 3

d u 1 +

d u 2 +

d u 3 ,

d x =

d u 1 +

d u 2 +

d u 3 ,

d y =

d u 1 +

d u 2 +

d u 3 ,

d z =

or in matrix form

       

    

∂ x ∂ u 3 ∂ y ∂ u 3 ∂ z ∂ u 3 ∂ x ∂ u 3 ∂ y ∂ u 3 ∂ z ∂ u 3

∂ x ∂ u 1 ∂ y ∂ u 1 ∂ z ∂ u 1 ∂ x ∂ u 1 ∂ y ∂ u 1 ∂ z ∂ u 1

   d x d y d z

   =

   .

  d u 1 d u 2 d u 3

The square matrix

∂ x ∂ u 2 ∂ y ∂ u 2 ∂ z ∂ u 2     is the variable transformation matrix. Its determinant is the so called functional determinant or Jacobian,, symbolically denoted by ∂ ( x , y , z ) ∂ ( u 1 , u 2 , u 3 ) . The elements of this determinant can be related to the tangent base vectors of the coordinate axes ∂ r ∂ u i = ∂ ∂ u i ( x i + y j + z k )= ∂ x ∂ u i i + ∂ y ∂ u i j + ∂ z ∂ u i k . As a mixed product can be expressed in terms of a formal determinant (see Example 15, p. 64), it follows that the functional determinant is equal to the mixed product ∂ 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 = ∂ r ∂ u 1 · ∂ r ∂ u 2 × ∂ r ∂ u 3 . (4.222) J =