Page 41 - Intro to Tensor Calculus

37

Coordinate Transformations

Consider a coordinate transformation from a set of coordinates (x, y, z)to (u, v, w) defined by a set of
transformation equations
x = x(u, v, w)
y = y(u, v, w) (1.2.8)
z = z(u, v, w)

It is assumed that these transformations are single valued, continuous and possess the inverse transformation

u = u(x, y, z)
v = v(x, y, z) (1.2.9)
w = w(x, y, z).

These transformation equations define a set of coordinate surfaces and coordinate curves. The coordinate
surfaces are defined by the equations
u(x, y, z)= c 1
(1.2.10)
v(x, y, z)= c 2
w(x, y, z)= c 3
where c 1 ,c 2 ,c 3 are constants. These surfaces intersect in the coordinate curves

r
~(u, c 2 ,c 3 ), ~(c 1 ,v,c 3 ), ~(c 1 ,c 2 ,w), (1.2.11)
r
r
where
~(u, v, w)= x(u, v, w) b e 1 + y(u, v, w) b e 2 + z(u, v, w) b e 3 .
r
The general situation is illustrated in the figure 1.2-1.
Consider the vectors

~ 1
~ 3
~ 2
E = gradu = ∇u, E = gradv = ∇v, E = gradw = ∇w (1.2.12)
evaluated at the common point of intersection (c 1 ,c 2 ,c 3 ) of the coordinate surfaces. The system of vectors
~ 1 ~ 2 ~ 3
(E , E , E ) can be selected as a system of basis vectors which are normal to the coordinate surfaces.
Similarly, the vectors
r
r
r
∂~ ∂~ ∂~
~
~
~
E 1 = , E 2 = , E 3 = (1.2.13)
∂u ∂v ∂w
~
~
~
when evaluated at the common point of intersection (c 1 ,c 2 ,c 3 ) forms a system of vectors (E 1 , E 2 , E 3 ) which
we can select as a basis. This basis is a set of tangent vectors to the coordinate curves. It is now demonstrated
~
~
~ 1 ~ 2 ~ 3
~
that the normal basis (E , E , E ) and the tangential basis (E 1 , E 2 , E 3 ) are a set of reciprocal bases.
Recall that ~ = x b e 1 + y b e 2 + z b e 3 denotes the position vector of a variable point. By substitution for
r
x, y, z from (1.2.8) there results
~ = ~(u, v, w)= x(u, v, w) b e 1 + y(u, v, w) b e 2 + z(u, v, w) b e 3 . (1.2.14)
r
r