Page 73 - Intro to Tensor Calculus

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EXAMPLE 1.3-3. (Cylindrical coordinates (r, θ, z))
The transformation equations from rectangular coordinates to cylindrical coordinates can be expressed
1
2
3
3
2
1
as x = r cos θ, y = r sin θ, z = z. Here y = x, y = y, y = z and x = r, x = θ, x = z, and the
r
position vector can be expressed ~ = ~(r, θ, z)= r cos θ b e 1 + r sin θ b e 2 + z b e 3 . The derivatives of this position
r
vector are calculated and we find
∂~ ∂~ ∂~
r
r
r
~
~
~
E 1 = =cos θ b e 1 +sin θ b e 2 , E 2 = = −r sin θ b e 1 + r cos θ b e 2 , E 3 = = b e 3 .
∂r ∂θ ∂z
From the results in equation (1.3.13), the metric components of this space are
 
1 0 0
g ij =  0 r 2 0  .
0 0 1
We note that since g ij =0 when i 6= j, the coordinate system is orthogonal.
Given a set of transformations of the form found in equation (1.3.10), one can readily determine the
metric components associated with the generalized coordinates. For future reference we list several differ-
ent coordinate systems together with their metric components. Each of the listed coordinate systems are
orthogonal and so g ij =0 for i 6= j. The metric components of these orthogonal systems have the form
 2 
h 1 0 0
g ij =  0 h 2 2 0 
0 0 h 2 3

and the element of arc length squared is

2
2
2 2
2
3 2
1 2
2
ds = h (dx ) + h (dx ) + h (dx ) .
3
1
2
1. Cartesian coordinates (x, y, z)
x = x h 1 =1
y = y h 2 =1
z = z h 3 =1
The coordinate curves are formed by the intersection of the coordinate surfaces
x =Constant, y =Constant and z =Constant.