Page 72 - Intro to Tensor Calculus

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r
r
∂~ ∂~ r ∂~
and consequently d~ = du + dv + dw, where
r
∂u ∂v ∂w
∂~ r ∂x ∂y ∂z
~
E 1 = = b e 1 + b e 2 + b e 3
∂u ∂u ∂u ∂u
r
∂~ ∂x ∂y ∂z
~
E 2 = = b e 1 + b e 2 + b e 3 (1.3.11)
∂v ∂v ∂v ∂v
r
∂~ ∂x ∂y ∂z
~
E 3 = = b e 1 + b e 2 + b e 3 .
∂w ∂w ∂w ∂w
are tangent vectors to the coordinate curves. The element of arc length in the curvilinear coordinates is
r
r
r
∂~ ∂~ r ∂~ ∂~ ∂~ ∂~ r
r
2
ds = d~r · d~r = · dudu + · dudv + · dudw
∂u ∂u ∂u ∂v ∂u ∂w
r
r
r
r
∂~ ∂~ ∂~ r ∂~ ∂~ ∂~
r
+ · dvdu + · dvdv + · dvdw (1.3.12)
∂v ∂u ∂v ∂v ∂v ∂w
∂~ ∂~ ∂~ ∂~ ∂~ ∂~
r
r
r
r
r
r
+ · dwdu + · dwdv + · dwdw.
∂w ∂u ∂w ∂v ∂w ∂w
Utilizing the summation convention, the above can be expressed in the index notation. Define the
quantities
r
r
∂~ ∂~ ∂~ r ∂~ r ∂~ r ∂~ r
g 11 = · g 12 = · g 13 = ·
∂u ∂u ∂u ∂v ∂u ∂w
∂~ ∂~ ∂~ r ∂~ ∂~ ∂~
r
r
r
r
r
g 21 = · g 22 = · g 23 = ·
∂v ∂u ∂v ∂v ∂v ∂w
∂~ ∂~ ∂~ r ∂~ ∂~ r ∂~
r
r
r
r
g 31 = · g 32 = · g 33 = ·
∂w ∂u ∂w ∂v ∂w ∂w
1
2
3
and let x = u, x = v, x = w. Then the above element of arc length can be expressed as
i
j
i
2
~
~
j
ds = E i · E j dx dx = g ij dx dx , i, j =1, 2, 3
where
∂~ r ∂~ r ∂y m ∂y m
~
~
g ij = E i · E j = · = , i, j free indices (1.3.13)
i
∂x i ∂x j ∂x ∂x j
are called the metric components of the curvilinear coordinate system. The metric components may be
thought of as the elements of a symmetric matrix, since g ij = g ji . In the rectangular coordinate system
2
2
2
2
x, y, z, the element of arc length squared is ds = dx + dy + dz . In this space the metric components are
 
10 0
g ij =  01 0  .
00 1