Page 62 - Intro to Tensor Calculus

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(c) Use the results from part (a) and the results from problems 6,7 to verify that

i j ∂x i
A = A for i =1, 2, 3.
∂x j
¯ i
j
(i.e. Substitute A from problem 6 to get A given by problem 7.)
I 9. Pick two arbitrary noncolinear vectors in the x, y plane, say

~ ~
V 1 =5 b e 1 + b e 2 and V 2 = b e 1 +5 b e 2
~
~
~
~
~
and let V 3 = b e 3 be a unit vector perpendicular to both V 1 and V 2 . The vectors V 1 and V 2 can be thought of
as defining an oblique coordinate system, as illustrated in the figure 1.2-6.
~ 1 ~ 2 ~ 3
(a) Find the reciprocal basis (V , V , V ).
(b) Let
r ~ ~ ~
~ = x b e 1 + y b e 2 + z b e 3 = αV 1 + βV 2 + γV 3
and show that
5x y
α = −
24 24
x 5y
β = − +
24 24
γ = z
(c) Show
x =5α + β
y = α +5β
z = γ

(d) For γ = γ 0 constant, show the coordinate lines are described by α = constant and β = constant,
and sketch some of these coordinate lines. (See figure 1.2-6.)
(e) Find the metrics g ij and conjugate metrices g ij associated with the (α, β, γ) space.

Figure 1.2-6. Oblique coordinates.