Page 97 - Intro to Tensor Calculus

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These components are needed to calculate the physical components associated with a covariant tensor of
1
order one. For example, in the x −direction, we have the covariant components
1
2
1
λ 1 = g 11 λ = h 1 = h 1 , λ 2 =0, λ 3 =0
h 1
and consequently the projection in V 3 can be represented
1 A 1
i j
i jm
jm
11
g ij A λ = g ij A g λ m = A j g λ m = A 1 λ 1 g = A 1 h 1 = = A(1).
h 2 h 1
1
In a similar manner we calculate the relations
A 2 A 3
A(2) = and A(3) =
h 2 h 3
3
2
for the other physical components in the directions x and x . These physical components can be represented
in the short hand notation
A (i) A (i)
A(i)= = √ , i =1, 2 or 3 no summation on i.
h (i) g (i)(i)
In an orthogonal coordinate system the physical components associated with both the contravariant and
i
covariant components are the same. To show this we note that when A g ij = A j is summed on i we obtain
1
2
3
A g 1j + A g 2j + A g 3j = A j .
Since g ij =0 for i 6= j this equation reduces to

(i)
A g (i)(i) = A (i) , i not summed.
Another form for this equation is

(i) √ A (i)
A(i)= A g (i)(i) = √ i not summed,
g (i)(i)
which demonstrates that the physical components associated with the contravariant and covariant compo-
nents are identical.
NOTATION The physical components are sometimes expressed by symbols with subscripts which represent
i
the coordinate curve along which the projection is taken. For example, let H denote the contravariant
components of a first order tensor. The following are some examples of the representation of the physical
i
components of H in various coordinate systems:
orthogonal coordinate tensor physical
coordinates system components components

3
2
1
general (x ,x ,x ) H i H(1),H(2),H(3)
rectangular (x, y, z) H i H x ,H y ,H z
cylindrical (r, θ, z) H i H r ,H θ ,H z
spherical (ρ, θ, φ) H i H ρ ,H θ ,H φ
general (u, v, w) H i H u ,H v ,H w