Page 96 - Intro to Tensor Calculus

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Physical Components For Orthogonal Coordinates

In orthogonal coordinates observe the element of arc length squared in V 3 is

j
2
2
i
2
3 2
2 2
1 2
2
ds = g ij dx dx =(h 1 ) (dx ) +(h 2 ) (dx ) +(h 3 ) (dx )
where  
(h 1 ) 2 0 0
g ij =  0 (h 2 ) 2 0  . (1.3.60)
0 0 (h 3 ) 2
In this case the curvilinear coordinates are orthogonal and
h 2 i not summed and g ij =0,i 6= j.
(i) = g (i)(i)
i
At an arbitrary point in this coordinate system we take λ ,i =1, 2, 3 as a unit vector in the direction
1
of the coordinate x . We then obtain
dx 1 2 3
1
λ = , λ =0, λ =0.
ds
This is a unit vector since
1 2
2
i j
1 1
1= g ij λ λ = g 11 λ λ = h (λ )
1
1
or λ = 1 . Here the curvilinear coordinate system is orthogonal and in this case the physical component
h 1
1
i
i
i
i
of a vector A , in the direction x , is the projection of A on λ in V 3 . The projection in the x direction is
determined from
1 1
2
1
1 1
i j
A(1) = g ij A λ = g 11 A λ = h A = h 1 A .
1
h 1
3
2
i
i
Similarly, we choose unit vectors µ and ν ,i =1, 2, 3in the x and x directions. These unit vectors
can be represented 2 3
1
µ =0, µ = dx = 1 , µ =0
2
ds h 2 dx 3 1
1
3
ν =0, 2 ν = =
ν =0, ds h 3
i
and the physical components of the vector A in these directions are calculated as
3
A(2) = h 2 A 2 and A(3) = h 3 A .
In summary, we can say that in an orthogonal coordinate system the physical components of a contravariant
tensor of order one can be determined from the equations
(i)
A(i)= h (i) A (i) = √ g (i)(i) A , i =1, 2 or 3 no summation on i,
3
2
1
which is a short hand notation for the physical components (h 1 A ,h 2 A ,h 3 A ). In an orthogonal coordinate
system the nonzero conjugate metric components are
1
(i)(i)
g = , i =1, 2,or 3 no summation on i.
g (i)(i)