Page 61 - Intro to Tensor Calculus

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Figure 1.2-5. Spherical coordinates (ρ, α, β).

I 7. For the spherical coordinates (ρ, α, β) illustrated in the figure 1.2-5.
(a) Write out the transformation equations from rectangular (x, y, z) coordinates to spherical (ρ, α, β)co-
ordinates. Also write out the equations which describe the inverse transformation.
(b) Determine the following basis vectors in spherical coordinates
~
~
~
(i) The tangential basis E 1 , E 2 , E 3 .
~ 1 ~ 2 ~ 3
(ii) The normal basis E , E , E .
(iii) ˆ e ρ , ˆ e α , ˆ e β which are normalized vectors in the directions of the tangential basis. Express all results
in terms of spherical coordinates.
~
(c) A vector A = A x b e 1 + A y b e 2 + A z b e 3 can be represented in any of the forms:
~ 1 ~ 2 ~ 3 ~
A = A E 1 + A E 2 + A E 3
~
~ 1
~ 3
~ 2
A = A 1 E + A 2 E + A 3 E
~
A = A ρ ˆ e ρ + A α ˆ e α + A β ˆ e β
depending upon the basis vectors selected . Calculate, in terms of the coordinates (ρ, α, β)and the
components A x ,A y ,A z
3
2
1
(i) The contravariant components A ,A ,A .
(ii) The covariant components A 1 ,A 2 ,A 3 .
(iii) The components A ρ ,A α ,A β which are called physical components.
1
3
2
I 8. Work the problems 6,7 and then let (x ,x ,x )= (r, β, z) denote the coordinates in the cylindrical
3
2
1
system and let (x , x , x )= (ρ, α, β) denote the coordinates in the spherical system.
(a) Write the transformation equations x → x from cylindrical to spherical coordinates. Also find the
inverse transformations. ( Hint: See the figures 1.2-4 and 1.2-5.)
(b) Use the results from part (a) and the results from problems 6,7 to verify that
∂x j
A i = A j i for i =1, 2, 3.
∂x
¯
(i.e. Substitute A j from problem 6 to get A i given in problem 7.)