Page 60 - Intro to Tensor Calculus

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Figure 1.2-4. Cylindrical coordinates (r, β, z).

I 6. For the cylindrical coordinates (r, β, z) illustrated in the figure 1.2-4.
(a) Write out the transformation equations from rectangular (x, y, z) coordinates to cylindrical (r, β, z)
coordinates. Also write out the inverse transformation.
(b) Determine the following basis vectors in cylindrical coordinates and represent your results in terms of
cylindrical coordinates.
~ ~ ~ ~ 1 ~ 2 ~ 3
(i) The tangential basis E 1 , E 2 , E 3 . (ii)The normal basis E , E , E . (iii) ˆ e r , ˆ e β , ˆ e z
where ˆ e r , ˆ e β , ˆ e z are normalized vectors in the directions of the tangential basis.
~
(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
~ 2
~ 3
~
~ 1
A = A 1 E + A 2 E + A 3 E
~
A = A r ˆ e r + A β ˆ e β + A z ˆ e z
depending upon the basis vectors selected . In terms of the components A x ,A y ,A z
2
3
1
(i) Solve for the contravariant components A ,A ,A .
(ii) Solve for the covariant components A 1 ,A 2 ,A 3 .
(iii) Solve for the components A r ,A β ,A z . Express all results in cylindrical coordinates. (Note the
components A r ,A β ,A z are referred to as physical components. Physical components are considered in
more detail in a later section.)