Problem 1B-05**: Transformation of three-dimensional rectangular coordinates into three-dimensional polar coordinates and the transformation Jacobian

A three-dimensional position vector can be represented by the rectangular coordinate system as (x, y, z) or by the polar coordinate system as (r, θ, φ).

(a) What are the relations between (x, y, z) and (r, θ, φ)?

(b) An integral with regard to x, y and z can be transformed into an integral with regard to r, θ and φ as

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Solution:

(a) A position vector is expressed as in the right figure with its rectangular coordinate representation and its polar coordinate represention indicated explicitly. From the figure we have

(b) Performing partial differenciations on the relations of Eqs.(1), we have

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