**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

Find the Jacobian J.

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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

Eqs.(1) can be manipulated to give

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

The Jacobian J is the determinant of the matrix as shown below:

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