**Problem 1B-4*:** Transformation from two-dimensional rectangular
coordinate representation to polar coordinate representation and the
transformation Jacobian

A two dimensional position vector can be expressed in the rectangular coordinate representation as (x, y) or in the polar coordinate representation as (r, θ).

(a) Derive the relations between the (x, y) pair and the (r, θ) pair.

(b) An integral expressed in the rectangular coordinate representation can be transformed into the polar coordinate representation as

Find the Jacobian, J, of this transformation.

Scroll down for solution

**Solution:**

(a) The relation between the rectangular representation of a position vector (x, y) and the polar coordinate representation (r, θ) is shown in the right picture. From the picture we see that

(b) From Eq. (1) we have

We know from calculus that the Jacobian, J, for the transformation from (x, y) to (r, θ) representation is the determinant

Thus

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