Question

Convert U = xyz + y + xz into Cylindrical coordinates.

(a) zρ^3sin⁡φ  cos⁡φ + ρsin⁡φ + zρcos⁡φ

(b) zρ^2sin⁡φ  cos⁡φ + ρsin⁡φ + zρcos⁡φ

(c) zρ^3sin⁡φ  cos⁡φ + ρ2sin⁡φ + zρcos⁡φ

(d) zρ^2sin⁡φ  cos⁡φ + ρ2sin⁡φ + zρcos⁡φ

I had been asked this question in an interview for internship.

This intriguing question comes from Conversion from Cartesian, Cylindrical and Spherical Coordinates topic in chapter Vector Differential Calculus of Engineering Mathematics

Answer 1

Correct answer is (b) zρ^2sin⁡φ  cos⁡φ + ρsin⁡φ + zρcos⁡φ

For explanation: x=ρcos⁡φ, y=ρsin⁡φ and z=z. Using this formula we should substitute the values for x and y and change it to Cylindrical coordinates.

U = xyz + y + xz

U = ρcos⁡φ ρsin⁡φ z+ ρsin⁡φ + ρcos⁡φz

U = zρ^2sin⁡φ  cos⁡φ + ρsin⁡φ + zρcos⁡φ.