Question

Find the gradient of A if A = ρ^2 + z^3 + cos(ϕ) + z and A is in cylindrical coordinates.

(a) \(2ρz^3 \, a_ρ – \frac{1}{ϕ} sin(ϕ) \, aΦ + 3ρ^2 z^2 \, a_z \)

(b) \(2ρz^3 \, a_ρ – \frac{1}{ρ} sin(ϕ) \, aΦ + 3ρ^2 z^2+1 \, a_z \)

(c) \(2ρz^3 \, a_ρ – \frac{1}{ϕ} sin(ϕ) \, aΦ + 3ρ^2 z^2+1 \, a_z \)

(d) \(2ρz^3 \, a_ρ – \frac{1}{ρ} sin(ϕ) \, aΦ + 3ρ^2 z^2 \, a_z \)

This question was addressed to me by my college professor while I was bunking the class.

My question is taken from Gradient of a Function and Conservative Field in chapter Vector Differential Calculus of Engineering Mathematics

Answer 1

The correct option is (b) \(2ρz^3 \, a_ρ – \frac{1}{ρ} sin(ϕ) \, aΦ + 3ρ^2 z^2+1 \, a_z \)

To explain: \(A = ρ^2 + z^3 + cos(ϕ) + z \)

Gradient of \(A = \frac{1}{ρ}  \frac{∂A}{∂ρ} a_ρ + \frac{1}{ρ}  \frac{∂A}{∂ϕ}a_y + \frac{∂A}{∂z}a_z \)

\(=2ρz^3 \, a_ρ – \frac{1}{ρ} sin(ϕ) \, aΦ + 3ρ^2 z^2 + 1 \, a_z \)