Question

Find the gradient of the function W if W = ρzcos(ϕ) if W is in cylindrical coordinates.

(a) zcos(ϕ)aρ – z sin(ϕ) aΦ + ρcos(ϕ) az

(b) zcos(ϕ)aρ – sin(ϕ) aΦ + cos(ϕ) az

(c) zcos(ϕ)aρ + z sin(ϕ) aΦ + ρcos(ϕ) az

(d) zcos(ϕ)aρ + z sin(ϕ) aΦ + cos(ϕ) az

I had been asked this question in quiz.

The origin of the question is Gradient of a Function and Conservative Field topic in portion Vector Differential Calculus of Engineering Mathematics

Answer 1

Right option is (a) zcos(ϕ)aρ – z sin(ϕ) aΦ + ρcos(ϕ) az

For explanation: \(W = ρzcos(ϕ) \)

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

\(= zcos(ϕ)a_ρ – z sin(ϕ) a_y + ρcos(ϕ) a_z \)