Question

Find gradient of B if B = rθϕ if X is in spherical coordinates.

(a) \(θϕ \, a_r – ϕ \,a_θ + \frac{θ}{sin(θ)} a_Φ \)

(b) \(rθϕ \, a_r – ϕ \,a_θ + r \frac{θ}{sin(θ)}  a_Φ \)

(c) \(θϕ \, a_r – ϕr \,a_θ +  \frac{θ}{sin(θ)} a_Φ \)

(d) \(θϕr \, a_r – ϕ \,a_θ +  r\frac{θ}{sin(θ)} a_Φ \)

I had been asked this question in an interview.

The above asked question is from Gradient of a Function and Conservative Field in division Vector Differential Calculus of Engineering Mathematics

Answer 1

Right choice is (a) \(θϕ \, a_r – ϕ \,a_θ + \frac{θ}{sin(θ)} a_Φ \)

The best explanation: Gradient of \(B =  \frac{∂B}{∂r} a_r + \frac{1}{r} \frac{∂B}{∂θ} a_θ + \frac{1}{rsin(θ)} \frac{∂B}{∂ϕ} a_Φ \)

\(B = rθϕ\)

Hence gradient of \(B = θϕ \, a_r -ϕ \, a_θ +  \frac{θ}{sin(θ)}  a_Φ \)