Question

Find gradient of B if B = ϕln(r) + r^2 ϕ if B is in spherical coordinates.

(a) \(\frac{ρ}{r}+ 2rθ \,a_r – r a_θ + \frac{lnr}{rsin(θ)}  a_Φ \)

(b) \(\frac{ρ}{r}+ 2rϕ \,a_r – r a_θ + \frac{lnr}{rsin(θ)}  a_Φ \)

(c) \(\frac{ρ}{r}+ 2rθ \,a_r – r^2 a_θ + \frac{lnr}{rsin(θ)}  a_Φ \)

(d) \( \frac{ρ}{r}+ 2rϕ \,a_r – r^2 a_θ + \frac{lnr}{rsin(θ)}  a_Φ \)

This question was posed to me by my school teacher while I was bunking the class.

I'd like to ask this question from Gradient of a Function and Conservative Field in chapter Vector Differential Calculus of Engineering Mathematics

Answer 1

Right option is (a) \(\frac{ρ}{r}+ 2rθ \,a_r – r a_θ + \frac{lnr}{rsin(θ)}  a_Φ \)

To elaborate: \(B = ϕln(r)+r^2 ϕ \)

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

\(= \frac{ρ}{r}+ 2rθ \,a_r – r a_θ + \frac{lnr}{rsin(θ)}  a_Φ \)