Suppose that w = x2 + y2 + z2 and x = ρ cos θ sin ?, y = ρ sin θ sin ?, z = ρ cos ?. (Note that the equations for x, y, and z in terms of ρ, ?, and θ are just the conversion relations from spherical to rectangular coordinates.)
(a) Use the chain rule to compute ∂w/∂ρ, ∂w/∂?, and ∂w/∂θ. Simplify your answers as much as possible.
(b) Substitute ρ, ?, and θ for x, y, and z in the original expression for w. Can you explain your answer in part (a)?
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