Consider the scalar field (equal to

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Vector calculus and field geometry · Vector calculus and field geometry

Conditions of Validity, Ch Gradient, divergence and curl without coordinates, and with any coordinates

Problem

Consider the scalar field f(r,θ)=r2sin⁡2θf(r,\theta) = r^2\sin^2\theta (equal to x2+y2x^2+y^2, with rr, xx and yy in metres) in spherical coordinates (r,θ,ϕ)(r,\theta,\phi). ((a)) Recall the volume element used for triple integrals in spherical coordinates (Chapter 2), dV=r2sin⁡θ dr dθ dϕ\mathrm dV = r^2\sin\theta\,\mathrm dr\,\mathrm d\theta\,\mathrm d\phi, and confirm that this matches g\sqrt g computed from the spherical metric grr=1g_{rr}=1, gθθ=r2g_{\theta\theta}=r^2, gϕϕ=r2sin⁡2θg_{\phi\phi}=r^2\sin^2\theta (Chapter 3). ((b)) Using the general Laplacian ∇2=(1/g)∂i(g gij∂j)\nabla^2 = (1/\sqrt g)\partial_i(\sqrt g\, g^{ij}\partial_j), compute ∇2f\nabla^2 f as a function of θ\theta only, then simplify to a single number. ((c)) Check the result against the Cartesian identity f=x2+y2f = x^2+y^2 using the ordinary Cartesian Laplacian.

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