Let on the cube

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

Mathematical Methods for Physics, Ch The integral theorems: Gauss, Stokes and Green

Problem

Let F=(x3,0,0)\mathbf{F} = (x^3, 0, 0) on the cube 0≤x,y,z≤L0\le x, y, z\le L, and cut the cube into NN slabs of equal thickness h=L/Nh = L/N by planes of constant xx, the kkth slab running from xk−1=(k−1)hx_{k-1} = (k-1)h to xk=khx_k = kh. ((a)) Find the outward flux of F\mathbf{F} out of the kkth slab. Show that the sum over all NN slabs equals the flux out of the whole cube for every NN, and say what became of the internal walls. ((b)) Estimate each slab's flux as (∇⋅F) ΔV(\nabla\cdot\mathbf{F})\,\Delta V, with the divergence taken on the slab's mid-plane and ΔV=hL2\Delta V = hL^2. Show that the estimate falls short by h2ΔV/4h^2\Delta V/4, so that the error vanishes faster than ΔV\Delta V as the slabs thin. ((c)) For N=10N = 10, find the sum of the estimates as a fraction of the true flux, and say what the sum becomes as NN grows without bound.

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