A vector field has the flux two-form .

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

Invariance, Ch Differential forms and the exterior derivative

Problem

A vector field F=(x2,y2,z2)\mathbf{F} = (x^2, y^2, z^2) has the flux two-form β=x2 dy∧dz+y2 dz∧dx+z2 dx∧dy\beta = x^2\,\mathrm{d}y\wedge\mathrm{d}z + y^2\,\mathrm{d}z\wedge\mathrm{d}x + z^2\,\mathrm{d}x\wedge\mathrm{d}y. ((a)) Compute dβ\mathrm{d}\beta, giving the coefficient of dx∧dy∧dz\mathrm{d}x\wedge\mathrm{d}y\wedge\mathrm{d}z. ((b)) Identify this coefficient with ∇⋅F\nabla\cdot\mathbf{F}.

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