Someone computes the wedge product of and as follows: 'Expanding,

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

Invariance, Ch Differential forms and the exterior derivative

Problem

Someone computes the wedge product of α=3 dx+4 dy\alpha=3\,\mathrm{d}x+4\,\mathrm{d}y and β=2 dx−5 dy\beta=2\,\mathrm{d}x-5\,\mathrm{d}y as follows: 'Expanding, α∧β=6 dx∧dx−15 dx∧dy+8 dy∧dx−20 dy∧dy\alpha\wedge\beta = 6\,\mathrm{d}x\wedge\mathrm{d}x-15\,\mathrm{d}x\wedge\mathrm{d}y+8\,\mathrm{d}y\wedge\mathrm{d}x-20\,\mathrm{d}y\wedge\mathrm{d}y. Since dx∧dx=dy∧dy=0\mathrm{d}x\wedge\mathrm{d}x=\mathrm{d}y\wedge\mathrm{d}y=0 and dy∧dx=dx∧dy\mathrm{d}y\wedge\mathrm{d}x=\mathrm{d}x\wedge\mathrm{d}y, this reduces to −15 dx∧dy+8 dx∧dy=−7 dx∧dy-15\,\mathrm{d}x\wedge\mathrm{d}y+8\,\mathrm{d}x\wedge\mathrm{d}y=-7\,\mathrm{d}x\wedge\mathrm{d}y.' ((a)) Identify the error in this working. ((b)) Give the correct value of α∧β\alpha\wedge\beta.

Browse the problem library