On the region consider the one-form

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

On the region x>0, y>0x>0,\ y>0 consider the one-form ω=y dx−x dy\omega = y\,\mathrm{d}x-x\,\mathrm{d}y, for which dω=−2 dx∧dy≠0\mathrm{d}\omega=-2\,\mathrm{d}x\wedge\mathrm{d}y\ne0, so ω\omega is not closed. Someone argues: 'Dividing by x+yx+y tidies ω\omega into yx+y dx−xx+y dy\dfrac{y}{x+y}\,\mathrm{d}x-\dfrac{x}{x+y}\,\mathrm{d}y, so 1/(x+y)1/(x+y) must be an integrating factor.' ((a)) Test the claim by computing d ⁣(ωx+y)\mathrm{d}\!\left(\dfrac{\omega}{x+y}\right) and deciding whether it vanishes. ((b)) Find a genuine integrating factor for ω\omega on this region (try a power of xx), and give the function whose differential is ω\omega times that factor.

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