# Exterior Derivatives

I am self-studying differentialgeometry and tried to compute the exterior derivatives of the following 2-forms:

$\omega_1 = x\,\mathrm{d}y\wedge \mathrm{d}z-y\,\mathrm{d}x\wedge \mathrm{d}z+z\,\mathrm{d}x\wedge \mathrm{d}y\\ \omega_2 = x_1x_2\,\mathrm{d}x_3\wedge \mathrm{d}x_4+x_3x_4\,\mathrm{d}x_1\wedge \mathrm{d}x_2\\ \omega_3 = e^{s_1s_3}\,\mathrm{d}s_2\wedge \mathrm{d}s_3-e^{s_1s_2}\,\mathrm{d}s_1\wedge \mathrm{d}s_2$

My results:

$\mathrm{d}\omega_1 = 3\,\mathrm{d}x\wedge\mathrm{d}y\wedge\mathrm{d}z\\ \mathrm{d}\omega_2 = x_2\,\mathrm{d}x_1\wedge\mathrm{d}x_3\wedge\mathrm{d}x_4+x_1\,\mathrm{d}x_2\wedge\mathrm{d}x_3\wedge\mathrm{d}x_4+x_4\,\mathrm{d}x_3\wedge\mathrm{d}x_1\wedge\mathrm{d}x_2+x_3\,\mathrm{d}x_4\wedge\mathrm{d}x_1\wedge\mathrm{d}x_2\\ \mathrm{d}\omega_3= e^{s_1s_3}s_3\,\mathrm{d}s_1\wedge\mathrm{d}s_2\wedge\mathrm{d}s_3$

Unfortunately, I have no solutions and need this to be checked, otherwise I might run into severe trouble when integrating over manifolds later.

• It's fine for me, except in the 2nd exterior derivative, I'd write the last two exterior product by increasing indices ($\mathrm dx_1\wedge\mathrm dx_2\wedge\mathrm dx_3$ and ($\mathrm dx_1\wedge\mathrm dx_2\wedge\mathrm dx_4).$ – Bernard Aug 14 '18 at 14:09