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I don't find arithemtic rules of the operators $\wedge$ and $d$. For example, why does this equality hold? $$ \\ (u^2\cos^2v+u^2\sin^2v)[\cos vdu-u\sin vdv]\wedge [\sin vdu+u\cos vdv] \ \\ +u\cos v[\cos v du-u\sin vdv]\wedge du \ \\ +u \sin v[\sin v du+u\cos v dv]\wedge dv\ \\ = u(u^2+1) du\wedge dv\ $$

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    $\begingroup$ $du\wedge du=0=dv\wedge dv$ and $dv\wedge du=-du\wedge dv$. $\endgroup$ – Lord Shark the Unknown Feb 11 at 17:32
  • $\begingroup$ And how can I handle $\sin$ and $\cos$? @LordSharktheUnknown $\endgroup$ – J. Doe Feb 11 at 17:37
  • $\begingroup$ Basically you have to use the fundamental equation $\sin^2 x + \cos^2 x = 1$, that holds for all $x\in \mathbb R$. $\endgroup$ – AlessioDV Feb 11 at 18:05
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    $\begingroup$ Sorry, also remember that $\wedge$ is multilinear. $\endgroup$ – AlessioDV Feb 11 at 18:06
  • $\begingroup$ The number of thing that go into an answer from what you might know is large, and you have made no mention of what you already know. One cannot simply write down a few "arithmetic rules" here -- there are many, and they are not simple. This question should be closed as being far too vague. $\endgroup$ – qman Mar 19 at 21:48

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