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Does it make sense to add two differential forms with different degrees like $dx+dx\bigwedge dy$? If yes, what's the arguments of it?

I ask this because in text book, the vector space, $\Omega^*(M)$, of $C^\infty$ differential forms on a manifold $M$ is defined as

$\Omega^*(M)=\bigoplus_{k=0}^n\Omega^k(M)$,

where $\Omega^k(M)$ is the vector space of k-forms. It means each element in $\Omega^*(M)$ is uniquely a sum $\sum_{k=0}^n \omega_k$, $\omega_k\in \Omega^k(M)$.

jizhou
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    I mean, sure, in the direct sum of those vector spaces you can write that. That's what a direct sum of vector spaces is. –  Jul 23 '16 at 06:13
  • But does $dx+dx\bigwedge dy$ make sense in terms of differential form? Let $\alpha= dx+dx\bigwedge dy$, then which one is correct: $\alpha(X)$ or $\alpha(X,Y)$? – jizhou Jul 23 '16 at 06:17
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    It is a formal sum of differential forms, or if you like a section of the bundle $\oplus \Lambda^k T^M$. There's no reason to believe there's some inherently, innately defined way of pairing it against vectors. (There is a reasonable pairing of $\oplus \Lambda^k(TM)$ with $\oplus \Lambda^k(T^M)$ - if you like, the pairing of a k-form with an m-vector is zero if $k \neq m$. But this is neither here nor there, and is unlikely to have the geometric content you're looking for.) –  Jul 23 '16 at 06:20
  • Ok. But is such expression like $dx + dx \bigwedge dy$ really used in any math branch, or should never be used? – jizhou Jul 23 '16 at 06:31
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    Sometimes certain operators are most naturally defined on the entire $\Omega^$ at once, or other direct sums of bundles; the signature operator is defined as a differential operator $\oplus \Lambda^{even} T^M \to \oplus \Lambda^{odd} T^*M$. So it's not impossible that it's useful. But it's not really anything to be concerned about. –  Jul 23 '16 at 06:33
  • I have the exact same question. Do you have the answer now? – rainman Mar 28 '20 at 10:19
  • I'm pretty sure spinors can be represented by certain combinations of differential forms of varying degrees – R. Rankin Dec 14 '23 at 06:47
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    These are sometimes called inhomogeneous differential forms. One can define a Clifford product on such elements and analyse its spinor representations, which leads to writing the Dirac equation (describing all massive spin-1/2 particles) in terms of differential forms. – AloneAndConfused Jan 20 '24 at 22:22

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