enter image description here Spivak says the definition of $dw$ for a k-form $w$ does not make sense on a manifold because $D_j(w_{i_1, \dots , i_p})$ has no meaning. Does it have no meaning because the function w_{i_1, \dots , i_p} is defined only on $M$ so that we have consider the tangent space but $dw$ is taking partial derivatives of the function based on the unit vectors of $\mathbb{R}^n$ instead of the tangent space?

  • $\begingroup$ I do not see any "previous" definition of $\mathrm{d}\omega$. I guess he defined the differential on $\mathbb{R}^n$ and he is now trying to make sense of $\mathrm{d}\omega$ on $M$. I say so because I think $D_j(\omega_{i_1,\dots,i_n})$ are partial derivatives on $\mathbb{R}^n$, so he should define partial derivatives on $M$. $\endgroup$ – Gibbs Mar 3 at 22:19
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    $\begingroup$ The "previous" definition refers to the definition when you have a form defined on an open subset of $\mathbb{R}^n$. Now he is trying to extend the concept to the more general situation of a form defined on a manifold. Since we do not know what partial derivatives mean on a manifold, we resort to coordinates to give a meaning to the concept. Not only we define the concept of differentiable using coordinates, but the derivative as well. After this paragraph, he will define $d\omega$ using partial derivatives of the functions $\omega_{i,j,k\dots}$ as functions on $W$. $\endgroup$ – GReyes Mar 3 at 23:25

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