k+1 Differential form

Question

Consider the k-form given by,

$ w = \sum_{i_{1}<i_{2}<...<i_{k}} w_{i_{1}i_{2}...i_{k}} dx^{i_{1}}\wedge dx^{i_{2}}\wedge...\wedge dx^{i_{k}}$

Define $k+1$ form $dw$ , the differential of $W$ by,

$ dw = \sum_{i_{1}<i_{2}<...<i_{n}} dw_{i_{1},i_{2}...i_{k}}\wedge dx^{i_{1}}\wedge dx^{i_{2}}\wedge...\wedge dx^{i_{k}} $

Can you explain me why this $k+1$ differential form is defined in this way.Why we are not applying the $d$ on $ dx^{i_{1}}\wedge dx^{i_{2}}\wedge...\wedge dx^{i_{k}} $.I am following Calculus on Manifolds by Spivak.The same definition is given in page no. 91 of Spivak.

Thanks in Advance.

Answer 1

You can apply the $d$ to $dx^i$ if you wish. Since $d(df)=0$ (after all, $x^i$ is a function), the product rule gives us the same result.