# Proof of the formula for the exterior derivative in terms of the covariant derivative.

SETUP

Everything is smooth.

Consider the ring $$\Omega^{p}(M)$$ of sections of the exterior powers of the cotangent bundle $$\Lambda^{p}(T^{*}M)$$ over a manifold $$M$$.

Purely in terms of the smooth structure, we define the exterior derivative as the map $$\operatorname{d}:\Omega^{p}(M)\to\Omega^{p+1}(M)$$ whose action on $$\phi\in\Omega^{p}(M)$$ is defined by $$\operatorname{d}\!\phi\bigg(\bigotimes_{k=0}^{p}X_{k}\bigg) = \sum_{i=0}^{p}(-1)^{i}X_{i}\left(\phi\Big(\bigotimes_{k\neq i}X_{k}\Big)\right) + \sum_{i where the $$X_{k}$$ are $$p+1$$ sections of the tangent bundle $$TM\to M$$, and the tensor products are taken in order.

On the other hand, if we have a connection on the frame bundle we may inherit a covariant derivative $$\nabla$$ in $$TM$$ and then in every tensor bundle $$T^{n}_{m}M$$, including the subbundles $$\Lambda^{p}(T^{*}M)$$. If the connection is torsion free, then we may express the exterior derivative in terms of the covariant derivative by (using abstract index notation)

$$(\operatorname{d}\phi)_{a_0\dots a_p} = (p+1)\nabla_{[a_0}\phi_{a_1\dots a_p]} \in\Omega^{p+1}(M)$$

for every $$\phi\in\Omega^{p}(M)$$. This construction is well defined for vector valued forms (elements of $$\Omega^{p}(M,E)$$), however at the moment I'm just interested in the following...

...QUESTION

Is there a coordinate-free proof (say, using abstract index notation) to see that the second definition is equivalent to the first, when $$\phi\in\Omega^{p}(M)$$?

I've just finished a proof using Penrose graphical notation, however it is rather lengthy. Maybe I was not ingenuous enough. Does anybody here know of a good one?

Althought there are better ways to do this, I wanted to close the question posting my own answer. This is a refined version of the proof I was refering to when I posted the question.

We start with $$(\operatorname{d}\phi)\left(\bigotimes_{k=0}^{p}X_{k}\right) = \sum_{i=0}^{p}(-1)^{i}X_{i}\left(\phi\left(\bigotimes_{k\neq i}X_{k}\right)\right) + \sum_{i

Let's denote the two terms in the RHS by $$(A)$$ and $$(B)$$.

Now we expand $$(A)$$ as follows

So at the end we recognize that

$$(A) = (p+1)\nabla_{[a_0}\phi_{a_1\dots a_p]}\left(\bigotimes_{k=0}^{p}X_{k}\right)^{a_0 \dots a_p} - (B)$$

Hence

$$(\operatorname{d}\phi)\left(\bigotimes_{k=0}^{p}X_{k}\right) = (A) + (B) = (p+1)\nabla_{[a_0}\phi_{a_1\dots a_p]}\left(\bigotimes_{k=0}^{p}X_{k}\right)^{a_0 \dots a_p}$$

Since the $$X_i$$ are arbitrary, we finally have

$$(\operatorname{d}\phi)_{a_0 \dots a_p} = (p+1)\nabla_{[a_0}\phi_{a_1\dots a_p]} \hspace{2cm} \square$$

• Ahahhahaha I see you've read roger penrose's road to reality :D. What is there some other book which discuseses the penrose notation O_O
– Babu
Commented Apr 19, 2022 at 19:09
• @Buraian Actually I haven't lol. I learned the notation from Spinors and spacetime and through playing around with it a lot. Now it's my favourite notation haha. Commented Apr 19, 2022 at 19:44
• @Buraian If you are asking about graphical notations, there are many surveys from category theorists. If you are interested you can look for "a prehistory of n-categorical physics" by Baez and Lauda. It turns out that Penrose graphical notation and Feynmann diagrams are special cases of string diagrams in 2-categories. Commented Apr 19, 2022 at 21:25
• What part of the book talks about this notation? Could you maybe explain it more @Jackozee Hakkiuz
– Babu
Commented Apr 29, 2022 at 10:55