definition of the exterior derivative I have a question concerning the definition of $d^*$. It is usually defined to be the (formally) adjoint of $d$? what is the meaning of formally?, is not just the adjoint of $d$? thanks

The formal adjoint of a differential operator with respect to a smooth density $\mu$ is, intuitively speaking, what should be adjoint according to integration by parts, or, equivalently, what actually is adjoint when restricted to, say, smooth compactly supported functions (or differential forms in your case, or sections of any vector bundle...)
Taking formal adjoint is the anti-automorphism of the algebra of differential operators given by $f^\ast = f$ for functions and $(v\partial)^\ast = -v\partial + \mu^{-1}\mathcal{L}_v \mu$ for vector fields, where $\mathcal L$ is Lie derivative.
The difference between "formal" and "actual" adjoint is a subtle issue of operator theory: when speaking of adjoint operators in Hilbert space we should bother with domains, extensions, ..., and here we just take the formal differential expression, not mentioning where exactly the operator acts.
