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

I will briefly answer two questions here.  First, what does the phrase "formal adjoint" mean in this context?  Second, how is the adjoint $d^*$ actually defined?
Definitions: $\Omega^k(M)$ ($M$ a smooth oriented $n$-manifold with a Riemannian metric) is a pre-Hilbert space with norm $$\langle \omega,\eta\rangle_{L^2} = \int_M \omega\wedge *\eta.$$
(Here, $*$ is the Hodge operator.)  
For the first question, the "formal adjoint" of the operator $d$ is the operator $d^*$ (if it exists, from some function space to some other function space) that has the property
$$\langle d\omega,\eta\rangle_{L^2} = \langle \omega,d^*\eta\rangle_{L^2}.$$
For the second question, the operator $d^*$ is actually defined as a map $\Omega^{k+1}(M)\to\Omega^k(M)$ by setting $$d^* = *d*.$$  
Adjointness is proven by using integration by parts and Stokes' theorem.
