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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

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this question might help: math.stackexchange.com/questions/12894/… –  Eric O. Korman Aug 21 '12 at 22:10
    
Aside from the answers, I am suggesting write down every integration by parts formula involving $d^*$ for the $k$-form in 3 dimensional space using $\nabla$, $\nabla\times$, and $\nabla \cdot$, this would help you better understand. –  Shuhao Cao Aug 22 '12 at 2:55
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up vote 4 down vote accepted

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.

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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.

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