> [!definition] > > Let $(X, g)$ be a smooth [[Riemannian Manifold|Riemannian n-manifold]]. For any rough [[Vector Field|vector fields]] $\xi$ and $\eta$, define an [[Inner Product|inner product]] > $ > \angles{\xi, \eta}_{\scl^2} = \int_X \angles{\xi, \eta}_g dV > $ > and $\mathscr L^2(X)$ as the [[Hilbert Space|Hilbert space]] of square-integrable vector fields. For any $f \in C^1(X)$, > $ > \angles{\grad f, \xi}_{\scl^2} = -\angles{f, \div{\xi}}_{L^2} > $ > Thus, for any $f \in L^2(X)$, if there exists $\eta \in \mathscr{L}^2(X)$ such that > $ > \angles{\eta, \xi}_{\scl^2} = -\angles{f, \div{\xi}}_{L^2} > $ > for all compactly supported $C^1$-vector fields. then $\eta = \Grad{f}$ is the **weak derivative** of $f$. The space $\mathscr H(X)$ refers to the subspace of $L^2(X)$ with weak derivatives in $\mathscr L^2(X)$, with the following inner product > $ > \angles{f, g}_\sch = \angles{f, g}_{L^2} + \angles{\Grad{f}, \Grad{g}}_{\scl^2} > $ > The [[Function on Manifold|smooth functions]] $C^\infty(X)$ is dense in $\sch(X)$. > > *Proof*. Since > $ > \div{f\div{\xi}} = f\div{\xi} + \angles{\grad{f}, \xi} > $ > By the [[Divergence Theorem|divergence theorem]], $\int_X f\div{\xi} dV = 0$. > [!definition] > > Let $f, g \in \sch(X)$, then the derivative part of the inner product > $ > D[f, g] = \angles{\Grad f, \Grad g}_{\scl^2} > $ > is known as the **energy integral**. > [!theorem] > > Let $\phi \in C^2(X)$ be a solution of the [[Closed Eigenvalue Problem|closed eigenvalue problem]], then > $ > \angles{\Delta \phi, f}_{L^2} = -D[\phi, f] > $ > for all $f \in \sch(X)$. Moreover, the mapping $F_\phi: \sch(X) \to \real$ defined by $f \mapsto -D[\phi, f]$ is a [[Bounded Linear Functional|bounded linear functional]]. > > *Proof*. Using [[Green's Formulas]], > $ > \int_X f\Delta \phi + \angles{\Grad f, \Grad{\phi}}dV = 0 > $ > for all $f \in C_c^\infty(X) = C^\infty(X)$ since $X$ is [[Compactness|compact]]. Moreover, $\norm{F_\phi} \le \norm{\phi}_{\sch}$. By the uniqueness part of the [[Linear Extension Theorem|linear extension theorem]], the equation holds for all $f \in \sch(X)$.