> [!theorem] > > Let $E$ be a [[Banach Space|Banach space]], $\Omega \subset E$ be an [[Open Set|open]] domain, $S \subset \Omega$ be a [[Hypersurface|hypersurface]] of class $C^1$, $A \in C^1(\Omega \times \real, E)$, and $b \in C^1(\Omega \times \real, \real)$. For any subset $F \subset \Omega$ and $f: F \to \real$, denote $A_f: F \to \real^d$ by $x \mapsto A(x, f(x))$, and $b_f: F \to \real$ by $x \mapsto b(x, f(x))$. > > Then for any $\phi \in C^1(S, \real)$ such that $A_\phi(x) \not \in T_xS$ for all $x \in S$, there exists $U \in \cn^o(S)$ and a unique $u \in C^1(U)$ such that > $ > \angles{A_u, u} - b_uu = 0 \quad u|_S = \phi > $ > > *Proof*. Let $p \in S$, $\mu \in E^*$ with $\mu \ne 0$, $F = \ker(\mu)$, and $(U, \psi)$ be a slice [[Atlas|chart]] at $p$ such that > $ > \psi: U \to E \quad \psi|_{U \cap S}: U \cap S \to F > $ > are $C^1$-isomorphisms and $\psi(p) = (0, 0)$. Define $\phi^*: \psi(U \cap S) \to \real$ by $x \mapsto \phi \circ \psi^{-1}(x)$, > $ > A^*: \psi(U) \times \real \to E \quad (x, y) \mapsto D\psi_{\psi^{-1}(x)} \cdot A(\psi^{-1}(x), y) > $ > and $b^*: \psi(U) \to E$ with $x \mapsto b \circ \psi^{-1}(x)$. Let > $ > F: \real \times \psi(U) \times \real \quad (t, x, y) \mapsto (A^*(x, y), b^*(x, y)) > $ > then by the [[Existence Theorem for Differential Equations]], there exists $\eps > 0$, a [[Neighbourhood|neighbourhood]] $V \in \cn^o(p)$, and a unique $C^1$-[[Flow|local flow]] > $ > \alpha: (-\eps, \eps) \times (\psi(V) \times \real) \to \psi(V) \times \real > $ > such that for each $(x, y) \in \psi(V \cap S) \times \real$, > $ > \alpha(0, x, y) = (x, \phi(\psi^{-1}(x))) \quad \frac{d\alpha_{(x, y)}}{dt}(t) = \begin{bmatrix}A^*(\alpha_{(x, y)}(t)_1, \alpha_{(x, y)}(t)_2) \\ b^*(\alpha_{(x, y)}(t)_1, \alpha_{(x, y)}(t)_2)\end{bmatrix} > $ > Observe that $\alpha$ does not depend on $y$. Moreover, the restriction of the first coordinate > $ > \alpha_1|_{(-\eps, \eps \times \psi(V \cap S))}: (-\eps, \eps) \times \psi(V \cap S) \to E \quad (t, x) \mapsto \pi_1 \circ \alpha(t, x, 0) > $ > has derivative > $ > D\alpha(0): \real \times F \to E \quad (t, x) \mapsto tA^*(p, \phi(p)) + x > $ > Given that $A_\phi(p) \not\in T_pS$, $A^*(p, \phi(p)) \not\in F$. Thus $D\alpha$ is an isomorphism. By the [[Inverse Function Theorem]], assume without loss of generality that > $ > \alpha_1|_{(-\eps, \eps) \times \psi(V \cap S)}: (-\eps, \eps) \times \psi(V \cap S) \to \psi(V) > $ > > is a $C^1$-isomorphism, where $\alpha_1(t, x) \in \psi(V \cap S)$ if and only if $t = 0$. Now define $u: V \to \real$ as the following composition > $ > \begin{CD} > V @>{\psi}>> \psi(V) @>\alpha_1^{-1}>> (-\eps, \eps) \times \psi(V \cap S) @>\pi_2 \circ \alpha>> \real > \end{CD} > $ > then $u|_S = \phi$ directly. Moreover, denote $t = \pi_2 \circ \alpha_1^{-1}(\psi^{-1}(y))$ and $x = \pi_1 \circ \alpha_1^{-1}(\psi^{-1}(y))$, then > $ > \begin{align*} > \anglesn{A^*, u \circ \psi^{-1}}(y) &= \braks{\frac{\partial u}{\partial x}\frac{\partial x}{\partial y} + \frac{\partial u}{\partial t}\frac{\partial t}{\partial y}} \cdot A^* \\ > &= \frac{\partial u}{\partial x}\frac{\partial x}{\partial y}\frac{\partial y}{\partial t} + \frac{\partial u}{\partial t}\frac{\partial t}{\partial y}\frac{\partial y}{\partial t} \\ > &= \frac{\partial u}{\partial t} = b^*(y, u(y)) > \end{align*} > $ > Therefore $u$ is a desired solution. Uniqueness follows from the [[Gluing Lemma|gluing lemma]].