[proofplan] The defining function $\Phi$ gives a normal covector to the hypersurface at $p$. We identify the tangent space $T_pS$ with the kernel of the linear functional $v \mapsto \nabla \Phi(p) \cdot v$. Thus the vector field value $a(p)$ is tangent exactly when this dot product vanishes, and negating tangency gives the noncharacteristic condition. [/proofplan] [step:Identify the tangent space as the kernel of the defining differential] Define the [linear map](/page/Linear%20Map) \begin{align*} D\Phi_p:\mathbb{R}^n \to \mathbb{R}, \qquad v \mapsto \nabla \Phi(p) \cdot v. \end{align*} Since $\Phi \in C^1(V;\mathbb{R})$ and $\nabla \Phi(p) \ne 0$, the level set $\Phi^{-1}(\{0\})$ is regular at $p$. For the $C^1$ hypersurface $S \cap V = \Phi^{-1}(\{0\})$, its tangent space at $p$ is therefore \begin{align*} T_pS = \ker D\Phi_p = \{v \in \mathbb{R}^n : \nabla \Phi(p) \cdot v = 0\}. \end{align*} [guided] The point of introducing $\Phi$ is that it supplies a normal direction to the hypersurface. Formally, define \begin{align*} D\Phi_p:\mathbb{R}^n \to \mathbb{R}, \qquad v \mapsto \nabla \Phi(p) \cdot v. \end{align*} This is the total derivative of $\Phi$ at $p$, written as a linear map. The hypothesis $\nabla \Phi(p) \ne 0$ says that this linear map is not the zero map, so the equation $\Phi(x)=0$ cuts out a regular codimension-one level set near $p$. For a $C^1$ curve $\gamma:(-\varepsilon,\varepsilon)\to S\cap V$ with $\gamma(0)=p$, differentiating the identity $\Phi(\gamma(t))=0$ at $t=0$ gives \begin{align*} D\Phi_p(\gamma'(0)) = 0. \end{align*} Since $D\Phi_p(v)=\nabla \Phi(p)\cdot v$, every velocity vector tangent to $S$ at $p$ lies in \begin{align*} \{v \in \mathbb{R}^n : \nabla \Phi(p)\cdot v=0\}. \end{align*} Conversely, because $\nabla \Phi(p)\ne 0$, the regular level-set description of the $C^1$ hypersurface identifies all vectors in this kernel as tangent vectors to $S$ at $p$. Hence \begin{align*} T_pS = \ker D\Phi_p = \{v \in \mathbb{R}^n : \nabla \Phi(p)\cdot v=0\}. \end{align*} [/guided] [/step] [step:Translate tangency of the vector field into a scalar equation] By definition, $S$ is characteristic at $p$ for $a$ exactly when $a(p)\in T_pS$. Using the tangent-space identification above, this is equivalent to \begin{align*} \nabla \Phi(p)\cdot a(p)=0. \end{align*} The Euclidean dot product is symmetric, so this is the same condition as \begin{align*} a(p)\cdot \nabla \Phi(p)=0. \end{align*} [/step] [step:Negate the characteristic condition to obtain the noncharacteristic criterion] By definition, $S$ is noncharacteristic at $p$ for $a$ exactly when $S$ is not characteristic at $p$, equivalently when $a(p)\notin T_pS$. From the preceding step, this holds exactly when \begin{align*} a(p)\cdot \nabla \Phi(p)\ne 0. \end{align*} This proves the stated equivalence. [/step]