In my book surfaces are defined as maps $\sigma\ \colon A \subseteq \mathbb R^2 \to \mathbb R^n$ ($n \geq 3$). Then the book goes on to define regular surfaces and the normal vector. The latter is defined as the cross product $$N_\sigma(u, v) = \frac{\partial\sigma}{\partial u} \times \frac{\partial\sigma}{\partial v}$$ However, this only holds for $\mathbb R^3$, because the cross product is defined only there. What about higher dimensions? I'm asking because I also have a theorem involving the normal vector and the dimension isn't $3$.
EDIT: Here is the theorem I'm talking about.
Def Let $\sigma\ \colon K \to \mathbb R^n$ e $\eta\ \colon K' \to \mathbb R^n$ be two regular surfaces. We say that $\sigma$ and $\eta$ differ by a change of variables if:
- there exist two open sets $A, A' \subseteq \mathbb R^2$ such that $K \subseteq A$ e $K' \subseteq A'$;
- there exist a change of variables $\alpha\ \colon A' \to A$ of class $C^1$ on $A'$ such that $\alpha(K') = K$;
- $\eta(s, t) = \sigma(\alpha(s, t))$ for every $(s, t) \in A'$.
Theorem If two surfaces differ by a change of variables, then:
- $\sigma(K) = \eta(K')$;
- The normal vector of $\sigma$ is $$N_\sigma(\alpha(s, t)) = N_\eta(\alpha(s, t))\det J_\alpha(s, t)$$ The proof is left to the reader.
I encountered this problem when trying to prove the above claim. I don't know how to calculate $N_\sigma$! Any hints?
A k-dimensional, smooth hypersurface in $ \mathbb{R}^n $, $ n > k $ is defined as a smooth map $ \phi : G \subset \mathbb{R}^k \rightarrow \mathbb{R}^n $, $ G $ open in $ \mathbb{R}^k $ and $ \text{rk} (D \phi) = k $ throughout $ G $. If the surface is parametrized, the normal vector is defined to be the k-wedge, $$ N(v_1, v_2, \cdots ,v_k) = \frac{ \partial \phi}{\partial v_1} \wedge \frac{ \partial \phi}{\partial v_2} \wedge \cdots \wedge \frac{ \partial \phi}{\partial v_k} $$ The above is simply the 'oriented volume' of the $ k $ - dimensional parallelopiped given by the vectors $ \left( \frac{ \partial \phi}{\partial v_i} \right )_{i=1}^{k} $. The norm of this vector, $ ||N(v_1, v_2, \cdots, v_k)|| $ is given by the determinant of the Grammian matrix of $ \left( \frac{ \partial \phi}{\partial v_i} \right )_{i=1}^{k} $, and this is used in calculating the surface area. For the case $ k=2, n=3 $ you can check that this is indeed the usual cross product in $ \mathbb{R}^3 $.