Let U be an open set in \mathbb R^{n+1} and f: U \to \mathbb R be a smooth function
Surface S = f^{-1}(c) where \forall_{p \in S} \nabla f(p) \ne 0, that is, level sets where all points are regular.
Let S_p = [\nabla f(p)]^\perp denote the tangent space to a surface S at p.
Let S = f^{-1}(c) be an n-surface where f: U \to \mathbb R. Suppose g: U \to \mathbb R is a smooth function and p \in S is an extreme point of g on S. Then there exists a real number \lambda (Lagrange multiplier) such that \nabla g(p) = \lambda \nabla f(p).
At each point p \in S vector is tangent to the surface. Denoted by T(p)
At each point p \in S vector is normal to the surface. Denoted by N(p)
An orientation is a smooth choice of a normal vector direction at every point p \in S.
A unit vector in \mathbb R^{n+1}_p is called the direction at p. Thus an orientation is a smooth choice of normal direction at each point of S.
By rotating the orientation normal direction at p through an angle of -{\pi \over 2} we get the positive tangent direction.
On a 2-surface orientation can be used to determine the direction of rotation at each point, R_\theta: S_p \to S_p
R_\theta(v) = (\cos \theta)v + (\sin \theta)N(p) \times v
A subset S of \mathbb R^{n+1} is called path-connected if for each pair of points p, q \in S there exists a continuous map \alpha: [a; b] \to S such that \alpha(a) = p and \alpha(b) = q.
Let S \subset \mathbb R^{n+1} be a connected n-surface, on S there exists exactly two unit normal vector fields \vec N_1(p) = -\vec N_2(p) for all p \in S.
Gauss map is the vectors translated to the origin of the smooth unit normal vector field N of an oriented n-surface.
N: S \to \mathbf S^n \subset \mathbb R^{n+1}
Note: \mathbf S^n is a unit sphere in n dimensions.
A subset K \subset \mathbb R^{n+1} is closed if for every convergent sequence of points \{a_n\}^\infty_{n=1} which is contained in K, the limit \lim_{n\to\infty}a_n belongs to K.
A subset K of a metric space is compact if every infinite sequence of points in K has a convergent subsequence whose limit belongs to K.
A subset of \mathbb R^{n+1} is compact iff it is closed and bounded.
A continuous function f defined on a compact set K achieves a minimum value m at x_m and a maximal value M at x_M.
Let S be a compact, connected, oriented n-surface in \mathbb R^{n+1} exhibited as a level set of some smooth function, then the Gauss map maps S onto the unit sphere \mathbf S^n in \mathbb R^{n+1}