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For closed surfaces of genus g ≥ 2, the moduli space of Riemann surfaces obtained as Γ varies over all such subgroups, has real dimension 6g − 6. [75] By Poincaré's uniformization theorem, any orientable closed 2-manifold is conformally equivalent to a surface of constant curvature 0, +1 or –1.
Surfaces that occur in two of the main theorems of vector calculus, Stokes' theorem and the divergence theorem, are frequently given in a parametric form. The curvature and arc length of curves on the surface, surface area , differential geometric invariants such as the first and second fundamental forms, Gaussian , mean , and principal ...
This theory was used by Lagrange, a co-developer of the calculus of variations, to derive the first differential equation describing a minimal surface in terms of the Euler–Lagrange equation. In 1760 Euler proved a theorem expressing the curvature of a space curve on a surface in terms of the principal curvatures, known as Euler's theorem.
Hypersurfaces share, with surfaces in a three-dimensional space, the property of being defined by a single implicit equation, at least locally (near every point), and sometimes globally. A hypersurface in a (Euclidean, affine, or projective) space of dimension two is a plane curve. In a space of dimension three, it is a surface.
A projective surface in a projective space of dimension three is the set of points whose homogeneous coordinates are zeros of a single homogeneous polynomial in four variables. More generally, a projective surface is a subset of a projective space, which is a projective variety of dimension two.
A study of limits and continuity in multivariable calculus yields many counterintuitive results not demonstrated by single-variable functions.. A limit along a path may be defined by considering a parametrised path (): in n-dimensional Euclidean space.
Ruled surface generated by two Bézier curves as directrices (red, green) A surface in 3-dimensional Euclidean space is called a ruled surface if it is the union of a differentiable one-parameter family of lines. Formally, a ruled surface is a surface in is described by a parametric representation of the form
The Willmore energy is always greater than or equal to zero. A round sphere has zero Willmore energy.. The Willmore energy can be considered a functional on the space of embeddings of a given surface, in the sense of the calculus of variations, and one can vary the embedding of a surface, while leaving it topologically unaltered.