Editing Sphere (section)

===Differential geometry===
The sphere is a [[smooth surface]] with constant [[Gaussian curvature]] at each point equal to {{math|1/''r''<sup>2</sup>}}.<ref name=MathWorld_Sphere /> As per Gauss's [[Theorema Egregium]], this curvature is independent of the sphere's embedding in 3-dimensional space. Also following from Gauss, a sphere cannot be mapped to a plane while maintaining both areas and angles. Therefore, any [[map projection]] introduces some form of distortion.

A sphere of radius {{mvar|r}} has [[area element]] <math>dA = r^2 \sin \theta\, d\theta\, d\varphi</math>. This can be found from the [[volume element]] in [[spherical coordinates]] with {{mvar|r}} held constant.<ref name=MathWorld_Sphere />

A sphere of any radius centered at zero is an [[integral surface]] of the following [[differential form]]:
:<math> x \, dx + y \, dy + z \, dz = 0.</math>
This equation reflects that the position vector and [[tangent plane (geometry)|tangent plane]] at a point are always [[Orthogonality|orthogonal]] to each other. Furthermore, the outward-facing [[normal vector]] is equal to the position vector scaled by {{mvar|1/r}}.

In [[Riemannian geometry]], the [[filling area conjecture]] states that the hemisphere is the optimal (least area) isometric filling of the [[Riemannian circle]].