example A.294 The sphere, made explicit

Rests on

• depends_on corollary A.293 Solving one scalar equation for one coordinate ¶
  • depends_on theorem A.286 Implicit function theorem ¶
    • depends_on definition 7.99 Differentiability at a point ¶
      • depends_on definition 7.16 Limit ¶
        • depends_on definition 7.2 Absolute value ¶
        • depends_on definition 7.9 Real function ¶
      • depends_on definition 5.37 Linear transformation ¶
        • depends_on definition 4.33 Vector space ¶
      • depends_on equation 6.12 eq:top-euclidean-metric ¶
    • depends_on proposition 7.104 Chain rule in several variables ¶
      • depends_on definition 7.13 Composite function ¶
        • depends_on definition 7.9 Real function ¶ ↺
      • depends_on definition 7.99 Differentiability at a point ¶ ↺
      • depends_on definition 7.102 Differential of a function ¶
        • depends_on definition 7.99 Differentiability at a point ¶ ↺
        • depends_on definition 7.97 Partial derivative; gradient ¶
      • proves proof ch:05-real-analysis@proof-63 ¶
    • depends_on theorem 7.100 $C^{1}$ implies differentiable ¶
      • depends_on definition 7.98 Functions of class $C^{1}$ ¶
        • depends_on definition 7.20 Continuity at a point ¶
        • depends_on definition 7.97 Partial derivative; gradient ¶ ↺
      • depends_on definition 7.99 Differentiability at a point ¶ ↺
      • depends_on theorem 7.35 Mean value theorem ¶
        • depends_on proposition 7.29 Linearity ¶
        • depends_on theorem 7.34 Rolle ¶
        • proves proof ch:05-real-analysis@proof-19 ¶
      • proves proof ch:05-real-analysis@proof-62 ¶
    • proves proof app:A-long-proofs@proof-185 ¶
  • proves proof app:A-long-proofs@proof-187 ¶
• depends_on example 13.61 The $2$-sphere in $\R^{3}$ ¶
  • depends_on definition 13.58 Hypersurface ¶
    • depends_on definition 13.54 Embedded submanifold ¶
      • depends_on definition 13.53 Immersion, submersion, embedding ¶
        • depends_on definition 13.52 Differential; pushforward ¶
        • depends_on definition 6.7 Homeomorphism ¶
      • depends_on definition 13.48 Differentiable manifold ¶
        • depends_on definition 13.47 Differentiable structure ¶
  • depends_on theorem 13.59 Regular value theorem ¶
    • depends_on definition 13.52 Differential; pushforward ¶ ↺
    • depends_on definition 13.58 Hypersurface ¶ ↺
    • depends_on definition 13.49 Smooth map between manifolds ¶
      • depends_on definition 13.44 Atlas ¶
        • depends_on definition 13.43 Coordinate system ¶
      • depends_on definition 13.45 Differentiable map on a topological space ¶
        • depends_on definition 13.44 Atlas ¶ ↺
        • depends_on definition 13.43 Coordinate system ¶ ↺
      • depends_on definition 13.48 Differentiable manifold ¶ ↺
    • proves proof ch:11-manifolds-tensors-curvature@proof-10 ¶

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