proposition 7.108 Euler's theorem on homogeneous functions

Rests on

• depends_on corollary 7.36 cor:ana-mvt-consequences ¶
  • depends_on theorem 7.35 Mean value theorem ¶
    • depends_on proposition 7.29 Linearity ¶
      • depends_on definition 7.11 Sum of functions ¶
        • depends_on definition 7.9 Real function ¶
      • depends_on equation 7.14 eq:ana-derivh ¶
      • depends_on proposition 7.6 Algebra of limits ¶
        • depends_on definition 7.4 Convergence ¶
        • depends_on proposition 7.3 Triangle inequality ¶
        • proves proof ch:05-real-analysis@proof-3 ¶
      • proves proof ch:05-real-analysis@proof-13 ¶
    • depends_on theorem 7.34 Rolle ¶
      • depends_on lemma 7.33 Fermat: interior extremum ¶
        • depends_on definition 7.26 Derivative of a function at a point ¶
        • depends_on proposition 7.18 Two-sided limit from one-sided limits ¶
        • proves proof ch:05-real-analysis@proof-17 ¶
      • depends_on theorem 7.24 Extreme value theorem ¶
        • depends_on axiom 7.1 Completeness of $\R$ ¶
        • depends_on proposition 7.22 Sequential characterization ¶
        • depends_on theorem 7.7 Bolzano–Weierstrass ¶
        • proves proof ch:05-real-analysis@proof-9 ¶
      • proves proof ch:05-real-analysis@proof-18 ¶
    • proves proof ch:05-real-analysis@proof-19 ¶
  • proves proof ch:05-real-analysis@proof-20 ¶
• 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 7.16 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 definition 4.8 Commutativity; abelian structure ¶
        • depends_on definition 4.32 Field ¶
        • depends_on definition 4.31 Module ¶
    • depends_on equation 6.12 eq:top-euclidean-metric ¶
  • 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 ¶
      • depends_on definition 7.26 Derivative of a function at a point ¶ ↺
      • depends_on definition 5.15 Basis ¶
        • depends_on definition 5.12 Subspace generated by a set of vectors ¶
        • depends_on definition 5.14 Linear independence ¶
  • proves proof ch:05-real-analysis@proof-63 ¶
• depends_on proposition 7.30 Leibniz rule ¶
  • depends_on definition 7.12 Product of functions ¶
    • depends_on definition 7.9 Real function ¶ ↺
  • depends_on equation 7.14 eq:ana-derivh ¶ ↺
  • depends_on proposition 7.27 Differentiable implies continuous ¶
    • depends_on definition 7.20 Continuity at a point ¶
      • depends_on definition 7.16 Limit ¶ ↺
      • depends_on equation 7.7 eq:ana-limit-left ¶
      • depends_on equation 7.5 eq:ana-limit-right ¶
    • depends_on definition 7.26 Derivative of a function at a point ¶ ↺
    • depends_on proposition 7.6 Algebra of limits ¶ ↺
    • proves proof ch:05-real-analysis@proof-11 ¶
  • depends_on proposition 7.6 Algebra of limits ¶ ↺
  • proves proof ch:05-real-analysis@proof-14 ¶
• proves proof ch:05-real-analysis@proof-66 ¶

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