proposition 5.95 The focal polar equation of a conic

Rests on

• depends_on corollary 7.44 Substitution and integration by parts ¶
  • depends_on proposition 7.31 Chain rule ¶
    • depends_on definition 7.13 Composite function ¶
      • depends_on definition 7.9 Real function ¶
        • depends_on definition 3.43 Map ¶
    • 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 definition 7.16 Limit ¶ ↺
        • depends_on definition 7.9 Real function ¶ ↺
      • 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-11 ¶
    • depends_on proposition 7.6 Algebra of limits ¶ ↺
    • proves proof ch:05-real-analysis@proof-15 ¶
  • 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 proposition 7.6 Algebra of limits ¶ ↺
    • proves proof ch:05-real-analysis@proof-14 ¶
  • depends_on theorem 7.43 Fundamental theorem of calculus, II ¶
    • depends_on definition 7.41 Antiderivative ¶
      • depends_on corollary 7.36 cor:ana-mvt-consequences ¶
        • depends_on theorem 7.35 Mean value theorem ¶
        • proves proof ch:05-real-analysis@proof-20 ¶
      • depends_on definition 7.26 Derivative of a function at a point ¶ ↺
    • depends_on theorem 7.42 Fundamental theorem of calculus, I ¶
      • depends_on definition 7.26 Derivative of a function at a point ¶ ↺
      • depends_on theorem 7.40 Continuous functions are integrable ¶
        • depends_on definition 7.39 Darboux sums and the definite integral ¶
        • depends_on theorem 7.24 Extreme value theorem ¶
        • depends_on theorem 7.25 Heine–Cantor: uniform continuity ¶
        • proves proof ch:05-real-analysis@proof-23 ¶
      • depends_on theorem 7.24 Extreme value theorem ¶ ↺
      • proves proof ch:05-real-analysis@proof-24 ¶
    • proves proof ch:05-real-analysis@proof-25 ¶
  • proves proof ch:05-real-analysis@proof-26 ¶
• depends_on equation 5.10 eq:lin-trig-power-cos ¶
• depends_on proposition 5.94 Principal axes of a real quadratic form ¶
  • depends_on definition 5.117 Signature ¶
    • depends_on definition 5.113 Non-degenerate form ¶
      • depends_on definition 5.110 Bilinear map ¶
        • depends_on definition 5.37 Linear transformation ¶
    • depends_on theorem 5.116 Sylvester's law of inertia ¶
      • depends_on definition 5.110 Bilinear map ¶ ↺
      • depends_on definition 5.111 Symmetric and antisymmetric forms ¶
        • depends_on definition 5.110 Bilinear map ¶ ↺
        • depends_on equation 5.137 eq:lin-bilinear-matrix ¶
      • depends_on equation 5.139 eq:lin-bilinear-congruence ¶
      • proves proof ch:03-linear-algebra-representations@proof-53 ¶
  • depends_on theorem 5.84 Spectral theorem for a real symmetric operator ¶
    • depends_on proposition 5.28 Gram–Schmidt ¶
      • depends_on definition 5.14 Linear independence ¶
        • depends_on definition 5.5 Linear combination ¶
      • depends_on definition 5.18 Inner product ¶
        • depends_on definition 4.33 Vector space ¶
      • depends_on definition 5.27 Orthonormal basis ¶
        • depends_on definition 5.26 Orthogonal basis ¶
        • depends_on definition 5.25 Unit vector ¶
      • depends_on equation 5.44 eq:lin-norm-assoc ¶
      • proves proof ch:03-linear-algebra-representations@proof-5 ¶
    • depends_on theorem 5.71 The eigenvalues are the roots of the characteristic polynomial ¶
      • depends_on definition 5.70 Characteristic polynomial ¶
        • depends_on definition 5.56 Endomorphism ¶
        • depends_on equation 5.19 eq:lin-leibniz-det ¶
        • depends_on equation 5.74 eq:lin-matrix-rep ¶
        • proves proof ch:03-linear-algebra-representations@proof-25 ¶
      • depends_on definition 5.69 Eigenvector, eigenvalue, eigenspace ¶
        • depends_on definition 5.56 Endomorphism ¶ ↺
        • depends_on definition 5.39 Kernel, image, nullity, rank ¶
      • depends_on proposition 5.49 Injective, surjective, invertible ¶
        • depends_on definition 5.39 Kernel, image, nullity, rank ¶ ↺
        • depends_on proposition 5.48 Existence, uniqueness, and linearity of the inverse ¶
        • depends_on theorem 5.40 Rank–nullity ¶
        • proves proof ch:03-linear-algebra-representations@proof-18 ¶
      • proves proof ch:03-linear-algebra-representations@proof-26 ¶
    • depends_on theorem 5.79 Spectral theorem for a self-adjoint operator ¶
      • depends_on corollary 5.72 Existence of an eigenvalue over $\C$ ¶
        • depends_on theorem 8.19 Fundamental theorem of algebra ¶
        • depends_on theorem 5.71 The eigenvalues are the roots of the characteristic polynomial ¶ ↺
        • proves proof ch:03-linear-algebra-representations@proof-27 ¶
      • depends_on definition 5.41 Adjoint ¶
        • depends_on definition 5.18 Inner product ¶ ↺
        • depends_on definition 5.37 Linear transformation ¶ ↺
      • depends_on theorem 5.40 Rank–nullity ¶ ↺
      • proves proof ch:03-linear-algebra-representations@proof-31 ¶
    • proves proof ch:03-linear-algebra-representations@proof-34 ¶
  • depends_on theorem 5.116 Sylvester's law of inertia ¶ ↺
  • proves proof ch:03-linear-algebra-representations@proof-41 ¶
• proves proof ch:03-linear-algebra-representations@proof-42 ¶

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