definition 7.98 Functions of class $C^{1}$

open in the book · parts/02-mathematical-methods/05-real-analysis.tex:2918 · p. 241

ground object -- no derivation owed

Rests on

depends_on definition 7.20 Continuity 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 3.43 Map ¶
      - depends_on definition 3.24 Quantifiers ¶
      - depends_on definition 3.28 Set ¶
- depends_on equation 7.7 eq:ana-limit-left ¶
- depends_on equation 7.5 eq:ana-limit-right ¶
depends_on definition 7.97 Partial derivative; gradient ¶
- 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 definition 5.15 Basis ¶
  - depends_on definition 5.12 Subspace generated by a set of vectors ¶
    - depends_on definition 5.5 Linear combination ¶
      - depends_on definition 4.33 Vector space ¶
    - depends_on definition 5.7 Vector subspace ¶
      - depends_on definition 5.5 Linear combination ¶ ↺
    - proves proof ch:03-linear-algebra-representations@proof-3 ¶
  - depends_on definition 5.14 Linear independence ¶
    - depends_on definition 5.5 Linear combination ¶ ↺

Supports

depends_on corollary 7.113 Inverse function theorem ¶
- depends_on definition A.503 The substitution property ¶
  - depends_on lemma A.506 Locality ¶
    - depends_on proposition A.511 The substitution property is universal ¶
      - depends_on remark A.516 The hypotheses of the global form ¶
      - depends_on theorem A.513 Change of variables ¶
  - depends_on lemma A.504 Transitivity ¶
- depends_on lemma 7.116 Functions vanishing on a regular zero set ¶
- depends_on lemma A.510 Every diffeomorphism factorises locally ¶
  - depends_on proposition A.511 The substitution property is universal ¶ ↺
- depends_on remark 7.114 Where the proof is, and why ¶
- depends_on theorem 7.129 Change of variables in a multiple integral ¶
  - depends_on example 7.130 The two Jacobians this treatise uses ¶
    - depends_on lemma A.518 The excised ball ¶
      - depends_on proposition A.521 Every derivative passes onto $g$ ¶
      - depends_on proposition A.519 The convolution exists ¶
  - depends_on lemma A.518 The excised ball ¶ ↺
  - depends_on proposition A.519 The convolution exists ¶ ↺
- depends_on theorem 7.115 Constant rank ¶
depends_on definition 7.119 Envelope of a family ¶
- depends_on proposition 7.120 The envelope touches every member it meets ¶
depends_on definition 7.126 Line and surface integrals ¶
- depends_on corollary A.313 Stokes' theorem for an antisymmetric tensor field ¶
- depends_on definition 30.17 Traction ¶
  - depends_on proposition 30.19 The stress tensor is symmetric ¶
    - depends_on definition 30.20 Pressure and deviatoric stress ¶
      - depends_on definition 30.31 The engineering moduli ¶
      - depends_on definition 30.79 Maxwell viscoelasticity ¶
      - depends_on proposition 30.74 The maximum shear stress, and its indifference to pressure ¶
      - depends_on remark 30.75 Two criteria, one uncited ¶
  - depends_on theorem 30.18 Cauchy: the stress tensor exists ¶
    - depends_on definition 30.24 The elasticity tensor ¶
      - depends_on definition 30.37 Voigt notation ¶
      - depends_on phenomenon 30.28 Hooke's law ¶
      - depends_on proposition 30.25 Twenty-one constants ¶
      - depends_on proposition 30.38 The Cauchy relations ¶
      - depends_on remark 30.78 Anharmonicity: what the quadratic energy also suppresses ¶
      - depends_on theorem A.675 Three constants for a cubic crystal, two for an isotropic solid ¶
    - depends_on definition 31.1 Fluid ¶
      - depends_on definition 31.21 Ideal fluid ¶
      - depends_on definition 31.2 Knudsen number ¶
      - depends_on definition 31.32 Newtonian fluid; dynamic and kinematic viscosity ¶
      - depends_on proposition 31.16 Hydrostatic balance ¶
    - depends_on proposition 30.19 The stress tensor is symmetric ¶ ↺
    - depends_on proposition 30.74 The maximum shear stress, and its indifference to pressure ¶ ↺
    - depends_on remark 30.72 Stress concentration ¶
      - depends_on proposition 30.76 Griffith's criterion ¶
    - depends_on theorem A.695 Boussinesq's point-force solution, quoted ¶
      - depends_on definition A.698 The Hertz contact problem ¶
      - depends_on lemma A.697 Surface displacement under a distributed pressure ¶
    - depends_on theorem 30.21 Cauchy's equation of motion ¶
      - depends_on definition 30.22 Boundary conditions of elastostatics ¶
      - depends_on proposition 30.23 Virtual work; the weak form ¶
      - depends_on proposition 31.16 Hydrostatic balance ¶ ↺
      - depends_on remark 30.69 Saint-Venant's principle, and its exact status ¶
      - depends_on theorem 30.40 Navier–Cauchy equations ¶
      - depends_on theorem 31.22 Euler's equations of motion ¶
depends_on definition 7.127 Simple regions ¶
- depends_on lemma A.512 The boundary strip is thin ¶
  - depends_on theorem A.513 Change of variables ¶ ↺
- depends_on remark A.516 The hypotheses of the global form ¶ ↺
- depends_on theorem 7.133 Gauss ¶
  - depends_on lemma A.708 The disturbance flux vanishes ¶
    - depends_on corollary A.711 The disturbance is a dipole at leading order ¶
  - depends_on lemma A.712 The force is a far-field integral ¶
  - depends_on lemma A.117 Laplacian in orthogonal coordinates ¶
    - depends_on theorem A.120 Stäckel and Robertson conditions ¶
      - depends_on example A.122 Cartesian ¶
      - depends_on example A.123 Circular cylindrical ¶
      - depends_on example A.124 Spherical ¶
      - depends_on proposition A.128 The ellipsoidal system separates ¶
  - depends_on lemma 44.10 Divergence theorem on $(M,g)$ ¶
    - depends_on proposition 44.11 The boundary term, and the Gibbons–Hawking–York action ¶
  - depends_on lemma 31.11 Transport theorem for a material volume ¶
    - depends_on theorem 31.12 Conservation of mass ¶
      - depends_on corollary 31.13 Incompressible flow ¶
      - depends_on phenomenon 31.88 Sound travels at the adiabatic speed ¶
      - depends_on proposition 52.24 The p-mode cavity ¶
      - depends_on theorem 31.91 Rankine–Hugoniot jump conditions ¶
    - depends_on theorem 31.22 Euler's equations of motion ¶ ↺
  - depends_on lemma 106.86 A shift of a divergent integral leaves a surface term ¶
    - depends_on proposition 106.88 The obstruction ¶
      - depends_on theorem 106.95 The cancellation condition ¶
  - depends_on lemma 10.44 Green's identities ¶
    - depends_on proposition 10.71 Green's representation formula ¶
    - depends_on proposition 10.67 The Newtonian potential is the fundamental solution ¶
      - depends_on example 10.49 The method of images ¶
      - depends_on lemma A.94 Fundamental solution of $\pp_{\bar z}$ ¶
      - depends_on proposition A.522 The Newtonian potential inverts the Laplacian ¶
      - depends_on proposition 10.71 Green's representation formula ¶ ↺
      - depends_on remark 10.73 What compact support is doing, and the rate that replaces it ¶
      - depends_on theorem 7.137 Helmholtz decomposition ¶
    - depends_on theorem 10.45 Symmetry of the Green's function ¶
  - depends_on lemma 10.58 Darboux's equation for spherical means ¶
    - depends_on theorem 10.59 Kirchhoff's formula ¶
      - depends_on corollary 10.60 Huygens' principle in three space dimensions ¶
    - depends_on theorem 10.69 Mean-value property ¶
      - depends_on corollary A.145 Harmonic functions are smooth ¶
      - depends_on theorem A.135 Converse of the mean-value property ¶
      - depends_on theorem 10.77 Strong maximum principle for harmonic functions ¶
  - depends_on proposition 23.54 The geometrical amplitude diverges ¶
    - depends_on proposition 23.55 The fold, and the quarter-power law ¶
      - depends_on proposition 23.56 The phase jump at a fold ¶
      - depends_on remark 23.58 The caustics that are not turning surfaces ¶
    - depends_on remark 23.59 What is observed ¶
  - depends_on proposition 10.57 Energy in a backward cone ¶
    - depends_on proposition 28.63 Energy density, flux, and the conservation law ¶
  - depends_on theorem 16.36 Euler–Lagrange equations for several independent variables ¶
    - depends_on definition 44.6 Stress–energy tensor ¶
      - depends_on definition 44.36 Energy conditions ¶
      - depends_on proposition 44.17 Stress–energy of the electromagnetic field ¶
      - depends_on theorem 44.19 Covariant conservation of stress–energy ¶
      - depends_on theorem 44.9 Variation of the Einstein–Hilbert action ¶
    - depends_on proposition 28.42 The wave equation of a stretched string ¶
  - depends_on theorem 32.6 Evolution of phase volume ¶
    - depends_on definition 32.7 Conservative and dissipative flows ¶
      - depends_on definition 32.62 The Lorenz system ¶
    - depends_on definition 32.62 The Lorenz system ¶ ↺
    - depends_on proposition 32.64 The Lorenz flow contracts volume ¶
      - depends_on phenomenon 32.68 Strange attractors ¶
    - depends_on proposition 32.76 The exponents sum to the mean divergence ¶
  - … 1 more
- depends_on theorem 7.131 Green ¶
  - depends_on proposition 32.33 Bendixson's negative criterion ¶
  - depends_on proposition 9.36 Bendixson–Dulac negative criterion ¶
  - depends_on theorem 7.132 Stokes ¶
    - depends_on definition 31.63 Circulation ¶
      - depends_on theorem 31.64 Kelvin's circulation theorem ¶
    - depends_on proposition 7.136 Properties of conservative fields ¶
      - depends_on example 31.15 The stream function of a plane flow ¶
      - depends_on proposition 31.28 Potential flow reduces to Laplace's equation ¶
      - depends_on theorem 16.63 Hilbert's invariant integral ¶
      - depends_on theorem 30.15 Compatibility is sufficient on a simply connected body ¶
  - depends_on theorem 8.12 Cauchy ¶
    - depends_on corollary 8.15 Deformation of contours ¶
      - depends_on lemma 106.1 The short-time kernel ¶
      - depends_on theorem 8.16 Cauchy integral formula ¶
      - depends_on theorem 8.21 Laurent expansion ¶
      - depends_on theorem 8.24 Residue theorem ¶
    - depends_on corollary 17.71 Residue evaluation and causality ¶
      - depends_on corollary 17.76 Heaviside's expansion theorem ¶
      - depends_on proposition 17.77 Poles and stability ¶
    - depends_on lemma 100.27 Wick rotation ¶
      - depends_on lemma 100.28 The $d$-dimensional loop integral ¶
    - depends_on proposition 106.14 The Feynman contour permits the rotation ¶
    - depends_on theorem 17.78 Causality implies dispersion relations ¶
      - depends_on remark 33.10 Fibre anelasticity, and why it biases one method and not the other ¶
      - depends_on remark 28.26 Causality is visible in the susceptibility ¶
      - depends_on remark 28.62 Neither velocity is a speed limit ¶
    - depends_on theorem 17.94 Mellin inversion ¶
      - depends_on proposition 17.96 Poles give asymptotics ¶
      - depends_on theorem 17.100 Moments diagonalise a convolution evolution ¶
  - depends_on theorem 13.41 Holonomy equals the enclosed curvature; local Gauss–Bonnet ¶
    - depends_on example 13.42 The sphere, the solid angle, and the pole ¶
  - depends_on theorem 10.96 Rankine–Hugoniot condition ¶
    - depends_on definition 10.99 Entropy condition ¶
      - depends_on definition 10.101 Entropy pair; entropy inequality ¶
    - depends_on example 10.97 Burgers shock ¶
    - depends_on example 10.98 Weak solutions are not unique ¶
    - depends_on proposition 10.102 Jump form of the entropy inequality ¶
depends_on definition A.508 Primitive map ¶
- depends_on lemma A.510 Every diffeomorphism factorises locally ¶ ↺
- depends_on lemma A.509 Primitive maps have the substitution property ¶
  - depends_on proposition A.511 The substitution property is universal ¶ ↺
depends_on definition A.116 Orthogonal curvilinear coordinates; scale factors ¶
- depends_on definition A.125 Ellipsoidal coordinates ¶
  - depends_on lemma A.126 Cartesian coordinates and scale factors ¶
    - depends_on proposition A.128 The ellipsoidal system separates ¶ ↺
- depends_on definition A.118 Simple separation of the Helmholtz equation ¶
  - depends_on theorem A.120 Stäckel and Robertson conditions ¶ ↺
- depends_on lemma A.117 Laplacian in orthogonal coordinates ¶ ↺
depends_on definition A.298 Half-space; smoothness on it ¶
- depends_on definition A.299 Manifold with boundary ¶
  - depends_on lemma A.300 The boundary is well defined, and is a manifold ¶
    - depends_on definition A.309 The induced orientation of the boundary ¶
      - depends_on theorem A.311 General Stokes theorem ¶
  - depends_on lemma A.307 Partition of unity on a compact manifold ¶
    - depends_on definition A.308 Integral over the manifold ¶
      - depends_on theorem A.311 General Stokes theorem ¶ ↺
depends_on definition 16.11 Admissible class; functional ¶
- depends_on definition 16.59 Weierstrass excess function ¶
- depends_on definition 16.12 The two norms; weak and strong neighbourhoods ¶
  - depends_on definition 16.13 Weak and strong extrema ¶
    - depends_on example 16.14 A weak minimum that is not strong ¶
    - depends_on proposition 16.16 Stationarity is necessary ¶
      - depends_on theorem 16.22 Euler–Lagrange ¶
      - depends_on theorem 16.51 Legendre's necessary condition ¶
    - depends_on theorem 16.60 Weierstrass' necessary condition ¶
- depends_on definition 16.15 Variation; the first variation ¶
  - depends_on definition 16.83 One-parameter transformation group; variational symmetry ¶
  - depends_on lemma 16.18 Fundamental lemma ¶
    - depends_on theorem A.688 The Kirchhoff plate equation ¶
      - depends_on remark A.690 Poisson's count, and why it cannot be right ¶
      - depends_on theorem A.691 Kirchhoff's edge conditions and the corner force ¶
    - depends_on theorem 16.36 Euler–Lagrange equations for several independent variables ¶ ↺
    - depends_on theorem 16.38 Euler–Poisson equation ¶
      - depends_on theorem 30.56 The beam equation ¶
      - depends_on theorem 30.64 The Kirchhoff plate equation ¶
    - depends_on theorem 16.41 Multiplier rule for pointwise constraints ¶
      - depends_on proposition 21.56 The two prescriptions and their difference ¶
    - depends_on theorem 16.31 Natural boundary condition ¶
      - depends_on remark 30.67 Germain, Kirchhoff, and the edge conditions ¶
      - depends_on theorem A.691 Kirchhoff's edge conditions and the corner force ¶ ↺
      - depends_on theorem 16.32 Transversality ¶
    - depends_on theorem 25.19 Hamilton's equations from the first-order action ¶
    - depends_on theorem 25.22 Faddeev–Jackiw equations and brackets ¶
      - depends_on example 25.24 A charged particle in a strong magnetic field ¶
  - depends_on proposition 16.16 Stationarity is necessary ¶ ↺
  - depends_on theorem 16.43 Euler's rule for integral constraints ¶
    - depends_on example 16.45 Two solved isoperimetric problems ¶
    - depends_on proposition 16.4 The hanging chain is a catenary ¶
    - depends_on proposition 16.6 The isoperimetric extremal is a circle ¶
- depends_on proposition 16.68 Weierstrass' counterexample ¶
depends_on definition 10.1 Partial differential equation; order ¶
- depends_on definition A.102 Normal Cauchy problem of order $k$ ¶
  - depends_on definition A.103 First-order quasilinear system with zero data ¶
    - depends_on lemma A.104 The recursion ¶
      - depends_on lemma A.109 Domination ¶
      - depends_on lemma A.105 Formal reduction of a normal problem ¶
    - depends_on proposition A.110 Explicit solution of the majorant problem ¶
      - depends_on theorem A.112 Cauchy–Kovalevskaya, first-order form ¶
  - depends_on lemma A.105 Formal reduction of a normal problem ¶ ↺
- depends_on definition 10.2 Linear, semilinear, quasilinear ¶
  - depends_on definition A.103 First-order quasilinear system with zero data ¶ ↺
  - depends_on definition 10.93 Scalar conservation law ¶
    - depends_on definition A.150 Kruzhkov entropy pair ¶
      - depends_on lemma A.157 Two elementary identities ¶
      - depends_on proposition A.156 The Kruzhkov inequality ¶
      - depends_on theorem A.151 $L^{1}$ contraction on a cone ¶
    - depends_on definition 10.95 Weak solution of a conservation law ¶
      - depends_on definition 10.99 Entropy condition ¶ ↺
      - depends_on definition 10.101 Entropy pair; entropy inequality ¶ ↺
      - depends_on example 10.98 Weak solutions are not unique ¶ ↺
      - depends_on theorem A.151 $L^{1}$ contraction on a cone ¶ ↺
      - depends_on theorem 10.103 Uniqueness in the entropy class ¶
      - depends_on theorem 10.96 Rankine–Hugoniot condition ¶ ↺
    - depends_on example 10.97 Burgers shock ¶ ↺
    - depends_on proposition 10.94 Gradient catastrophe ¶
  - depends_on definition 10.3 Principal part and principal symbol ¶
    - depends_on definition 10.10 Characteristic surface ¶
      - depends_on proposition 10.13 Canonical form in two variables ¶
      - depends_on proposition 10.11 What a characteristic surface is ¶
    - depends_on definition 10.4 The second-order operator ¶
      - depends_on definition 10.10 Characteristic surface ¶ ↺
      - depends_on definition 10.42 Green's function ¶
      - depends_on definition 10.6 Type of a second-order operator ¶
    - depends_on definition 10.83 Weak solution ¶
      - depends_on definition 10.87 Weak form of an elliptic problem ¶
      - depends_on definition 10.85 Sobolev space ¶
      - depends_on definition 10.95 Weak solution of a conservation law ¶ ↺
      - depends_on proposition 10.84 Consistency ¶
  - depends_on lemma 10.28 The separation constant ¶
    - depends_on definition A.118 Simple separation of the Helmholtz equation ¶ ↺
    - depends_on example 10.31 Cartesian separation ¶
      - depends_on example A.122 Cartesian ¶ ↺
    - depends_on example 10.32 Cylindrical separation ¶
      - depends_on example A.123 Circular cylindrical ¶ ↺
    - depends_on example 10.33 Spherical separation ¶
      - depends_on example A.124 Spherical ¶ ↺
    - depends_on proposition 23.9 Separation of the time ¶
      - depends_on proposition 23.10 The characteristic function is the abbreviated action ¶
      - depends_on remark 23.12 Two functions, two roles ¶
    - depends_on proposition 10.29 Time separation reduces the canonical equations to Helmholtz ¶
      - depends_on example 10.38 The vibrating string ¶
      - depends_on phenomenon 28.53 Chladni's figures ¶
      - depends_on theorem 10.36 Solution of the heat problem on an interval ¶
    - depends_on theorem 10.34 Separable systems for the Helmholtz operator ¶
      - depends_on remark 23.19 The same conditions govern the Helmholtz equation ¶
  - depends_on proposition 28.44 Superposition ¶
  - depends_on proposition 10.15 Method of characteristics, first order ¶
    - depends_on definition 10.93 Scalar conservation law ¶ ↺
    - depends_on proposition 10.16 Cauchy's characteristic strips ¶
    - depends_on proposition 10.94 Gradient catastrophe ¶ ↺
    - depends_on remark 31.90 Why a compression wave must steepen ¶
    - depends_on remark 23.3 What Part II owes this chapter ¶
  - depends_on proposition 10.43 Representation of the solution ¶
  - depends_on theorem 10.22 Cauchy–Kovalevskaya ¶
  - depends_on theorem 10.52 Duhamel's principle ¶
    - depends_on corollary 10.53 Duhamel for the wave equation ¶
- depends_on definition 10.24 Well-posed problem ¶
  - depends_on corollary A.152 Uniqueness ¶
  - depends_on corollary 10.76 Uniqueness and stability for the Dirichlet problem ¶
  - depends_on corollary 10.81 Uniqueness for the heat equation ¶
  - depends_on definition 30.22 Boundary conditions of elastostatics ¶ ↺
  - depends_on proposition 10.25 The backward heat problem is ill posed ¶
  - depends_on theorem 44.45 Local existence and uniqueness; Choquet-Bruhat; imported ¶
    - depends_on theorem 44.46 Maximal globally hyperbolic development; Choquet-Bruhat and Geroch; imported ¶
depends_on definition 10.66 Harmonic function ¶
- depends_on theorem A.135 Converse of the mean-value property ¶ ↺
- depends_on theorem A.491 Dirichlet problem for the disc; quoted ¶
- depends_on theorem 10.69 Mean-value property ¶ ↺
- depends_on theorem 10.75 Weak maximum principle ¶
  - depends_on corollary 10.76 Uniqueness and stability for the Dirichlet problem ¶ ↺
  - depends_on theorem A.525 Helmholtz decomposition: uniqueness ¶
    - depends_on remark A.527 The hypotheses, and what happens without them ¶
  - depends_on theorem 10.80 Parabolic maximum principle ¶
    - depends_on corollary 10.81 Uniqueness for the heat equation ¶ ↺
    - depends_on proposition A.163 The viscous problem produces the entropy inequality ¶
depends_on definition 9.30 Hyperbolic equilibrium ¶
- depends_on theorem 9.32 Hartman–Grobman; quoted ¶
  - depends_on theorem 32.15 Hartman–Grobman, restated from Part II ¶
    - depends_on definition 32.22 Bifurcation ¶
      - depends_on proposition 32.23 Saddle-node bifurcation ¶
      - depends_on proposition 32.24 Transcritical and pitchfork bifurcations ¶
      - depends_on theorem 32.25 Hopf bifurcation, quoted ¶
    - depends_on example 32.18 A linear centre that is really a stable focus ¶
depends_on lemma A.520 A $C^{1}$ limit ¶
- depends_on proposition A.521 Every derivative passes onto $g$ ¶ ↺
… 8 more

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Edges

type	direction	node	provenance	where
`depends_on`	→	Continuity at a point	declared	parts/02-mathematical-methods/05-real-analysis.tex:2922
`depends_on`	→	Partial derivative; gradient	declared	parts/02-mathematical-methods/05-real-analysis.tex:2922
`depends_on`	←	Inverse function theorem	declared	parts/02-mathematical-methods/05-real-analysis.tex:3545
`depends_on`	←	Envelope of a family	declared	parts/02-mathematical-methods/05-real-analysis.tex:3823
`depends_on`	←	Line and surface integrals	declared	parts/02-mathematical-methods/05-real-analysis.tex:4215
`depends_on`	←	Simple regions	declared	parts/02-mathematical-methods/05-real-analysis.tex:4235
`depends_on`	←	Primitive map	declared	appendices/A-long-proofs.tex:24823
`depends_on`	←	Orthogonal curvilinear coordinates; scale factors	declared	appendices/A-long-proofs.tex:6707
`depends_on`	←	Half-space; smoothness on it	declared	appendices/A-long-proofs.tex:14894
`depends_on`	←	Admissible class; functional	declared	parts/02-mathematical-methods/14-calculus-of-variations.tex:386
`depends_on`	←	Partial differential equation; order	declared	parts/02-mathematical-methods/08-pdes.tex:72
`depends_on`	←	Harmonic function	declared	parts/02-mathematical-methods/08-pdes.tex:1787
`depends_on`	←	Hyperbolic equilibrium	declared	parts/02-mathematical-methods/07-odes-sturm-liouville.tex:1154
`depends_on`	←	A $C^{1}$ limit	declared	appendices/A-long-proofs.tex:25405
`depends_on`	←	The Newton map contracts	declared	appendices/A-long-proofs.tex:14408
`depends_on`	←	The flat exponential	declared	appendices/A-long-proofs.tex:7665
`depends_on`	←	Differentiation under the integral sign	declared	appendices/A-long-proofs.tex:7587
`depends_on`	←	Green's identities	declared	parts/02-mathematical-methods/08-pdes.tex:1204
`depends_on`	←	Darboux's equation for spherical means	declared	parts/02-mathematical-methods/08-pdes.tex:1585
`depends_on`	←	$C^{1}$ implies differentiable	declared	parts/02-mathematical-methods/05-real-analysis.tex:2967
`depends_on`	←	Cauchy–Riemann equations	declared	parts/02-mathematical-methods/06-complex-analysis.tex:135
`depends_on`	←	Constant rank theorem	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:2938