definition 7.97 Partial derivative; gradient

open in the book · parts/02-mathematical-methods/05-real-analysis.tex:2900 · p. 241

ground object -- no derivation owed

Rests on

depends_on definition 7.26 Derivative of a function 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 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 4.8 Commutativity; abelian structure ¶
      - depends_on definition 4.32 Field ¶
      - depends_on definition 4.31 Module ¶
  - 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 definition 7.98 Functions of class $C^{1}$ ¶
- 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 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 lemma A.518 The excised ball ¶ ↺
    - depends_on proposition A.519 The convolution exists ¶
      - depends_on proposition A.524 Rate of decay ¶
  - 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 theorem 30.18 Cauchy: the stress tensor exists ¶
      - depends_on definition 30.24 The elasticity tensor ¶
      - depends_on definition 31.1 Fluid ¶
      - 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 theorem A.695 Boussinesq's point-force solution, quoted ¶
      - depends_on theorem 30.21 Cauchy's equation 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 corollary A.514 Degeneracy on a negligible set ¶
      - depends_on remark A.516 The hypotheses of the global form ¶
  - 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 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 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 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 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 theorem 10.69 Mean-value property ¶
    - depends_on proposition 23.54 The geometrical amplitude diverges ¶
      - depends_on proposition 23.55 The fold, and the quarter-power law ¶
      - 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 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 proposition 32.64 The Lorenz flow contracts volume ¶
      - 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 proposition 7.136 Properties of conservative fields ¶
    - depends_on theorem 8.12 Cauchy ¶
      - depends_on corollary 8.15 Deformation of contours ¶
      - depends_on corollary 17.71 Residue evaluation and causality ¶
      - depends_on lemma 100.27 Wick rotation ¶
      - depends_on proposition 106.14 The Feynman contour permits the rotation ¶
      - depends_on theorem 17.78 Causality implies dispersion relations ¶
      - depends_on theorem 17.94 Mellin inversion ¶
    - 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 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 lemma A.307 Partition of unity on a compact manifold ¶
      - depends_on definition A.308 Integral over the manifold ¶
- 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 16.60 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 theorem 16.36 Euler–Lagrange equations for several independent variables ¶ ↺
      - depends_on theorem 16.38 Euler–Poisson equation ¶
      - depends_on theorem 16.41 Multiplier rule for pointwise constraints ¶
      - depends_on theorem 16.31 Natural boundary condition ¶
      - 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 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 proposition A.110 Explicit solution of the majorant problem ¶
    - 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 definition 10.95 Weak solution of a conservation law ¶
      - 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 definition 10.4 The second-order operator ¶
      - depends_on definition 10.83 Weak solution ¶
    - 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 10.32 Cylindrical separation ¶
      - depends_on example 10.33 Spherical separation ¶
      - depends_on proposition 23.9 Separation of the time ¶
      - depends_on proposition 10.29 Time separation reduces the canonical equations to Helmholtz ¶
      - depends_on theorem 10.34 Separable systems for the Helmholtz operator ¶
    - 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 30.50 Love waves need a slow surface layer ¶
      - depends_on proposition 30.48 Rayleigh's secular equation ¶
      - depends_on proposition 30.23 Virtual work; the weak form ¶
    - 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 corollary A.146 Locally uniform limits of harmonic functions ¶
    - depends_on corollary A.145 Harmonic functions are smooth ¶
  - 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 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 $g$ ¶
    - depends_on proposition A.524 Rate of decay ¶ ↺
    - depends_on proposition A.522 The Newtonian potential inverts the Laplacian ¶
      - depends_on theorem A.523 Helmholtz decomposition: existence ¶
- … 8 more
depends_on definition 7.102 Differential of a function ¶
- depends_on proposition 7.104 Chain rule in several variables ¶
  - depends_on corollary 7.113 Inverse function theorem ¶ ↺
  - depends_on corollary 7.110 Variable limits of integration ¶
  - depends_on corollary A.292 Inverse function theorem ¶
    - depends_on lemma A.549 The action is transitive ¶
      - depends_on proposition A.552 The component is a torus ¶
    - depends_on lemma A.300 The boundary is well defined, and is a manifold ¶ ↺
    - depends_on proposition A.535 Existence of a slice ¶
      - depends_on lemma A.536 A slice is a chart domain downstairs ¶
      - depends_on remark A.542 Where each hypothesis is spent ¶
    - depends_on proposition 22.32 Duality of the two brackets ¶
      - depends_on remark 22.33 What the duality is good for ¶
    - depends_on proposition 22.3 Invertibility of the Legendre map ¶
      - depends_on definition 26.3 Singular Lagrangian ¶
      - depends_on remark 22.4 When the Legendre map is singular ¶
    - depends_on proposition 13.132 Simultaneous straightening of commuting fields ¶
      - depends_on theorem 13.133 Frobenius ¶
      - depends_on theorem 13.137 Commuting complete fields on a compact manifold ¶
    - depends_on remark 22.19 Which form exists ¶
    - depends_on theorem 13.63 Constant rank theorem ¶
      - depends_on corollary 13.64 The image of a constant-rank map, locally ¶
      - depends_on lemma A.533 The orbit map has constant rank $d$ ¶
      - depends_on lemma A.538 Submersions have smooth local sections ¶
      - depends_on theorem 13.67 Quotient manifold theorem ¶
  - depends_on definition 30.4 Material and spatial descriptions ¶
  - depends_on definition 31.4 Material derivative ¶
    - depends_on definition 31.6 Streamline, pathline, streakline ¶
      - depends_on proposition 31.7 The three coincide in steady flow ¶
    - depends_on lemma 31.11 Transport theorem for a material volume ¶ ↺
  - depends_on definition 22.26 Lagrange bracket ¶
  - depends_on definition 10.18 Complete integral; envelope ¶
    - depends_on proposition 10.19 An envelope of solutions is a solution ¶
  - depends_on lemma A.510 Every diffeomorphism factorises locally ¶ ↺
  - depends_on lemma A.504 Transitivity ¶ ↺
  - depends_on lemma A.306 The standard smooth bump ¶
    - depends_on lemma A.307 Partition of unity on a compact manifold ¶ ↺
  - depends_on proposition 7.136 Properties of conservative fields ¶ ↺
  - depends_on proposition 7.120 The envelope touches every member it meets ¶ ↺
  - … 9 more
- depends_on remark 7.103 rem:ana-differential-scalar ¶
depends_on definition 7.119 Envelope of a family ¶ ↺
depends_on definition A.70 Pullback ¶
- depends_on definition 24.17 Symplectomorphism ¶
  - depends_on definition 25.6 Infinitesimal canonical transformation ¶
    - depends_on definition 25.11 Generators of the Galilei transformations ¶
      - depends_on proposition 25.12 The algebra of generators closes ¶
  - depends_on definition 24.28 Ball and cylinder ¶
    - depends_on definition 24.33 Symplectic capacity ¶
      - depends_on proposition 24.34 Capacities exist if and only if non-squeezing holds ¶
    - depends_on example 24.29 A volume-preserving squeeze ¶
    - depends_on proposition 24.30 Linear non-squeezing ¶
      - depends_on theorem 24.31 Gromov's non-squeezing theorem ¶
    - depends_on theorem 24.31 Gromov's non-squeezing theorem ¶ ↺
  - depends_on definition 24.43 Momentum map ¶
    - depends_on definition A.561 Coadjoint action and equivariance ¶
      - depends_on lemma A.567 The differential of the momentum map along the orbit ¶
      - depends_on proposition A.566 The isotropy group acts, and the quotient is smooth ¶
      - depends_on theorem A.563 Marsden–Weinstein reduction ¶
    - depends_on example 24.44 Linear and angular momentum ¶
      - depends_on example A.572 Rotational reduction of the central-force problem ¶
    - depends_on lemma 24.46 The level set of the momentum map is the symplectic orthogonal of the orbit ¶
      - depends_on proposition A.565 Freeness makes every value regular ¶
      - depends_on theorem A.563 Marsden–Weinstein reduction ¶ ↺
      - depends_on theorem A.568 The radical is the isotropy orbit ¶
      - depends_on theorem 24.47 Marsden–Weinstein reduction ¶
    - depends_on theorem 24.45 Noether, symplectic form ¶
      - depends_on proposition A.571 Invariant Hamiltonians descend with their flows ¶
  - depends_on theorem 24.18 Symplectic form of the transformation condition ¶
    - depends_on corollary 24.19 Canonical transformations preserve phase volume ¶
- depends_on lemma A.75 Differentiating a pullback along a flow ¶
  - depends_on lemma A.77 Poincaré lemma, converse form ¶
- depends_on lemma A.71 The pullback is an algebra map commuting with $\dd$ ¶
depends_on definition 32.19 Lyapunov function ¶
- depends_on theorem 32.20 Lyapunov's direct method ¶
  - depends_on example 32.21 The damped pendulum ¶
depends_on definition 30.2 Representative volume element; continuum field ¶
- depends_on definition 30.4 Material and spatial descriptions ¶ ↺
- depends_on definition 30.17 Traction ¶ ↺
- depends_on example 30.3 How small a piece of copper is still a continuum ¶
- depends_on remark 30.5 Where the hypothesis fails, and how one can tell ¶
depends_on definition 22.29 Poisson bracket ¶
- depends_on definition 26.20 Dirac bracket ¶
  - depends_on definition 26.23 Gauge conditions ¶
    - depends_on proposition 26.24 The Faddeev–Popov determinant ¶
      - depends_on remark 26.26 What the factor is doing ¶
  - depends_on example 26.22 Holonomic constraints ¶
  - depends_on theorem A.617 Jacobi identity for the Dirac bracket ¶
  - depends_on theorem 26.21 Properties of the Dirac bracket ¶
    - depends_on postulate 26.47 Dirac quantization ¶
      - depends_on definition 26.50 The BRST charge ¶
      - depends_on proposition 26.48 Consistency of the state conditions ¶
- depends_on definition 26.12 First and second class ¶
  - depends_on proposition 26.13 The first-class functions close ¶
    - depends_on proposition 26.43 The hypersurface-deformation algebra ¶
    - depends_on proposition 26.37 Constraint structure of Yang–Mills theory ¶
  - depends_on proposition 26.19 Second-class constraints come in pairs ¶
    - depends_on definition 26.20 Dirac bracket ¶ ↺
    - depends_on proposition 26.24 The Faddeev–Popov determinant ¶ ↺
    - depends_on proposition 26.25 The Liouville measure of a second-class surface ¶
      - depends_on proposition A.619 The Dirac bracket is a projected Hamiltonian flow ¶
      - depends_on remark 26.26 What the factor is doing ¶ ↺
    - depends_on theorem 26.17 Counting the physical degrees of freedom ¶
      - depends_on corollary A.642 The constraints are first class ¶
      - depends_on definition 26.23 Gauge conditions ¶ ↺
      - depends_on proposition 26.32 Constraint structure of the free Maxwell field ¶
      - depends_on proposition 26.34 Proca constraints are second class ¶
      - depends_on proposition 26.29 Reparametrization invariance and the mass-shell constraint ¶
      - depends_on theorem 26.42 The Hamiltonian structure of general relativity ¶
    - depends_on theorem 26.21 Properties of the Dirac bracket ¶ ↺
  - depends_on proposition 26.34 Proca constraints are second class ¶ ↺
- depends_on definition 26.8 Total Hamiltonian ¶
  - depends_on definition 26.16 Extended Hamiltonian ¶
  - depends_on theorem 26.21 Properties of the Dirac bracket ¶ ↺
  - depends_on theorem 26.14 First-class constraints generate gauge transformations ¶
    - depends_on definition 26.16 Extended Hamiltonian ¶ ↺
    - depends_on postulate 26.47 Dirac quantization ¶ ↺
    - depends_on theorem 26.17 Counting the physical degrees of freedom ¶ ↺
- depends_on definition 25.11 Generators of the Galilei transformations ¶ ↺
- depends_on definition 25.6 Infinitesimal canonical transformation ¶ ↺
- depends_on definition 25.2 Poisson algebra ¶
- depends_on definition 24.36 Poisson manifold ¶
  - depends_on definition 25.2 Poisson algebra ¶ ↺
  - depends_on definition 24.37 Casimir function ¶
    - depends_on proposition 24.41 The free rigid body is a Lie–Poisson system ¶
      - depends_on remark 24.48 Reduction is what physicists do without saying so ¶
      - depends_on remark 24.42 The intermediate axis ¶
    - depends_on remark 29.27 Euler's equations as a Lie–Poisson system ¶
    - depends_on theorem 24.38 Symplectic foliation, quoted ¶
  - depends_on definition 24.40 Lie–Poisson bracket ¶
    - depends_on proposition 24.41 The free rigid body is a Lie–Poisson system ¶ ↺
    - depends_on remark 29.27 Euler's equations as a Lie–Poisson system ¶ ↺
  - depends_on theorem 24.38 Symplectic foliation, quoted ¶ ↺
- depends_on notation 26.28 Field systems ¶
  - depends_on definition 26.40 The $3+1$ decomposition ¶
    - depends_on lemma A.623 The adapted frame of the $3+1$ split ¶
      - depends_on lemma A.626 Decomposition of $\nabla n$, and the acceleration ¶
    - depends_on proposition 26.41 The ADM form of the Einstein–Hilbert Lagrangian ¶
      - depends_on theorem 26.42 The Hamiltonian structure of general relativity ¶ ↺
  - depends_on notation A.636 Densities and smearing ¶
    - depends_on lemma A.637 Closed form of $H_{\parallel}$, and what it generates ¶
      - depends_on lemma A.638 Naturality ¶
  - depends_on notation 26.36 Gauge fields and their SI dimensions ¶
    - depends_on proposition 26.37 Constraint structure of Yang–Mills theory ¶ ↺
- depends_on notation 26.1 Weak and strong equality ¶
  - depends_on notation 26.28 Field systems ¶ ↺
- depends_on proposition 24.15 Poisson bracket from the symplectic form ¶
  - depends_on definition 24.37 Casimir function ¶ ↺
  - depends_on theorem 24.45 Noether, symplectic form ¶ ↺
depends_on lemma A.73 Differentiation under the integral sign ¶
- depends_on lemma A.77 Poincaré lemma, converse form ¶ ↺

Neighborhood

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Edges

type	direction	node	provenance	where
`depends_on`	→	Derivative of a function at a point	declared	parts/02-mathematical-methods/05-real-analysis.tex:2915
`depends_on`	→	Basis	declared	parts/02-mathematical-methods/05-real-analysis.tex:2915
`depends_on`	←	Functions of class $C^{1}$	declared	parts/02-mathematical-methods/05-real-analysis.tex:2922
`depends_on`	←	Differential of a function	declared	parts/02-mathematical-methods/05-real-analysis.tex:3082
`depends_on`	←	Envelope of a family	declared	parts/02-mathematical-methods/05-real-analysis.tex:3823
`depends_on`	←	Pullback	declared	appendices/A-long-proofs.tex:4507
`depends_on`	←	Lyapunov function	declared	parts/03-classical-mechanics/15-nonlinear-dynamics-chaos.tex:454
`depends_on`	←	Representative volume element; continuum field	declared	parts/03-classical-mechanics/13-continuum-elasticity.tex:89
`depends_on`	←	Poisson bracket	declared	parts/03-classical-mechanics/05-hamiltonian-mechanics.tex:962
`depends_on`	←	Differentiation under the integral sign	declared	appendices/A-long-proofs.tex:4644