theorem 31.91 Rankine–Hugoniot jump conditions

open in the book · parts/03-classical-mechanics/14-fluid-dynamics.tex:3117 · p. 1077

Rests on

depends_on theorem 31.12 Conservation of mass ¶
- depends_on lemma 31.11 Transport theorem for a material volume ¶
  - depends_on definition 31.4 Material derivative ¶
    - depends_on equation 7.124 eq:ana-nabla-gradient ¶
    - depends_on proposition 7.104 Chain rule in several variables ¶
      - depends_on definition 7.13 Composite function ¶
      - depends_on definition 7.99 Differentiability at a point ¶
      - depends_on definition 7.102 Differential of a function ¶
      - proves proof ch:05-real-analysis@proof-63 ¶
    - proves proof ch:14-fluid-dynamics@proof-1 ¶
  - depends_on theorem 7.133 Gauss ¶
    - depends_on definition 7.127 Simple regions ¶
      - depends_on definition 7.98 Functions of class $C^{1}$ ¶
      - depends_on definition 6.9 Compact set ¶
    - depends_on remark 7.128 What the derivations below take as given ¶
      - depends_on definition 7.125 Multiple integral ¶
      - depends_on theorem 7.40 Continuous functions are integrable ¶
      - depends_on theorem 7.25 Heine–Cantor: uniform continuity ¶
    - depends_on theorem 7.43 Fundamental theorem of calculus, II ¶
      - depends_on definition 7.41 Antiderivative ¶
      - depends_on theorem 7.42 Fundamental theorem of calculus, I ¶
      - proves proof ch:05-real-analysis@proof-25 ¶
    - proves proof ch:05-real-analysis@proof-79 ¶
  - proves proof ch:14-fluid-dynamics@proof-4 ¶
- proves proof ch:14-fluid-dynamics@proof-5 ¶
depends_on theorem 31.22 Euler's equations of motion ¶
- depends_on definition 31.21 Ideal fluid ¶
  - depends_on definition 31.1 Fluid ¶
    - depends_on theorem 30.18 Cauchy: the stress tensor exists ¶
      - depends_on definition 30.17 Traction ¶
      - depends_on definition 13.5 Tensor under orthogonal transformations ¶
      - proves proof ch:13-continuum-elasticity@proof-7 ¶
- depends_on lemma 31.11 Transport theorem for a material volume ¶ ↺
- depends_on theorem 30.21 Cauchy's equation of motion ¶
  - depends_on theorem 7.133 Gauss ¶ ↺
  - depends_on theorem 30.18 Cauchy: the stress tensor exists ¶ ↺
  - proves proof ch:13-continuum-elasticity@proof-9 ¶
- proves proof ch:14-fluid-dynamics@proof-11 ¶
proves proof ch:14-fluid-dynamics@proof-45 ¶

Supports

depends_on phenomenon 31.93 The Mach cone ¶
depends_on remark 31.92 Entropy selects the physical branch ¶

Neighborhood

Every logical edge within two steps of this node.

declared and complete
partly declared
a check failed
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inferred from structure

Edges

type	direction	node	provenance	where
`cites`	→	Mémoire sur la propagation du mouvement dans les corps et spécialement dans les gaz parfaits (deuxième partie)	derived	parts/03-classical-mechanics/14-fluid-dynamics.tex:3127
`cites`	→	On the thermodynamic theory of waves of finite longitudinal disturbance	derived	parts/03-classical-mechanics/14-fluid-dynamics.tex:3127
`depends_on`	→	Conservation of mass	declared	parts/03-classical-mechanics/14-fluid-dynamics.tex:3132
`depends_on`	→	Euler's equations of motion	declared	parts/03-classical-mechanics/14-fluid-dynamics.tex:3132
`depends_on`	←	The Mach cone	declared	parts/03-classical-mechanics/14-fluid-dynamics.tex:3185
`depends_on`	←	Entropy selects the physical branch	declared	parts/03-classical-mechanics/14-fluid-dynamics.tex:3172
`proves`	←	ch:14-fluid-dynamics@proof-45	declared	parts/03-classical-mechanics/14-fluid-dynamics.tex:3135