proposition 10.29 Time separation reduces the canonical equations to Helmholtz
open in the book ·
parts/02-mathematical-methods/08-pdes.tex:820
· p. 348
Rests on
- depends_on equation 10.25 eq:pde-helmholtz-equation ¶
-
depends_on
lemma 10.28
The separation constant
¶
-
depends_on
definition 10.2
Linear, semilinear, quasilinear
¶
-
depends_on
definition 10.1
Partial differential equation; order
¶
-
depends_on
definition 7.98
Functions of class $C^{1}$
¶
- depends_on definition 7.20 Continuity at a point ¶
- depends_on definition 7.97 Partial derivative; gradient ¶
-
depends_on
definition 6.2
Open set
¶
- depends_on definition 6.1 Topological space ¶
-
depends_on
definition 7.98
Functions of class $C^{1}$
¶
-
depends_on
definition 10.1
Partial differential equation; order
¶
- proves proof ch:08-pdes@proof-9 ¶
-
depends_on
definition 10.2
Linear, semilinear, quasilinear
¶
- proves proof ch:08-pdes@proof-10 ¶
Supports
- 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 ¶
Neighborhood
Every logical edge within two steps of this node.
- declared and complete
- partly declared
- a check failed
- not graded
- declared in the source
- inferred from structure
Edges
| type | direction | node | provenance | where |
|---|---|---|---|---|
depends_on |
→ | eq:pde-helmholtz-equation | declared | parts/02-mathematical-methods/08-pdes.tex:836 |
depends_on |
→ | The separation constant | declared | parts/02-mathematical-methods/08-pdes.tex:836 |
depends_on |
← | The vibrating string | declared | parts/02-mathematical-methods/08-pdes.tex:1072 |
depends_on |
← | Chladni's figures | declared | parts/03-classical-mechanics/11-oscillations-waves.tex:1932 |
depends_on |
← | Solution of the heat problem on an interval | declared | parts/02-mathematical-methods/08-pdes.tex:1017 |
proves |
← | ch:08-pdes@proof-10 | declared | parts/02-mathematical-methods/08-pdes.tex:839 |