theorem 10.62 The heat kernel

open in the book · parts/02-mathematical-methods/08-pdes.tex:1692 · p. 358

Rests on

depends_on equation 17.42 eq:ft-gaussian ¶
depends_on proposition 10.39 Fourier reduction on the line ¶
- depends_on definition 17.29 Fourier transform; the treatise convention ¶
- depends_on equation 17.38 eq:ft-derivative ¶
- proves proof ch:08-pdes@proof-12 ¶
depends_on theorem 17.53 Convolution theorem ¶
- depends_on definition 17.50 Convolution ¶
- depends_on definition 17.29 Fourier transform; the treatise convention ¶ ↺
- depends_on proposition 17.51 Algebra of convolution ¶
  - depends_on definition 17.50 Convolution ¶ ↺
  - proves proof ch:15-fourier-integral-transforms@proof-32 ¶
- depends_on theorem 17.32 Fourier inversion ¶
  - depends_on definition 17.29 Fourier transform; the treatise convention ¶ ↺
  - depends_on equation 17.42 eq:ft-gaussian ¶ ↺
  - proves proof ch:15-fourier-integral-transforms@proof-20 ¶
- proves proof ch:15-fourier-integral-transforms@proof-34 ¶
proves proof ch:08-pdes@proof-25 ¶

Supports

depends_on proposition 10.63 Infinite propagation speed ¶

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:ft-gaussian	declared	parts/02-mathematical-methods/08-pdes.tex:1705
`depends_on`	→	Fourier reduction on the line	declared	parts/02-mathematical-methods/08-pdes.tex:1705
`depends_on`	→	Convolution theorem	declared	parts/02-mathematical-methods/08-pdes.tex:1705
`depends_on`	←	Infinite propagation speed	declared	parts/02-mathematical-methods/08-pdes.tex:1741
`proves`	←	ch:08-pdes@proof-25	declared	parts/02-mathematical-methods/08-pdes.tex:1709