PNAS-1952-Kasner-The Fourier Heat Equation in Riemannian Space
5
MATHEMATICS: K SNER AND DE CICCO PRoc. N . A . S . 1 1 Larkin, R . A . , MacMasters, M. M . , Cull, I . M . , Wolf, M. J ., a n d Rist, C . E. Stain Tech., 2 7 , 1 07 (1952). 1 2 Randolph, L. F . , Ibid., 1 5 , 4 5 (1940). THE FOURIER HEAT EQUATION IN RIEMANNIAN SPACE* B y EDWARD KASNER an d JOHN D E CiCco DEPARTMENTS O F MATHEMATICS, COLUMBIA UNIVERSITY, N E W YORK, AN D D E PAUL UNIVERSITY, CHICAGO Communicated July 2 , 1952 1 . L e t a certain body, capable o f absorbing heat, occupy a region o f n dimensional Riemannian space V , whose metric i s defined b y t h e positive quadratic differential form d s 2 = g j j d x l dxJ. Suppose that this body i s heated b y conduction i n any manner. Th e situation c a n b e visualized a s a flow o f heat from t h e warmer t o t h e cooler parts o f t h e body. Th e rate o f flow o f heat within t h e body c a n be depicted a s a vector field XI(x; t ) o r X1(x; t ) , where X i = g JX, and Xi = gijX . At t h e position x within t h e body f o r th e particular instant o f time t , t he vector X i s i n t he direction o f t h e flow o f heat a nd i ts magnitude i s t h e rate o f flow o f heat. I t i s assumed that this vector field XI x; t ) o r X,(x; t ) i s single valued a n d continuous with continuous partial derivatives o first order over t h e given region a n d i n a given interval o f time. Th e lines o f flow o f heat ar e t h e intergral solutions o f t h e system o f n ordinary differential equations o f t h e first order dxi 1(x; t ). ( 1) I n general, there a r e c o lines o f flow o f heat. 2 . There a r e t wo fundamental assumptions which characterize a given flow o f heat.1 These m a y b e described i n t h e following manner. Assumption I.-The velocity o f t h e flow o f heat a t any position x i n t h e body f o r an y given instant o f time t , i s proportional t o t h e rate o f decrease o f t h e temperature U -at t h e position x f o r t h e given instant o f time t . That i s a u Xi k . ( 2 ) Th e factor o f proportionality k i s t h e conductivity o f t h e body. I n cer- tain bodies such a s crystals, this k depends n o t only on t h e position o f t h e point -x a n d t h e time t , b u t also o n t h e direction through t h e point x . W e 8 2 2
Transcript of PNAS-1952-Kasner-The Fourier Heat Equation in Riemannian Space
7/27/2019 PNAS-1952-Kasner-The Fourier Heat Equation in Riemannian Space
http://slidepdf.com/reader/full/pnas-1952-kasner-the-fourier-heat-equation-in-riemannian-space 1/4
MATHEMATICS: KASNER AND DE CICCO P R o c . N . A . S .
1 1 L a r k i n , R . A . , M a c M a s t e r s , M. M . , C u l l , I . M . , W o l f , M. J . , a n d R i s t , C . E .S t a i n T e c h . , 2 7 , 1 0 7 ( 1 9 5 2 ) .
1 2 R a n d o l p h , L . F . , I b i d . , 1 5 , 4 5 ( 1 9 4 0 ) .
THE FOURIER HEAT EQUATION IN RIEMANNIAN SPACE*
By EDWARD KASNER a n d JOHN D E C i C c o
DEPARTMENTS OF M A T H E M A T I C S , COLUMBIA U N I V E R S I T Y , NEW Y O R K , AND D E PAUL
U N I V E R S I T Y , C H I C A G O
C o m m u n i c a t e d J u l y 2 , 1 9 5 2
1 . L e t a c e r t a i n b o d y , c a p a b l e o f a b s o r b i n g h e a t , o c c u p y a r e g i o n o f n
d i m e n s i o n a l R i e m a n n i a n s p a c e V , w h o s e m e t r i c i s d e f i n e d b y t h e p o s i t i v eq u a d r a t i c d i f f e r e n t i a l f o r m d s 2 = g j j d x l d x J . S u p p o s e t h a t t h i s b o d y i sh e a t e d b y c o n d u c t i o n i n a n y m a n n e r . Th e s i t u a t i o n c a n b e v i s u a l i z e d a s
a f l o w o f h e a t f r o m t h e warmer t o t h e c o o l e r p a r t s o f t h e b o d y .
Th e r a t e o f f l o w o f h e a t w i t h i n t h e b o d y c a n b e d e p i c t e d a s a v e c t o r f i e l d
X I ( x ; t )o r
X 1 ( x ; t ) ,w h e r e Xi =
g ' J X , a n d X i=
g i j X ' . Att h e
p o s i t i o nx w i t h i n t h e b o d y f o r t h e p a r t i c u l a r i n s t a n t o f t i m e t , t h e v e c t o r X i s i n t h e
d i r e c t i o n o f t h e f l o w o f h e a t and i t s m a g n i t u d e i s t h e r a t e o f f l o w o f h e a t .I t i s a s s u m e d t h a t t h i s v e c t o r f i e l d X I ( x ; t ) o r X , ( x ; t ) i s s i n g l e v a l u e d a n d
c o n t i n u o u s w i t h c o n t i n uo u s p a r t i a l d e r i v a t i v e s o f f i r s t o r d e r o v e r t h e g i v e n
r e g i o n a n d i n a g i v e n i n t e r v a l o f t i m e .Th e l i n e s o f f l o w o f h e a t a r e t h e i n t e r g r a l s o l u t i o n s o f t h e s y s t e m o f n
o r d i n a r y d i f f e r e n t i a l e q u a t i o n s o f t h e f i r s t o r d e r
d x i-=1 ( x ; t ) .
( 1 )
I n g e n e r a l , t h e r e a r e c o ' l i n e s o f f l o w o f h e a t .
2 . T h e r e a r e t w o f u n d a m e n t a l a s s u m p t i o n s w h i c h c h a r a c t e r i z e a g i v e n
f l o w o f h e a t . 1 T h e s e m a y b e d e s c r i b e d i n t h e f o l l o w i n g m a n n e r .
A s s u m p t i o n I . - T h e v e l o c i t y o f t h e f l o w o f h e a t a t a n y p o s i t i o n x i n t h e
b o d y f o r a n y g i v e n i n s t a n t o f t i m e t , i s p r o p o r t i o n a l t o t h e r a t e o f d e c r e a s e
o f t h e t e m p e r a t u r e U - a t t h e p o s i t i o n x f o r t h e g i v e n i n s t a n t o f t i m e t .
T h a t i s
a uX i -k-. ( 2 )
Th e f a c t o r o f p r o p o r t i o n a l i t y k i s t h e c o n d u c t i v i t y o f t h e b o d y . I n c e r -t a i n b o d i e s s u c h a s c r y s t a l s , t h i s k d e p e n d s n o t o n l y o n t h e p o s i t i o n o f t h e
p o i n t - x a n d t h e t i m e t , b u t a l s o o n t h e d i r e c t i o n t h r o u g h t h e p o i n t x . W e
8 2 2
7/27/2019 PNAS-1952-Kasner-The Fourier Heat Equation in Riemannian Space
http://slidepdf.com/reader/full/pnas-1952-kasner-the-fourier-heat-equation-in-riemannian-space 2/4
V O L . 3 8 , 1 9 5 2 MATHEMATICS: KASNER AND DE CICCO
s h a l l s t u d y o n l y t h e r m a l l y i s o t o p i c b o d i e s , w h e r e k d e p e n d s o n l y o n t h ep o s i t i o n o f t h e p o i n t x a n d t h e t i m e t . O f c o u r s e , i t i s a l w a y s s u p p o s e d
t h a t k > 0 .
Th e t e m p e r a t u r e U = U ( x ; t ) i s a s s u m e d t o b e s i n g l e v a l u e d a n d c o n -
t i n u o u s w i t h c o n t i n u o u s p a r t i a l d e r i v a t i v e s o f a t l e a s t t h e s e c o n d o r d e r
o v e r t h e r e g i o n a n d i n t h e g i v e n i n t e r v a l o f t i m e .A s s u m p t i o n I I . - F o r e v e r y n d i m e n s i o n a l r e g i o n Rn w i t h i n t h e b o d y , t h e
r a t e o f f l o w o f h e a t i n t o R . i s e q u a l t o t h e r a t e o f i n c r e a s e o f t h e t e m p e r a -
t u r e o f R n .L e t ' y b e t h e u n i t n o r m a l v e c t o r o f t h e c l o s e d ( n - 1 ) d i m e n s i o n a l h y p e r -
s u r f a c e R n - 1 , w h i c h b o u n d s a n n d i m e n s i o n a l r e g i o n R n w i t h i n t h e b o d y .
Th e v e c t o r y 9 p o i n t s o u t w a r d s f r o m R . . Th e n o r m a l c o m p o n e n t o f t h e
v e c t o r f i e l d X I ( x ; t ) a l o n g R . - , i s N = X ' 7 y . Th e r a t e o f f l o w o f h e a t i n t o
R n i s t h e n e g a t i v e o f t h e f l u x o f t h e v e c t o r f i e l d X i ( x ; t ) o u t f r o m t h e b o u n d -
i n g ( n - 1 ) d i m e n s i o n a l h y p e r s u r f a c e R , , , n a m e l y
- R. -1N d v n - I = - f R n -. X T i Y d V n l I = - f R d i v ( X ) d v n . ( 3 )
T h i s f o l l o w s f r o m t h e D i v e r g e n c e T h e o r e m o f G r e e n a n d G a u s s .
I f c i s t h e s p e c i f i c h e a t o f t h e s u b s t a n c e , t h e n u m b e r o f h e a t u n i t s a c -
q u i r e d b y t h e b o d y f o r e a c h u n i t o f m a s s i s c a u . I f p ( x ) > 0 , i s t h e
d e n s i t y f u n c t i o n o f t h e b o d y , t h e n t h e t o t a l n u m b e r o f h e a t u n i t s a c q u i r e d
b y t h e p o r t i o n o f t h e b o d y i n t h e r e g i o n R . p e r u n i t i n t e r v a l o f t i m e i s
r a uc p-v n . ( 4 )
A p p l y i n g A s s u m p t i o n I I t o ( 3 ) a n d ( 4 ) , i t i s f o u n d t h a t
X C P a U + d i v . ( X ) ] d v n =0. ( 5 )
2 . 1 . T h e t e m p e r a t u r e U o f t h e b o d y o b e y s t h e F o u r i e r h e a t e q u a t i o n
d i v (kaU) a u ( 6 )
I f k , c , p a r e a l l c o n s t a n t s o t h a t t h e d i f f u s i v i t y a 2 = k / c p , i s c o n s t a n t , t h e
F o u r i e r h e a t e q u a t i o n i s
au 1 au
t a x ) a 2 a t ( 7 )
T h i s r e s u l t i s a c o n s e q u e n c e o f e q u a t i o n s ( 2 ) a n d ( 5 ) .I f a f t e r a c e r t a i n l a p s e o f t i m e , t h e f l o w o f h e a t b e c o m e s s t a t i o n a r y s o
t h a t t h e t e m p e r a t u r e U i s i n d e p e n d e n t o f t h e t i m e t , t h e n t h e F o u r i e r h e a t
8 2 3
7/27/2019 PNAS-1952-Kasner-The Fourier Heat Equation in Riemannian Space
http://slidepdf.com/reader/full/pnas-1952-kasner-the-fourier-heat-equation-in-riemannian-space 3/4
4MATHEMATICS: KASNER AND D E CICCO P R o c . N . A . S .
e q u a t i o n ( 7 ) r e d u c e s t o t h e L a p l a c e e q u a t i o n . I n t h i s s t a t i o n a r y c a s e , i tf o l l o w s f r o m ( 1 ) t h a t t h e f l o w o f h e a t t h r o u g h o u t t h e b o d y i s s t a t i o n a r y ,a n d t h e r e a r e On - l i n e s o f f l o w .
3 . By a p p r o p r i a t e l y c h a n g i n g t h e u n i t o f t i m e , t h e F o u r i e r h e a t e q u a -
t i o n ( 7 ) m ay b e w r i t t e n i n t h e f o r m
a x 1x -= t * ( 8 )
I f U ( x ; t ) i s a n o n - c o n s t a n t s o l u t i o n o f t h i s e q u a t i o n ( 8 ) , t h e h e a t h y p e r -
s u r f a c e s , e a c h o f ( n - 1 ) d i m e n s i o n s , a r e d e f i n e d b y U = U ( x ; t ) , w h e r eU a n d t a r e t w o a r b i t r a r y c o n s t a n t s . T h u s a l o n g a n y o n e o f t h e h e a t h y -
p e r s u r f a c e s , b o t h t h e t e m p e r a t u r e U a n d t h e t i m e t a r e c o n s t a n t . 2 I n
g e n e r a l , t h e r e a r e c o 2 h e a t h y p e r s u r f a c e s .
F o r a n y g i v e n f i x e d i n s t a n t o f t i m e t , t h e h y p e r s u r f a c e s d e f i n e d by t h e
e q u a t i o n U = U ( x ; t ) , w h e r e U i s c o n s t a n t , a r e s a i d t o f o r m a n i s o t h e r m a lf a m i l y . I n g e n e r a l , f o r a n y g i v e n i n s t a n t o f t i m e , t h e r e a r e c o 1 i s o t h e r m a l
h y p e r s u r f a c e s , e a c h o f ( n - 1 ) d i m e n s i o n s . Th e s y s t e m o f o o 2 h e a t h y p e r -
s u r f a c e s c o n s i s t s o f 0 o 0 i s o t h e r m a l f a m i l i e s o f X 1 h y p e r s u r f a c e s .
3 . 1 . F o r a n y g i v e n i n s t a n t o f t i m e , t h e f l o w o f h e a t i s o r t h o g o n a l t o t h ef a m i l y o f X I i s o t h e r m a l h y p e r s u r f a c e s .
T h i s i s a c o n s e q u e n c e o f e q u a t i o n s ( 1 ) a n d ( 2 ) .
3 . 2 . L e t F ( x ; t , U ) b e a s i n g l e v a l u e d a n d c o n t i n u o u s f u n c t i o n w i t h c o n -t i n u o u s p a r t i a l d e r i v a t i v e s o f a t l e a s t t h e f i r s t a n d s e c o n d o r d e r s o v e r a r e g i o no f t h e x ' s , t , U , s u c h t h a t Z F / 4 U i s n o w h e r e z e r o i n t h e r e g i o n . T h e e q u a t i o n
F ( x ; t , U ) = 0 , ( 9 )
r e p r e s e n t s t h e i m p l i c i t f o r m o f t h e r e l a t i o n b e t w e e n t h e c o o r d i n a t e s x i o f t h ep o i n t , t h e t i m e t , a n d t h e t e m p e r a t u r e U i f a n d o n l y i f F o b e y s t h e p a r t i a l d i f -
f e r e n t i a l e q u a t i o n o f s e c o n d o r d e r
( FA 2 g - ? / a 1 / 2 i i a F _ F ' -2 a F 6 ia 2 F a F\ a N L x i \ a j x . ? a t a u a u a x l a x }
a 2 F a F a F
a ) U 2 g j a x l a X i 1 0
w h e r e p a r t i a l d i f f e r e n t i a t i o n w i t h r e s p e c t t o a n x i o r t s i g n i f i e s t h a t U i s h e l df i x e d .
4 . I f t h e n u m b e r o f h e a t h y p e r s u r f a c e s i n a c e r t a i n f l o w o f h e a t i sm e r e l y X 1 , t h e n t h e s e t o f h y p e r s u r f a c e s i s c a l l e d a d e g e n e r a t e h e a t f a m i l y . 'F o r a d e g e t e r a t e h e a t f a m i l y , t h e e q u a t i o n ( 9 ) c a n b e w r i t t e n i n t h e f o r m
f ( x ) = C ( t , U ) , ( 1 1 )
w h e r e f i s s i n g l e v a l u e d a n d c o n t i n u o u s w i t h c o n t i n u o u s p a r t i a l d e r i v a t i v e s
8 2 4
7/27/2019 PNAS-1952-Kasner-The Fourier Heat Equation in Riemannian Space
http://slidepdf.com/reader/full/pnas-1952-kasner-the-fourier-heat-equation-in-riemannian-space 4/4
V O L . 3 8 , 1 9 5 2 MATHEMATICS: KASNER AND D E CICCO
o f a t l e a s t t h e f i r s t a n d s e c o n d o r d e r s o v e r a r e g i o n o f t h e x ' s a n d C i ss i n g l e v a l u e d a n d c o n t i n u o u s w i t h c o n t i n u o u s p a r t i a l d e r i v a t i v e s o f a tl e a s t t h e f i r s t a n d s e c o n d o r d e r s s u c h t h a t a C / U n e v e r v a n i s h e s o v e r a
r e g i o n o f v a l u e s ( t , U ) .4 . 1 . A o n e - p a r a m e t e r s y s t e m o f h y p e r s u r f a c e s , e a c h o f ( n - 1 ) d i m e n -
s i o n s , i n a R i e m a n n i a n s p a c e V n o f n > 2 d i m e n s i o n s , i s a d e g e n e r a t e h e a tf a m i l y o f h y p e r s u r f a c e s i f a n d o n l y i f i t s e q u a t i o n c a n b e w r i t t e n i n t h e f o r m
f ( x ) = c o n s t . , s u c h t h a t f o b e y s a p a r t i a l d i f f e r e n t i a l e q u a t i o n o f s e c o n d o r d e ro f t h e f o r m
* g - i / 2 ( g / 2 g i i . L ) = a f + b , ( 1 2 )
w h e r e a a n d b a r e c o n s t a n t s .
T h i s c a n b e d e d u c e d a f t e r a l o n g d i s c u s s i o n f r o m e q u a t i o n s ( 9 ) , ( 1 0 ) a n d
( 1 1 ) .4 . 2 . U n d e r h o m o t h e t i c r e p r e s e n t a t i o n s o f R i e m a n n i a n m a n i f o l d s , a n y
d e g e n e r a t e h e a t f a m i l y c a n b e g i v e n b y a n e q u a t i o n o f t h e f o r m f ( x ) = c o n s t . ,w h e r e f i s a s o l u t i o n o f o n e o f t h e t h r e e t y p e s o f p a r t i a l d i f f e r e n t i a l e q u a t i o n s o f
s e c o n d o r d e r :T y p e I . T h e L a p l a c e e q u a t i o n
g - l / A a x ( g / 2 g f - 0 .°(3
T y p e I I . T h e s p e c i a l P o i s s o n e q u a t i o n
g - 1 / 2 i ( g l ti f ( 1 4 )
T y p e I I I . T h e H e l m h o l t z - P o c k e l s e q u a t i o n
a g / 2 g i b x q =f. ( 1 5 )
T h i s r e s u l t f o l l o w s f r o m e q u a t i o n ( 1 2 ) .
* P r e s e n t e d t o t h e A m e r i c a n M a t h e m a t i c a l S o c i e t y , S e p t e m b e r 1 9 5 2 .1 K e l l o g g , 0 . D . , F o u n d a t i o n s o f P o t e n t i a l T h e o r y , F r e d e r i c k U n g e r P u b l i s h i n g Com-
p a n y , Ne w Y o r k .2 K a s n e r , " G e o m e t r y o f t h e H e a t E q u a t i o n , ( f i r s t p a p e r ) , " P R O C . N A T L . A C A D . S C I . ,
1 8 ,4 7 5 - 4 8 0
( 1 9 3 2 ) .3 K a s n e r , " G e o m e t r y o f t h e Heat E q u a t i o n ( s e c o n d p a p e r ) , - T h e T h r e e D e g e n e r a t e
T y p e s o f L a p l a c e , P o i s s o n a n d H e l m h o l t z , " I b i d . , 1 9 , 2 5 7 - 2 6 2 ( 1 9 3 3 ) . A l s o K a s n e r
a n d D e C i c c o , " G e o m e t r y o f t h e F o u r i e r Heat E q u a t i o n , " T r a n s . Am . M a t h . S o c . , 6 0 ,1 1 9 - 1 3 2 ( 1 9 4 6 ) .
8 2 5
![Riemannian GeometryRiemannian Geometry Dr Emma Carberry Semester 2, 2015 Lecture 15 [Riemannian Geometry – Lecture 15]Riemannian Geometry – Lecture 15 Riemann Curvature Tensor](https://static.fdocuments.net/doc/165x107/5eaeb2e645c7213d450b3d20/riemannian-riemannian-geometry-dr-emma-carberry-semester-2-2015-lecture-15-riemannian.jpg)
