A Series Illustrating Innovative Forms of the Organization & Exposition of Mathematics by Walter Gottschalk

Transcription

1 Math Medley #45 of Gottschalk s Gestalts A Seres Illustratg Iovatve Forms of the Orgazato & Exposto of Mathematcs by Walter Gottschalk Ifte Vstas Press PVD RI 2001 GG45-1 (30)

2 2001 Walter Gottschalk 500 Agell St #414 Provdece RI permsso s grated wthout charge to reproduce & dstrbute ths tem at cost for educatoal purposes; attrbuto requested; o warraty of fallblty s posted GG45-2

3 ufed product rule for dervatves D to dfferetate a product, dfferetate each factor separately & add D applcatos product of ay umber of real/complex fuctos scalar product of two vectors vector product of two 3-vectors trple scalar product of three 3-vectors trple vector product of three 3-vectors multple vector products of 3-vectors determat of ay order (op o rows or cols) etc D the uformty of ths product rule comes from the multlearty of these products GG45-3

4 T. let f Œreal -valued fucto o a real terval the f Œdfferetable fl f Œcotuous fl f Œtegrable & all coverses fal GG45-4

5 D. a Lebz seres a alteratg seres a a - a + a - a + of postve real umbers st, a, a, a, a ( Œoeg t var) & df a + 1 a Æ0 as Æ T. every Lebz seres coverges GG45-5

6 T. the harmoc seres dverges P. otherwse s Ê ˆ Ë + Ê1 Ë + 1ˆ + Ê 1 Ë + 1ˆ > Ê ˆ Ë + Ê 1 Ë + 1ˆ + Ê 1 Ë + 1ˆ s & \ s>s the harmoc seres dverges QED GG45-6

7 a terestg equalty T. 1  ( t 2) k > Œ k1 P. ote that 1 > k k - k -1 for k 2 & a telescopg sum results from substtuto QED GG45-7

8 a theorem whose proof from the aalytc POV s ot mmedate but whch s evdet from the geometrc POV T. let a,b Œreal os st a < b y y(x), 0 & x a x x(y), 0 y b Œverse strctly creasg cotuous real fuctos the Ú a Ú ydx + xdy ab 0 0 b GG45-8

9 T. the fudametal theorem for space curves let s arclegth k curvature t torso k k() s 0 & t t( s) Œgve cotuous real - valued fuctos o a real terval the there exsts a class C space curve, uque up to rgd moto, wth the gve curvature k k() s & wth the gve torso t t(s) 2 GG45-9

10 four forms of the Drchlet box / drawer / pgeohole prcple (1) f +1 objects are placed boxes where s a postve teger, the at least oe box cotas at least two objects (2) f X ad Y are sets of ay cardalty such that crd X > crd Y, ad f f:x Æ Y, the there exsts y Œ Y such that crd f () y > 1-1 () 3 f ftely may objects are placed ftely may boxes, the at least oe box cotas ftely may objects (4) f X s a fte set, f Y s a fte set, ad f f: X Æ Y, the there exsts y ŒY such that f () y sa fte set -1 (2) s a set - theoretc geeralzato of (1); (4) s a set - theoretc equvalet of (3) GG45-10

11 Questo. Is the followg statemet mathematcally true? If there are more trees the world tha there are leaves o ay oe tree, the there are at least two trees wth the same umber of leaves. Aswer. Not as t stads. It s true uder three addtoal codtos, amely: there are oly ftely may trees the world, there s at least oe tree the world, every tree has at least oe leaf. GG45-11

12 Lagrage's detty for vectors T. Lagrage's detty for two 2 - vectors let ab, Π2-vectors over a com rg the ab, 2 a a b a a b b b GG45-12

13 T. Lagrage' s detty for two 3 - vectors let ab, Π3-vectors over a com rg the ( a b) 2 a a b a a b b b GG45-13

14 T. Lagrage's detty for two vectors let ab, Œ-vectors over a com rg ( Œpos t) the scalar otato Âa 2 2 a a  j 1 b j, b 1 j  < j 1 ab  1  1 a b b 2 vector otato 1 2 ( ab - ba) 2 a a b a a b b b GG45-14

15 T. Lagrage' s detty for four 2 - vectors let abcd,,, Π2-vectors over a com rg the ab, cd, a c b c a d b d GG45-15

16 T. Lagrage' s detty for four 3 - vectors let abcd,,, Π3-vectors over a com rg the ( a b) ( c d) a c b c a d b d GG45-16

17 T. Lagrage's detty for four vectors let abcd,,, Œ-vectors over a com rg ( Œ pos t) the scalar otato Â, j 1 < j a b a b j j c d c j d j  1  1 ac b c  1  1 ad b d vector otato ( ab - ba) cd - dc ( ) 2 a c b c a d b d GG45-17

18 T. Cauchy - Rema equatos CRE let x, y, u, v Œ real var z, w Œ complex var x + y z u + v w w w( z ) $ dw dz the Ïu Ì Óu x y v y - v x GG45-18

19 P. three parts (1) dw dz w x ( x u + v ) u + v x x dw ( 2) dz w ( y) y u + ( ) v 1 ( y u + v ) 1 ( uy + vy ) v - u y ( ) y GG45-19

20 () 3 ux + v dw dz v - u \ u v & u - qed y x y y x y v x GG45-20

21 K. the Jacoba of ( u, v ) wrt ( x, y ) (u,v) det (x,y) u v u v u u u v x x dw dz 0 u v - y y u v x y y x 2 x + 2 vx 2 y + 2 vy 2 x + 2 uy 2 x + 2 vy 2 GG45-21

22 the law of cluso - excluso for fte sets D law of cluso - excluso for two fte sets A,B crd (A» B) crd A + crd B - crd ( A «B) D law of cluso - excluso for three fte sets A, B, C crd (A» B» C) crd A + crd B + crdc - crd ( A «B) - crd ( A «C) - crd ( B «C) + crd ( A «B «C) GG45-22

23 D law of cluso - excluso for fte sets A, A,, A ( Œ pos t) 1 2 crd - +  1  j, 1 < j jk,, 1 < j< k -  U A crd A crd ( A A ) crd ( A A A ) + (-1) crd ( A A A ) j j k 1 2 GG45-23

24 the geeral law of cluso-excluso ot oly holds for fte cardalty but also holds for the measure of ftely may sets of fte measure a ftely addtve measure space ad also holds for the dcator appled to ftely may sets (these sets fte or ot); the cases of 2 & 3 sets are gve below GG45-24

25 the law of cluso - excluso for the measure of two sets A, B & of three sets A, B, C, all of fte measure a ftely addtve measure space m(a» B) ma + mb - ma ( «B) m(a» B» C) ma+mb+mc - ma ( «B) - ma ( «C) - mb ( «C) + ma ( «B «C) GG45-25

26 the law of cluso - excluso for the dcator appled to two sets A, B & appled to three sets A, B, C I(x,A» B) I(x,A) + I(x,B) - IxA (, «B) I(x,A» B» C) I(x,A) + I(x,B) + I(x,C) - IxA (, «B) - IxA (, «C) - IxB (, «C) + IxA (, «B «C) GG45-26

27 relatve stregths of the cluso-excluso laws the law for the dcator mples the law for measure mples the law for cardalty GG45-27

28 the Heseberg determacy / ucertaty prcple apples to a partcle Dq Dp h/2p & Dt DE h/2p wh D error measuremet of qposto coordate p mometum t tme E eergy h Plack' s costat bole Werer Karl Heseberg Germa theoretcal physcst GG45-28

29 a gudg prcple the drve toward brevty: to say more & more less & less tme & space because we have lots to say & we have lttle tme & space to say t GG45-29

30 the exposto of mathematcs mathematcal exposto mathematcs exposto s a art form & should be a work of art GG45-30