THEORETICAL MECHANICS FOR SIXTH FORMS
CONTENTS MATHEMATICAL PREPARATIONS ELEMENTS OF DIFFERENTIAL CALCULUS SET OF NUMBERS SEQUENCE OF NUMBERS AND LIMITING VALUES SERIES AND LIMITING VALUES FUNCTIONS AND LIMITS CONTINUITY TRIGONOMETRIC FUNCTIONS EXPONENTIAL FUNCTION AND LOGARITHM DIFFERENTIAL QUOTIENT RULES OF DIFFERENTIATION TAYLOR EXPANSION LIMITING VALUES OF INDETERMINATE EXPRESSIONS EXTREME VALUES EXERCISES ELEMENTS OF INTEGRAL CALCULUS NOTIONS FIRST RULES OF INTEGRATION FUNDAMENTAL THEOREM OF CALCULUS THE TECHNIQUE OF INTEGRATION MULTIPLE INTEGRALS VECTORS ELEMENTARY MATHEMATICAL OPERATIONS SCALAR PRODUCT VECTOR (OUTER, CROSS) PRODUCT ‘HIGHER’ VECTOR PRODUCTS BASIS VECTORS COMPONENT REPRESENTATIONS VECTOR-VALUED FUNCTIONS PARAMETRIZATION OF SPACE CURVES DIFFERENTIATION OF VECTOR-VALUED FUNCTIONS ARC LENGTH MOVING TRIHEDRON FIELDS CLASSI?CATION OF THE FIELDS PARTIAL DERIVATIVES GRADIENT DIVERGENCE AND CURL (ROTATION) MATRICES AND DETERMINANTS MATRICES CALCULATION RULES FOR MATRICES TRANSFORMATION OF COORDINATES (ROTATIONS) DETERMINANTS CALCULATION RULES FOR DETERMINANTS SPECIAL APPLICATIONS COORDINATE SYSTEMS TRANSFORMATION OF VARIABLES, JACOBIAN DETERMINANT CURVILINEAR COORDINATES CYLINDRICAL COORDINATES SPHERICAL COORDINATES SELF-EXAMINATION QUESTIONS MECHANICS OF THE FREE MASS POINT KINEMATICS VELOCITY AND ACCELERATION SIMPLE EXAMPLES FUNDAMENTAL LAWS OF DYNAMICS NEWTON’S LAWS OF MOTION FORCES INERTIAL SYSTEMS, GALILEAN TRANSFORMATION ROTATING REFERENCE SYSTEMS, PSEUDO FORCES (FICTITIOUS FORCES) ARBITRARILY ACCELERATED REFERENCE SYSTEMS SIMPLE PROBLEMS OF DYNAMICS MOTION IN THE HOMOGENEOUS GRAVITATIONAL FIELD LINEAR DIFFERENTIAL EQUATIONS MOTION WITH FRICTION IN THE HOMOGENEOUS GRAVITATIONAL FIELD SIMPLE PENDULUM COMPLEX NUMBERS LINEAR HARMONIC OSCILLATOR FREE DAMPED LINEAR OSCILLATOR DAMPED LINEAR OSCILLATOR UNDER THE IN?UENCE OF AN EXTERNAL FORCE ARBITRARY ONE-DIMENSIONAL SPACE-DEPENDENT FORCE FUNDAMENTAL CONCEPTS AND THEOREMS WORK, POWER, AND ENERGY POTENTIAL ANGULAR MOMENTUM AND TORQUE (MOMENT) CENTRAL FORCES INTEGRATION OF THE EQUATIONS OF MOTION PLANETARY MOTION MECHANICS OF MANY-PARTICLE SYSTEMS CONSERVATION LAWS PRINCIPLE OF CONSERVATION OF LINEAR MOMENTUM (CENTER OF MASS THEOREM) CONSERVATION OF ANGULAR MOMENTUM CONSERVATION OF ENERGY VIRIAL THEOREM TWO-PARTICLE SYSTEMS RELATIVE MOTION TWO-BODY COLLISION ELASTIC COLLISION INELASTIC COLLISION PLANETARY MOTION AS A TWO-PARTICLE PROBLEM COUPLED OSCILLATIONS THE RIGID BODY MODEL OF A RIGID BODY ROTATION AROUND AN AXIS ANGULAR-MOMENTUM LAW PHYSICAL PENDULUM STEINER’S THEOREM ROLLING MOTION ANALOGY BETWEEN TRANSLATIONAL AND ROTATIONAL MOTION INERTIAL TENSOR KINEMATICS OF THE RIGID BODY KINETIC ENERGY OF THE RIGID BODY PROPERTIES OF THE INERTIAL TENSOR ANGULAR MOMENTUM OF THE RIGID BODY THEORY OF THE SPINNING TOP EULER’S EQUATIONS EULER’S ANGLES ROTATIONS AROUND FREE AXES FORCE-FREE SYMMETRIC SPINNING TOP A SOLUTIONS OF THE EXERCISES CHAPTER SET OF NUMBERS ONE DE?NES THE FOLLOWING TYPES OF NUMBERS: N D F1; 2; 3; : : :G NATURAL NUMBERS Z D F: : : ; 2; 1; 0; 1; 2; 3; : : :G INTEGER NUMBERS Q D N XI X D P QI P 2 Z; Q 2 N O RATIONAL NUMBERS R D FXI CONTINUOUS NUMBER LINEG REAL NUMBERS : THEREFORE N Z Q R MATHEMATICAL PREPARATIONS THE BODY OF COMPLEX NUMBERS C WILL BE INTRODUCED AND DISCUSSED LATER IN SECT 2 3 5 FOR THE ABOVE-MENTIONED SET OF NUMBERS THE BASIC OPERATIONS ADDITION AND MULTIPLICATION ARE DE?NED IN THE WELL-KNOWN MANNER WE WILL REMIND HERE ONLY SHORTLY TO THE PROCESS OF RAISING TO A POWER FOR AN ARBITRARY REAL NUMBER A THE N-TH POWER IS DE?NED AS: AN D A A A : : : A „ ?‚ … N-FOLD N 2 N : (1 1) THERE ARE THE FOLLOWING RULES: 1 A B/N D A B/ A B/ : : : A B/ „ ?‚ … N-FOLD D AN BN (1 2) 2 AK AN D A A : : : A „ ?‚ … K-FOLD A A : : : A „ ?‚ … N-FOLD D AKCN (1 3) 3 AN/K D AN AN : : : AN „ ?‚ … K-FOLD D ANK : (1 4) EVEN NEGATIVE EXPONENTS ARE DE?NED AS CAN BE SEEN BY THE FOLLOWING CONSIDERATION: AN D ANCKK D AN AK AK Õ AK AK D 1 : THEREFORE WE HAVE: AK AK 8A 2 R A ¤ 0/ : (1 5) FURTHERMORE, WE RECOGNIZE THE IMPORTANT SPECIAL CASE: AKK A0 D 8A 2 R : (1 6) THIS RELATION IS VALID ALSO FOR A D 0 ANALOGOUSLY AND AS AN EXTENSION OF (1 4) SPLIT EXPONENTS CAN BE DE?NED: BN D A D A N N Õ B D A ONE DENOTES A N NPA W N-TH ROOT OF A (1 7) THUS IT IS A NUMBER THE N-TH POWER OF WHICH IS JUST A EXAMPLES 2P 4 2 D BECAUSE: 22 D 2 2 D 3P 27 3 D BECAUSE: 33 D 3 3 3 D SEQUENCE OF NUMBERS AND LIMITING VALUES BY A SEQUENCE OF NUMBERS WE WILL UNDERSTAND A SEQUENCE OF (INDEXED) REAL NUMBERS: A1; A2; A3; ; AN; AN 2 R : (1 9) WE HAVE ?NITE AND IN?NITE SEQUENCES OF NUMBERS IN CASE OF A ?NITE SEQUENCE THE INDEX N IS RESTRICTED TO A ?NITE SUBSET OF N THE SEQUENCE IS FORMALLY DENOTED BY THE SYMBOL FANG AND REPRESENTS A MAPPING OF THE NATURAL NUMBERS N ON THE BODY OF REAL NUMBERS R: F W N 2 N ! AN 2 R N ! AN/ : EXAMPLES 1 AN D N ! A1 D 1; A2 D 2; A3 D 3; A4 D (1 10) MATHEMATICAL PREPARATIONS 2 AN D N N C 1/ ! A1 D 1 2; A2 D 2 3; A3 D 3 4; (1 11) 3 AN D 1 C N ! A1 D 2; A2 D ! A : (1 13) THE MATHEMATICAL DE?NITION READS: FANG CONVERGES TO A ” 8" > 0 9 N" 2 N SO THAT JAN AJ < " 8N > N" : (1 14) DOES SUCH AN A NOT EXIST THEN THE SEQUENCE IS CALLED DIVERGENT IN CASE FANG CONVERGES TO A, THEN FOR EACH " > 0 ONLY A ?NITE NUMBER OF SEQUENCE ELEMENTS HAS A DISTANCE GREATER THAN " TO A EXAMPLES 1 FANG D N ! NULL SEQUENCE/ (1 15) 2 FANG D N N C (1 16) BECAUSE: N N C 1 D 1 C 3 FANG D FQNG ! 0 ; IF JQJ < 1 : (1 17) THE PROOF OF THIS STATEMENT IS PROVIDED ELEGANTLY BY THE USE OF THE SPECIAL FUNCTION LOGARITHM, WHICH, HOWEVER, WILL BE INTRODUCED ONLY WITH EQ (1 65) THUS WE PRESENT THE JUSTI?CATION OF (1 17) AFTER THE DERIVATION OF (1 70) 4 AN D 1 C N N ! E D 2:71828 : : : EULER NUMBER : (1 18) THE LIMITING VALUE OF THIS SEQUENCE, WHICH IS VERY IMPORTANT FOR APPLICATIONS, IS GIVEN HERE WITHOUT PROOF FOR DETAILS THE READER IS REFERRED TO SPECIAL TEXTBOOKS ON MATHEMATICS AGAIN WITHOUT PROOF WE LIST UP THE FOLLOWING RULES FOR SEQUENCES OF NUMBERS THE EXPLICIT, RATHER STRAIGHTFORWARD DERIVATION OF WHICH SHALL BE LEFT TO THE READER ASSUMING THE CONVERGENCE OF THE TWO SEQUENCES FANG AND FBNG: LIM N ! 1 AN D A I LIM N ! 1 BN D B : WE GET: LIM N ! 1 AN ? BN/ D A ? B (1 19) LIM N ! 1 C AN/ D C A C 2 R/ (1 20) LIM N ! 1 AN BN/ D A B (1 21) LIM N ! SERIES AND LIMITING VALUES ADDING UP THE TERMS OF AN IN?NITE SEQUENCE OF NUMBERS LEADS TO WHAT IS CALLED A SERIES: A1; A2; A3; ; AN; Õ A1 C A2 C A3 C C AN C D X M D AM : (1 23) MATHEMATICAL PREPARATIONS STRICTLY, THE SERIES IS DE?NED AS LIMITING VALUE OF A SEQUENCE OF (?NITE) PARTIAL SUMS: SR D R X M D M D 1 AM TO BE CONVERGENT IS LIM M ! 1 AM D M D 1 AM IS INDEED CONVERGENT THEN IT MUST HOLD: LIM M ! 1 AM D LIM M ! 1 SM SM1/ D LIM M ! 1 SM LIM M ! 1 SM1 D S S D 0 : HOWEVER, EQ (1 26) IS NOT A SUF?CIENT CONDITION A PROMINENT COUNTER-EXAMPLE REPRESENTS THE HARMONIC SERIES M D 1 C 2 C 3 C : (1 27) IT IS DIVERGENT, ALTHOUGH LIMM ! 1 M D 0! THE PROOF OF THIS IS GIVEN AS AN EXERCISE 1 1 3 IN MATHEMATICS (ANALYSIS) ONE LEARNS OF DIFFERENT NECESSARY AND SUF?CIENT CONDITIONS OF CONVERGENCE FOR IN?NITE SERIES: COMPARISON CRITERION , RATIO TEST , ROOT TEST IN THE COURSE OF THIS BOOK WE DO NOT NEED THESE CRITERIA EXPLICITLY AND THUS RESTRICT OURSELVES TO ONLY MAKING A REMARK THE GEOMETRIC SERIES TURNS OUT TO BE AN IMPORTANT SPECIAL CASE OF AN IN?NITE SERIES BEING DE?NED AS Q0 C Q1 C Q2 C C QM C D CAN EASILY BE CALCULATED ANALYTICALLY FOR THIS PURPOSE WE MULTIPLY THE LAST EQUATION BY Q, Q SR D Q1 C Q2 C C QR AND BUILD THE DIFFERENCE: SR Q SR D SR 1 Q/ D Q0 QR D 1 QR : THEN WE GET THE IMPORTANT RESULT: SR D 1 QR 1 Q : (1 29) INTERESTING IS THE LIMIT: LIM R ! 1 SR D 1 LIMR ! 1 QR 1 Q : FOR THIS, EQS (1 19) AND (1 20) HAVE BEEN EXPLOITED BECAUSE OF (1 17) WE ARRIVE AT: S D LIM R ! 1 SR D ˆ< ˆ 1Q , IF JQJ < NOT EXISTENT, IF JQJ FUNCTIONS AND LIMITS BY THE TERM FUNCTION F X/ ONE UNDERSTANDS THE UNIQUE ATTRIBUTION OF A DEPENDENT VARIABLE Y FROM THE CO-DOMAIN W TO AN INDEPENDENT VARIABLE X FROM THE DOMAIN OF DE?NITION D OF THE FUNCTION F X/: Y D F X/ I D R F ! W R : (1 31) WE ASK OURSELVES HOW F X/ CHANGES WITH X ALL ELEMENTS OF THE SEQUENCE FXNG D X1; X2; X3; ; XN; X3 C X I LIM X ! 1 F X/ D ? (1 34) THIS EXPRESSION CAN BE REFORMULATED FOR ALL X ¤ 0: F X/ D FOR ALL SEQUENCES FXNG, WHICH TEND TO 1, X2 AND X3 BECOME NULL SEQUENCES THAT MEANS: LIM X ! X3 C X 1 D 1 : 2 F X/ D 1 C X/ X I LIM X ! 0 F X/ D ? (1 35) FOR THE SPECIAL NULL SEQUENCE FXNG D F NG ACCORDING TO (1 18) WE KNOW THE LIMIT OF THIS FUNCTION IT CAN BE SHOWN, HOWEVER, THAT THE SAME IS TRUE FOR ARBITRARY NULL SEQUENCES: LIM X ! 0 1 C X/ IN CASE OF A ONE-TO-ONE MAPPING X F ! Y (1 37) ONE CAN DE?NE THE SO-CALLED ‘INVERSE FUNCTION’ F BELONGING TO F WHICH COMES OUT BY SOLVING Y D F X/ WITH RESPECT TO X: F 1 F X// D X : (1 38) EXAMPLE Y D F X/ D AX C B A; B 2 R Õ X D F 1 Y/ D A Y B A : LATER WE WILL ENCOUNTER SOME FURTHER EXAMPLES NOTE THAT IN GENERAL F 1 X/ F X/ : IT IS IMPORTANT TO STRESS ONCE MORE THE UNIQUENESS OF F 1, BECAUSE ONLY THEN F CONTINUITY WE ARE NOW COMING TO THE VERY IMPORTANT CONCEPT CONTINUITY Y D F X/ IS CALLED CONTINUOUS AT X0 FROM THE DOMAIN OF DE?NITION OF F IF FOR ALL " > A I > 0 EXISTS SO THAT FOR EACH X WITH JX X0J < I HOLDS: JF X/ F X0/J < " MATHEMATICAL PREPARATIONS ALTERNATIVE FORMULATION: Y D F X/ IS CONTINUOUS AT X0 FROM THE DOMAIN OF DE?NITION OF F IF FOR EACH SEQUENCE FXNG ! X0 FOLLOWS: LIM X ! X0 F X/ D F X0/ D F0 : THE LIMITING VALUE F0 IS THEREFORE JUST THE FUNCTION VALUE F X0/ WE ELUCIDATE THE TERM OF CONTINUITY BY TWO EXAMPLES: F X/ D X W X 1 W X < 1 : (1 39) THE FUNCTION (1 39), REPRESENTED IN FIG 1 1, IS OBVIOUSLY CONTINUOUS, IN CONTRARY TO THE FUNCTION FROM FIG 1 2: F X/ D X 1 W X W X < 1 : (1 40) WHICH IS APPARENTLY DISCONTINUOUS AT X D 1: LIM X ! 1 F X/ D C1 ¤ LIM X ! 1C F X/ D 0 : FIG 1 1 EXAMPLE OF A CONTINUOUS FUNCTION X F FIG 1 2 EXAMPLE OF A DISCONTINUOUS FUNCTION X F TRIGONOMETRIC FUNCTIONS IT CAN BE ASSUMED THAT THE TRIGONOMETRIC FUNCTIONS ARE WELL-KNOWN FROM SCHOOL- MATHEMATICS THEREFORE, ONLY THE MOST IMPORTANT RELATIONS SHALL BE COMPILED IN THIS SUBSECTION RADIAN MEASURE FIGURE 1 3 ILLUSTRATES THAT THE ANGLE ' CAN BE EXPRESSED NOT ONLY BY ANGULAR DEGREES I, BUT EQUALLY UNIQUELY ALSO VIA THE ARC OF THE CIRCLE S: S D S '/ W S 360I/ D 2R I S 180I/ D R I S 90I/ D 2 R I : : : ONE INTRODUCES THE DIMENSIONLESS QUANTITY ' D S R ‘RADIAN’ (1 41) ' I/ D 360 180; 90; 45; 1/ ! ; 2 ; 4 ; RAD : (1 42) TRIGONOMETRIC FUNCTIONS IN THE RIGHT-ANGLED TRIANGLE IN FIG 1 4 A AND B, ADJACENT AND OPPOSITE TO ANGLE ?, RESPECTIVELY, ARE CALLED THE LEG (SIDE, CATHETUS) AND C THE HYPOTHENUSE WITH THESE TERMS ONE DE?NES: SIN ? D B C (1 43) COS ? D A C (1 44) TAN ? D SIN ? COS ? D B A (1 45) COT ? D COS ? SIN ? D TAN ? D A B : (1 46) FIG 1 3 TO THE DE?NITION OF RADIAN MEASURE C2 D 1 : THAT LEADS TO THE IMPORTANT AND FREQUENTLY USED FORMULA: SIN2 ? C COS2 ? D 1 : (1 47) SINE FUNCTION THE SINE FUNCTION CAN BE GRAPHICALLY ILLUSTRATED AS IN FIG 1 5 THEREBY ONE SHOULD NOTICE THAT THE ANGLE ? HAS TO BE COUNTED IN THE MATHEMATICALLY POSITIVE SENSE, I E COUNTERCLOCKWISE THE SINE IS PERIODIC WITH THE PERIOD 2 IT IS AN ODD FUNCTION OF THE ANGLE ?: SIN ?/ D SIN ?/ : (1 48) AS AN INSERTION, LET US INVESTIGATE A SPECIAL LIMITING CASE IN CONNECTION WITH THE SINE: F X/ D SIN X X I LIM X ! 0 F X/ D ? (1 49) AT ?RST GLANCE THE LIMIT APPEARS TO BE UNDE?NED (00=00) WE TRY A GRAPHIC SOLUTION BY USE OF FIG 1 6 X SHALL BE A PIECE (FROM B TO C) OF A CIRCLE WITH RADIUS R D X 2R D X R 2 D X 2 : FURTHERMORE, ONE READS FROM THE SKETCH: OB D OC D 1 I OA D COS X I BA D SIN X : IN ADDITION THE INTERCEPT THEOREM YIELDS: DC BA D OC OA Õ DC D SIN X COS X D TAN X : OBVIOUSLY, THE FOLLOWING ESTIMATION FOR THE AREAS HOLDS: F OBA/ < F OBC/ < F ODC/ : THAT MEANS 2 COS X SIN X < X 2 < 2 TAN X Õ COS X < X SIN X < COS X SIN X > 0/ Õ COS X > SIN X X > COS X : EVENTUALLY WE CAN EXPLOIT THAT FOR THE LIMITING PROCESS X ! 0 IT FOLLOWS COS X ! AND COS X ! 1 LEADING THEREWITH TO: LIM X ! SIN X X D 1 : (1 50) MATHEMATICAL PREPARATIONS FIG 1 7 GRAPHICAL REPRESENTATION OF THE COSINE FUNCTION IN (1 94) WE SHALL DERIVE A SERIES EXPANSION FOR THE SINE: SIN ? D ? 3Š ?3 C 5Š ?5 C : : : D X N D 2N C 1/Š : (1 51) HERE WE USED THE TERM NŠ D 1 2 3 : : : N I 0Š D 1Š D FOR ? THIS INVERSE FUNCTION DELIVERS THE VALUE OF THE ANGLE ? IN RADIAN MEASURE, WHOSE SINE-VALUE IS JUST Y COSINE-FUNCTION WHILE, ACCORDING TO FIG 1 5, THE SINE IS ?XED BY THE SIDE OPPOSITE TO THE ANGLE IN THE RIGHT-ANGLED TRIANGLE THE COSINE-FUNCTION IS DETERMINED IN A ANALOGOUS MANNER BY THE ADJACENT SIDE (FIG 1 7) ONE RECOGNIZES FROM THE RIGHT-ANGLED TRIANGLES IN THE FIGS 1 5 AND 1 7 THAT THE COSINE IS NOTHING ELSE BUT THE =2-SHIFTED SINE: COS ?/ D SIN ? C 6Š C : : : D 1/N ?2N 2N/Š : (1 58) FROM THIS EXPANSION WE CONCLUDE THAT FOR SMALL ANGLES ? (RADIAN MEASURE!) APPROXIMATELY HOLDS: COS ? EXPONENTIAL FUNCTION AND LOGARITHM EXPONENTIAL FUNCTION BY THIS ONE UNDERSTANDS THE FOLLOWING FUNCTION: Y D AX : (1 62) A IS CALLED THE ‘BASIS’ AND X THE ‘EXPONENT’ HERE A MAY BE AN ARBITRARY REAL NUMBER VERY OFTEN ONE USES EULER’S NUMBER E (1 18) WRITING: Y D Y0 E?X Y0 EXP ?X/ : (1 63) THIS FUNCTION IS OF GREAT IMPORTANCE IN THEORETICAL PHYSICS AND APPEARS OFTEN IN A VARIETY OF CONTEXTS (RATE OF GROWTH, INCREASE OF POPULATION, LAW OF RADIOACTIVE DECAY, CAPACITOR CHARGE AND DISCHARGE, ) (FIG 1 8) MATHEMATICAL PREPARATIONS FIG 1 8 SCHEMATIC BEHAVIOR OF THE EXPONENTIAL FUNCTION IN SECT 1 1 10 WE WILL BE ABLE TO PROVE, BY USING THE TAYLOR EXPANSION, THE FOLLOWING IMPORTANT SERIES EXPANSION OF THE EXPONENTIAL FUNCTION: EX D LN YA/ D LN EX/A/ D LN EAX/ D A X D A LN Y : (1 69) ONE STILL RECOGNIZES THE SPECIAL CASES: LN 1/ D LN E0/ D I LN X < IF 0 < X < 1 : (1 70) FINALLY, LET US STILL WORK OUT THE PROOF OF (1 17) WHICH WE HAD TO POSTPONE BECAUSE IT EXPLOITS PROPERTIES OF THE LOGARITHM EQUATION (1 17) IS CONCERNED WITH THE FOLLOWING STATEMENT ABOUT THE LIMIT OF THE SEQUENCE FANG D FQNG ! 0 ; IF JQJ < 1 : WE ASSUME JAN 0J < " < 1 : THAT MEANS (FIG 1 9): JQNJ D JQJN < " < , LN JQJN < LN " < , N LN JQJ „?‚… < < LN " < ) N > LN " LN JQJ > 0 : IF N" IS THE SMALLEST NATURAL NUMBER (INTEGER) WITH N" LN " LN JQJ ; THEN THE STARTING INEQUALITY IS FUL?LLED FOR ALL N N" AND 0 IS INDEED THE LIMIT OF THE SEQUENCE FOR ALL JQJ < 1 FIG 1 9 SCHEMATIC BEHAVIOR OF THE NATURAL LOGARITHM FIG 1 11 TO THE DE?NITION OF THE DERIVATIVE OF A FUNCTION Y D F X/ CURVE F X/ AT P (BROKEN LINE IN FIG 1 11), TAN ? D LIM ?0!? TAN ?0 D LIM X ! Y X AND ONE ARRIVES AT THE ‘DIFFERENTIAL QUOTIENT’ LIM X ! X D 2X C X : THUS THE ?RST DERIVATIVE IS: F 0 X/ D 2X : ALL THE DIFFERENTIAL QUOTIENTS DO NOT EXHIBIT A UNIQUE LIMIT EVERYWHERE! THE CURVE IN FIG 1 12 IS CONTINUOUS AT P, BUT HAS THERE DIFFERENT SLOPES IF WE COME, RESPECTIVELY, MATHEMATICAL PREPARATIONS FIG 1 12 EXAMPLE OF A FUNCTION Y D F X/ BEING NOT DIFFERENTIABLE IN THE POINT P FROM THE LEFT AND THE RIGHT HAND SIDE ONE SAYS THAT F X/ IS ‘NOT DIFFERENTIABLE’ AT THE POINT P DE?NITION Y D F X/ IS DIFFERENTIABLE AT X0 IF AND ONLY IF F X0/ IS DE?NED AND A UNIQUE LIMITING VALUE OF THE DIFFERENCE QUOTIENT EXISTS: F 0 X0/ D LIM X ! THAT CAN BE SEEN AS FOLLOWS: X C X/N D N XN C N XN1X C : : : C N N XN N R D NŠ RŠ N R/Š Õ Y X D C X C X/N XN X N 2/ D C X N XN1X C N XN2X2 C : : : : : : C N N XN D C N XN1 C N Õ LIM X ! Y X D C N XN1 : FOR N D 0 (OR N D 1) THE DIFFERENCE QUOTIENT IS ALREADY IDENTICAL TO ZERO (OR C), I E INDEPENDENT OF X, SO THAT THE ASSERTION IS IMMEDIATELY FUL?LLED 2 Y D F X/ D C I C 2 R H) F 0 X/ (1 78) BECAUSE: Y X D C C X D 0 : THIS IS OF COURSE SIMPLY THE N D 0-SPECIAL CASE OF THE ?RST EXAMPLE 3 Y D F X/ D EX H) F 0 X/ D EX : (1 79) THE EXPONENTIAL FUNCTION IS DIFFERENTIABLE FOR ALL X AS CAN BE SEEN AS FOLLOWS: Y X D EXCX EX X D EX EX X D EX 1 C X C 2X2 C MATHEMATICAL PREPARATIONS D EX 1 C 2X C 6X2 C : : : Õ LIM X ! Y X D EX : HERE WE HAVE USED THE ANTICIPATED SERIES EXPANSION (1 64) OF THE EXPONENTIAL FUNCTION, WHICH WILL BE EXPLICITLY DERIVED IN (1 95) 4 Y D F X/ D SIN X H) F 0 X/ D COS X : (1 80) SIN X IS DIFFERENTIABLE FOR ALL REAL X, BECAUSE: Y X D SIN X C X/ SIN X X D SIN X COS X C COS X SIN X SIN X X D SIN X COS X 1/ X C COS X SIN X X : IN THE SECOND STEP WE HAVE APPLIED THE ADDITION THEOREM (1 60) IF WE FURTHERMORE USE THE RELATION PROVED AS EXERCISE 1 1 5, 1 COS X D 2 SIN2 X 2 ; THEN IT REMAINS TO CALCULATE: F 0 X/ D LIM X ! SIN X SIN X SIN X EXAMPLES F X/ D X3 Õ F 0 X/ D 3X2 Õ F 00 X/ D 6X Õ F 3/ X/ D 6 Õ F 4/ D 0 Õ F N/ X/ 0 8 N RULES OF DIFFERENTIATION WE LIST SOME OF THE CENTRAL RULES FOR DIFFERENTIATING FUNCTIONS OF ONE INDEPENDENT VARIABLE: 1 CONSTANT FACTOR: Y D C F X/ H) Y0 D C F 0 X/ ; (1 82) PROOF: Y0 D LIM X ! C F X C X/ C F X/ X D C LIM X ! F X C X/ F X/ X D C F 0 X/ : 2 SUM: Y D F X/ C G X/ H) Y0 D F 0 X/ C G0 X/ : (1 83) THIS CAN DIRECTLY BE READ OFF FROM THE DE?NITION MATHEMATICAL PREPARATIONS 3 PRODUCT: Y D F X/ G X/ H) Y0 D F 0 X/ G X/ C F X/ G0 X/ ; (1 84) PROOF: Y0 D LIM X ! X F X C X/ G X C X/ F X/ G X// D LIM X ! X F X C X/ F X// G X C X/ C G X C X/ F X/ F X/ G X/ D LIM X ! F X C X/ F X/ X G X C X/ C LIM X ! 0 F X/ G X C X/ G X/ X D F 0 X/ G X/ C F X/ G0 X/ : IN THE LAST STEP WE HAVE EXPLOITED THE FACT THAT THE FUNCTIONS G AND F OF COURSE HAVE TO BE CONTINUOUS SINCE OTHERWISE THE DERIVATIVES WOULD NOT EXIST EXAMPLE SUPPOSE N 2 N, THEN: XN XN D Õ XN/0 XN C XN Õ NXN1 XN D XN XN/0 : AS AN EXTENSION TO (1 77) WE NOW HAVE A CODE FOR HOW TO DIFFERENTIATE A POWER OF X WITH NEGATIVE EXPONENT: XN/0 D N X NC1/ : (1 85) 4 QUOTIENT Y D F X/ G X/ I G X/ ¤ H) Y0 D F 0 X/ G X/ F X/ G0 X/ G2 X/ ; (1 86) PROOF: FIRST WE INVESTIGATE THE DERIVATIVE OF H X/ D WHERE WE CAN AGAIN PRESUME THE CONTINUITY OF G X/: H0 X/ D LIM X ! G X C X/ G X/ D LIM X ! G X C X/ G X/ X G X C X/ G X/ D G0 X/ G2 X/ : WITH THE PRODUCT RULE (1 84) THE ASSERTION IS THEN PROVEN 5 CHAIN RULE: Y D F G X// H) Y0 D DF DG G0 X/ ; (1 87) PROOF LET U D G X/ BE DIFFERENTIABLE IN X AND Y D F U/ DIFFERENTIABLE IN U D G X/, THEN IT CAN BE WRITTEN WITH G X C X/ D U C U (CONTINUITY!): F G X C X// F G X// X D F U C U/ F U/ U G X C X/ G X/ X : UTILIZING ONCE MORE THE CONTINUITY OF U D G X/ X ! 0 Õ U ! 0/ WE GET: LIM X ! F G X C X// F G X// X D D DU F U/ D DXG X/ : FORMALLY WE THUS OBTAIN A RESULT WHICH APPEARS TO BE TAKEN FROM ‘NORMAL FRACTIONAL ARITHMETIC’: DY DX D DY DU DU DX : EXAMPLE WE DEMONSTRATE THE CHAIN RULE IN CONNECTION WITH AN IMPORTANT APPLI- CATION FOR THIS PURPOSE WE CALCULATE THE ?RST DERIVATIVE OF Y D F X/ D LN X ; WHICH EXISTS FOR ALL POSITIVE X WE USE THE CHAIN RULE TOGETHER WITH (1 79) TO DIFFERENTIATE THE EXPRESSION X D ELN X WITH RESPECT TO X: 1 D ELN X D DX LN X MATHEMATICAL PREPARATIONS OBVIOUSLY THIS YIELDS: D DX LN X D ELN X D X ; (1 88) NOW WE CAN GENERALIZE ONCE MORE THE RULES OF DIFFERENTIATION (1 77) AND (1 85), RESPECTIVELY SUPPOSE THAT ? IS NOW AN ARBITRARY REAL NUMBER THEN WE HAVE: X? D ELN X? D E? LN X Õ DX? DX D DEU DU ???? U D ? LN X D ? LN X/ DX D E? LN X ? X D X?? X : THAT YIELDS THE GENERALIZATION OF (1 77) AND (1 85), RESPECTIVELY, DX? DX D ? X?1 ; (1 89) WHICH IS THUS PROVEN NOW FOR ARBITRARY REAL NUMBERS ? 6 FINALLY, WE WILL CONSIDER THE INVERSE FUNCTION (1 38): F 1 F X// D X : WITH THE CHAIN RULE WE HAVE: D DF F F/ F 0 X/ D 1 : THAT MEANS: D DF F F/ D F 0 X/ : (1 90) WITH Y D F X/ Õ X D F 1 Y/ Õ D DY F 1 Y/ D DX DY WE GET AN EXPRESSION WHICH AGAIN SEEMS TO STEM FROM ELEMENTARY FRACTIONAL ARITHMETIC: DX DY D EXAMPLES TO 1 : F X/ D A SIN X I A 2 R H) F 0 X/ D A COS X TO 2 : F X/ D X5 3 LN X H) F 0 X/ D 5X4 TAYLOR EXPANSION OCCASIONALLY IT IS UNAVOIDABLE FOR A PHYSICIST TO DIGRESS FROM RIGOROUS MATHEMATICAL EXACTNESS IN ORDER TO COME BY ADOPTING SOME ‘REASONABLE’ MATHEMATICAL SIMPLI?CA- TIONS TO CONCRETE PHYSICAL RESULTS IN THIS RESPECT, THE SO-CALLED ‘TAYLOR EXPANSION (SERIES)’ OF A MATHEMATICAL FUNCTION Y D F X/ REPRESENTS A VERY IMPORTANT AND FREQUENTLY USED AUXILIARY MEANS WE ASSUME THAT THIS FUNCTION POSSESSES ARBITRARILY MANY CONTINUOUS DERIVATIVES AT X D X0 THEN THE FOLLOWING POWER SERIES EXPANSION IS VALID WHAT IS EXPLICITLY PROVED AS EXERCISE 1 1 9: F X/ D F X0/ C F 0 X0/ 1Š X X0/ C F 00 X0/ 2Š X X0/2 C : : : D F N/ X0/ NŠ X X0/N (1 92) F N/ X0/ D F N/ X/ ?? X D X0 : THE ASSUMPTION JX X0J < 1 GUARANTEES THE CONVERGENCE OF THE SERIES THEN ONE CAN ASSUME THAT THE TERMS OF THE SERIES BECOME SMALLER AND SMALLER WITH INCREASING MATHEMATICAL PREPARATIONS INDEX N, SO THAT IT SHOULD BE ALLOWED, IN THE SENSE OF A CONTROLLED APPROXIMATION, TO CUT THE SERIES AFTER A ?NITE NUMBER OF SUMMANDS THE ERROR CAN STRICTLY BE ESTIMATED AS WILL BE DEMONSTRATED IN SECT 1 2 OF VOLUME 3 HOWEVER, THE TAYLOR EXPANSION CAN ALSO BE USED FOR THE DERIVATION OF EXACT SERIES AS IS SHOWN BY THE FOLLOWING EXAMPLES: 1 F X/ D 1 C X I X0 D 0 I JXJ < 1 : WE USE F 0/ D 1 I F 0 0/ D 1 1 C 0/2 D 1 I F 00 0/ D 2 1 C 0/3 D 2 : : : Õ F N/ 0/ D 1/N NŠ I X X0 D X : THAT MEANS 1 C X D X/N : (1 93) COMPARE THIS RESULT WITH (1 30)! 2 F X/ D SIN X I X0 D 0 : NOW WE APPLY THE FOLLOWING TERMS IN THE TAYLOR SERIES (1 92): F 0/ D 0 I F 0 0/ D COS 0/ D 1 I F 00 0/ D SIN 0/ D 0 I F 000 0/ D COS 0/ D 1 I : : : Õ F 2N/ 0/ D 0 I F 2NC1/ 0/ D 1/N : THUS WE ?ND IN THIS CASE: SIN X D X 3Š X3 C 5Š X5 C : : : D WITH (1 79) IT HOLDS: E0 D 1 I D DX EX D EX Õ DN DXN EX D EX Õ DN DXN EX ???? X D D 1 : THEREWITH WE GET: EX D LN X FOR EXPRESSIONS OF THIS KIND WE HAVE THE VERY USEFUL L’HOSPITAL’S RULE, WHICH, HOWEVER, HAS TO BE PRESENTED HERE WITHOUT PROOF IF THE FUNCTION F X/ D ' X/ X/ GIVES FOR X ! A AN INDETERMINED EXPRESSION OF THE ABOVE KIND THEN ONE CAN USE LIM X ! A F X/ D LIM X ! A '0 X/ 0 X/ : (1 96) MATHEMATICAL PREPARATIONS IF THE RIGHT-HAND SIDE PERSISTS TO BE NOT DE?NED ONE REPLACES THE ?RST BY THE SECOND DERIVATIVES IF EVEN THEN THE QUOTIENT ON THE RIGHT-HAND SIDE CONTINUES TO BE UNDETERMINED ONE TAKES THE THIRD DERIVATIVES, AND SO ON HENCE, THE ABOVE EXAMPLES ARE CALCULATED AS FOLLOWS: LIM X ! LN 1 C X/ X D LIM X ! 1CX (1 97) LIM X ! SIN X X D LIM X ! COS X (1 98) LIM X ! X D LIM X ! D LIM X ! 0 X/ D HAVING THERE A (LOCAL) EXTREMUM, THEN IT MUST HOLD: F 0 X0/ D AND MONOTONICALLY INCREASING IF X > X0 THAT MEANS: F 00 X/ AND THEREWITH: F 00 X0/ D (IN?ECTION POINT) (1 101) MATHEMATICAL PREPARATIONS FIG 1 15 FUNCTION F X/ WITH A MAXIMUM AT X D X0 AND ITS DERIVATIVE F 0 X/ FIG 1 16 FUNCTION F X/ WITH A MINIMUM AT X D X0 AND ITS DERIVATIVE F 0 X/ A SUF?CIENT CRITERION FOR AN EXTREMUM AT THE POINT X D X0 CAN EASILY BE READ OFF FROM FIGS 1 15 AND 1 16: F 0 X0/ D AND F 00 X0/ > W MINIMUM < W MAXIMUM : (1 102) REGARDING (1 101) IT IS TO BE NOTED THAT IT IS ALSO ONLY A NECESSARY CONDITION FOR AN IN?ECTION POINT, WHILE A SUF?CIENT CONDITION WOULD BE: F 0 X0/ D F 00 X0/ D AND F 000 X0/ ¤ 0 : (1 103) THE GENERAL CASE, HOWEVER, MUST BE ACCEPTED HERE AGAIN WITHOUT PROOF: LET US ASSUME A SUF?CIENTLY OFTEN DIFFERENTIABLE FUNCTION F X/ WITH THE FOLLOWING PROPERTIES AT X D X0: F 0 X0/ D F 00 X0/ D : : : D F N/ X0/ D 0 WITH F NC1/ X0/ ¤ FIG 1 17 SCHEMATIC BEHAVIOR OF THE FUNCTIONS F1 X/ D X3 AND F2 X/ D X4 IN THE VICINITY OF X D AT X D ONE IMMEDIATELY ?NDS: F 1 0/ D F 1 0/ D I F IN THIS CASE HOLDS: F 2 0/ D F 2 0/ D EXERCISES EXERCISE 1 1 1 DETERMINE THE LIMITING VALUES OF THE SEQUENCES FANG FOR N ! (N 2 N) MATHEMATICAL PREPARATIONS 1 AN D PN N 2 AN D N3 C 2N3 C N2 C N 3 AN D N2 N C 1/2 C 1 CALCULATE THE FOLLOWING SUMS: S3 D M I S D M : 2 IS 1:111 : : : A RATIONAL NUMBER? IF YES, WHICH ONE? EXERCISE 1 1 3 SHOW THAT THE HARMONIC SERIES (1 27) DOES NOT CONVERGE IN SPITE OF LIMM ! M D 0! EXERCISE 1 1 4 TRY TO SIMPLIFY THE FOLLOWING EXPRESSIONS FOR TRIGONOMETRIC FUNC- TIONS: COS2 ' TAN2 ' C COS2 ' 1 COS2 ' SIN ' COS ' COS2 ' 1 SIN ' C SIN '1 C '2/ C SIN '1 '2/ COS '1 C '2/ C COS '1 '2/ COS2 ' SIN 2' EXERCISE 1 1 5 PROVE THE FORMULA 1 COS ' D 2 SIN2 ' 2 : WHICH HAS BEEN USED FOR THE DERIVATION OF (1 80) EXERCISE 1 1 6 VERIFY THE FOLLOWING RELATIONS FOR THE ?RST DERIVATIVES OF THE TRIGONO- METRIC FUNCTIONS: 1 D DX COS X D SIN X 2 D DX TAN X D COS2 X 3 D DX COT X D 2 F2 X/ D 7X3 4X 6 F6 X/ D 3 COS 6X/ 7 F7 X/ D SIN X2/ 8 F8 X/ D EXP 2X3 4/ 9 F9 X/ D LN 2X C 1/ : EXERCISE 1 1 8 USE THE RULE OF DIFFERENTIATION FOR THE INVERSE FUNCTION (1 90) IN ORDER TO ?ND THE DERIVATIVES OF THE ARC FUNCTIONS (INVERSES OF THE TRIGONOMETRIC FUNCTIONS): 1 D DX ARCSIN X D 2 D DX ARCCOS X D 3 D DX ARCTAN X D 4 D DX ARCCOTX D EXERCISE 1 1 9 ASSUME THAT THE FUNCTION F X/ IS ARBITRARILY OFTEN DIFFERENTIABLE IN ADDITION, AN EXPANSION INTO A POWER SERIES SHALL EXIST: F X/ D AN XN : ALL X FOR WHICH THE SERIES CONVERGES CONSTITUTE THE SO-CALLED REGION OF CONVERGENCE OF THE FUNCTION F X/ 1 DETERMINE THE COEF?CIENTS AN FROM THE BEHAVIOR OF THE FUNCTION F AND ITS DERIVATIVES AT X D 0 2 VERIFY EQ (1 92): F X/ D F N/ X0/ NŠ X X0/N : EXERCISE 1 1 10 WHY CAN THE FUNCTION F X/ D 1 C X/N FOR X 1 BE REPLACED TO A GOOD APPROXIMATION BY F X/ 1 C N X C N N 1/ X2 ‹ EXERCISE 1 1 11 VERIFY THE SERIES EXPANSION OF THE LOGARITHM (JXJ < 1): LN 1 C X/ D N XN D X 2 X2 C 3 X3 MATHEMATICAL PREPARATIONS 2 G X/ D SIN BECAUSE OF THE CONSTANT D THE RESULT COMES OUT AS A FULL FAMILY OF CURVES FIXING D NEEDS THE INTRODUCTION OF ‘BOUNDARY CONDITIONS’ WE ACCEPT THAT: ‘INTEGRATION’ : SEARCHING FOR THE ANTIDERIVATIVE (PRIMITIVE FUNCTION) TO GENERATE A GRAPHIC IMAGE, THE INTEGRAL CAN BE INTERPRETED AS THE AREA UNDER THE CURVE Y D F X/ IF THE CURVE Y D F X/ IS GIVEN THEN WE ASK OURSELVES HOW WE CAN DETERMINE THE AREA F IN FIG 1 18 UNDER THE CURVE BETWEEN THE LIMITS X D A AND X D B THIS CAN EASILY BE DONE FOR THE SPECIAL CASE THAT F X/ REPRESENTS A STRAIGHT LINE HOWEVER, HOW CAN WE CALCULATE THE AREA UNDER AN ARBITRARY (CONTINUOUS) FUNCTION F X/? IN A ?RST STEP, WE APPROACH THE CALCULATION OF THE AREA BY DECOMPOSING THE INTERVAL ŒA; B IN N EQUAL SUB-INTERVALS XN, XN D B A N N 2 N=0 ; (1 107) WHERE XI IS THE CENTER OF THE I-TH PARTIAL INTERVAL: XI D A C I XN I I D 1; 2 : : : ; N : (1 108) THEN F XI/ XN IS THE AREA OF THE I-TH PILLAR IN FIG 1 19 HENCE IT HOLDS APPROXIMATELY FOR THE AREA F: F N X I D F XI/ XN : (1 109) FIG 1 18 INTERPRETATION OF THE INTEGRAL AS AREA UNDER THE CURVE Y D F X/ FIG 1 19 RIEMANN SUM FOR CALCULATING THE INTEGRAL MATHEMATICAL PREPARATIONS FOR N ! 1 THE SUB-INTERVALS BECOME ARBITRARILY SMALL (XN ! 0), AND IT APPEARS OBVIOUS THAT THE MISTAKE WHICH RESULTS FROM APPROXIMATING F BY THE SUM OF THE ‘PILLAR AREAS’ ALSO BECOMES ARBITRARILY SMALL THE LIMITING VALUE FOR N ! 1 COMES OUT AS A REAL NUMBER AND IS CALLED: ‘DE?NITE (RIEMANN) INTEGRAL’ F D LIM N ! N X I D FIRST RULES OF INTEGRATION SOME IMPORTANT RULES FOLLOW DIRECTLY FROM THE DE?NITION OF THE INTEGRAL: IDENTICAL BOUNDS OF INTEGRATION: Z A A F X/ DX D A F X/ DX > F X/ DX < F X/ DX > CONSTANT FACTOR C 2 R: Z B A C F X/ DX D LIM N ! C F XI/XN D C LIM N ! F XI/XN : HENCE IT HOLDS: Z B A C F X/ DX D C Z B A F X/ DX : (1 112) SUM: ASSUME F X/ G X/ C H X/ : THEN IT FOLLOWS FROM THE DE?NITION OF THE RIEMANN INTEGRAL: Z B A F X/ DX D LIM N ! G XI/ C H XI// XN D LIM N ! G XI/XN C LIM N ! N THEREWITH WE CAN WRITE: Z B A F X/ DX D LIM N ! F X 1/ I X 1/ N C LIM N ! F X 2/ I X 2/ N MATHEMATICAL PREPARATIONS FIG 1 21 PARTITIONING THE INTERVAL OF INTEGRATION X 1;2/ I ARE DE?NED AS IN (1 108) WITH CORRESPONDING X 1;2/ N X 1/ I D A C I 2/ X0A N ; X 2/ I D X0 C I THE AREA UNDER THE CURVE F T/ IN THIS CASE IS NOT CONSTANT BUT A FUNCTION OF X (FIG 1 22) IF THE UPPER BOUND OF INTEGRATION IS SHIFTED BY X THE AREA WILL CHANGE BY: F D F X C X/ F X/ D Z XCX A F T/ DT Z X A F T/ DT D Z XCX X F T/ DT : IN THE LAST STEP WE HAVE USED THE RULE (1 114) WITHOUT EXPLICIT PROOF WE ACCEPT THE IMPORTANT ‘MEAN VALUE THEOREM OF INTEGRAL CALCULUS’ THIS THEOREM IMPLIES: 9 OX 2 ŒX; X C X WITH F D X F OX/ : (1 117) ALTHOUGH NOT EXACTLY PROVEN THE THEOREM APPEARS RATHER PLAUSIBLE ACCORDING TO FIG 1 23 SO WE CAN FURTHER CONCLUDE: F0 X/ D LIM X ! F X D LIM X ! 0 F OX/ D F X/ : THUS AFTER (1 105), THE AREA FUNCTION IS THE ANTIDERIVATIVE OF F X/! FURTHERMORE, THE EQUIVALENCE OF THE DE?NITIONS (1 105) AND (1 110) FOR THE ANTIDERIVATIVE, WHICH FIG 1 22 DE?NITION OF THE AREA FUNCTION FIG 1 23 TO THE MEAN VALUE THEOREM (1 117) FIG 1 25 TO THE DE?NITE INTEGRAL OF THE COSINE THE ‘DE?NITE INTEGRAL’, ALREADY KNOWN TO US, CAN ALSO BE EXPRESSED BY THE ANTIDERIVATIVE: F X/ C ? D Z X A F T/ DT Õ F A/ C ? D Õ F A/ D ? : THEREWITH IT FOLLOWS THAT WHEN WE TAKE X D B: Z B A F X/ DX D F B/ C ? D F B/ F A/ F X/ ??? B A : (1 121) AT THE EXTREME WE HAVE INTRODUCED THE USUAL SYMBOL FOR THE DE?NITE INTEGRAL EXAMPLE F X/ D COS X Õ F X/ D SIN X C C C 2 R/ : (1 122) THE ANTIDERIVATIVE CAN EASILY BE GUESSED WITH (1 80) (FIG 1 25) HENCE WE OBTAIN THE FOLLOWING DE?NITE INTEGRALS FOR THE COSINE: Z C COS X DX D SIN X ??C 2 D 1 1/ D Z C COS X DX D SIN X ?? 0 D 1 0 D 2 D 1/ 1 D (SIGN OF THE AREA!) Z 2 D 0 1/ D MATHEMATICAL PREPARATIONS Z COS X DX D SIN X ?? 0 D 0 0 D F3 X/ D THE LAST THREE RELATIONS CAN OF COURSE BE PROVEN ALSO DIRECTLY VIA CORRESPONDING SERIES EXPANSIONS WE BRIE?Y DEMONSTRATE THEM: Z SIN X DX D