PARTIAL DIFFERENTIAL EQUATIONS: FOUNDATIONS AND INTEGRAL REPRESENTATIONS
CONTENTS OF VOLUME FOUNDATIONS AND INTEGRAL REPESENTATIONS I DI?ERENTIATION AND INTEGRATION ON MANIFOLDS THE WEIERSTRAß APPROXIMATION THEOREM PARAMETER-INVARIANT INTEGRALS AND DI?ERENTIAL FORMS THE EXTERIOR DERIVATIVE OF DI?ERENTIAL FORMS THE STOKES INTEGRAL THEOREM FOR MANIFOLDS THE INTEGRAL THEOREMS OF GAUß AND STOKES CURVILINEAR INTEGRALS THE LEMMA OF POINCAR´E CO-DERIVATIVES AND THE LAPLACE-BELTRAMI OPERATOR SOME HISTORICAL NOTICES TO CHAPTER I II FOUNDATIONS OF FUNCTIONAL ANALYSIS DANIELL’S INTEGRAL WITH EXAMPLES EXTENSION OF DANIELL’S INTEGRAL TO LEBESGUE’S INTEGRAL MEASURABLE SETS MEASURABLE FUNCTIONS RIEMANN’S AND LEBESGUE’S INTEGRAL ON RECTANGLES BANACH AND HILBERT SPACES THE LEBESGUE SPACES LP(X) BOUNDED LINEAR FUNCTIONALS ON LP(X) AND WEAK CONVERGENCE SOME HISTORICAL NOTICES TO CHAPTER II THE WINDING NUMBER THE DEGREE OF MAPPING IN RN GEOMETRIC EXISTENCE THEOREMS THE INDEX OF A MAPPING THE PRODUCT THEOREM THEOREMS OF JORDAN-BROUWER – IV GENERALIZED ANALYTIC FUNCTIONS THE CAUCHY-RIEMANN DI?ERENTIAL EQUATION HOLOMORPHIC FUNCTIONS IN CN GEOMETRIC BEHAVIOR OF HOLOMORPHIC FUNCTIONS IN C ISOLATED SINGULARITIES AND THE GENERAL RESIDUE THEOREM THE INHOMOGENEOUS CAUCHY-RIEMANN DI?ERENTIAL EQUATION PSEUDOHOLOMORPHIC FUNCTIONS CONFORMAL MAPPINGS BOUNDARY BEHAVIOR OF CONFORMAL MAPPINGS SOME HISTORICAL NOTICES TO CHAPTER IV V POTENTIAL THEORY AND SPHERICAL HARMONICS POISSON’S DI?ERENTIAL EQUATION IN RN POISSON’S INTEGRAL FORMULA WITH APPLICATIONS DIRICHLET’S PROBLEM FOR THE LAPLACE EQUATION IN RN THEORY OF SPHERICAL HARMONICS: FOURIER SERIES THEORY OF SPHERICAL HARMONICS IN N VARIABLES VI LINEAR PARTIAL DI?ERENTIAL EQUATIONS IN RN THE MAXIMUM PRINCIPLE FOR ELLIPTIC DI?ERENTIAL EQUATIONS QUASILINEAR ELLIPTIC DI?ERENTIAL EQUATIONS THE HEAT EQUATION CHARACTERISTIC SURFACES THE WAVE EQUATION IN RN FOR N = 1, 3, 2 THE WAVE EQUATION IN RN FOR N ?2 THE INHOMOGENEOUS WAVE EQUATION AND AN INITIAL-BOUNDARY- VALUE PROBLEM CLASSI?CATION, TRANSFORMATION AND REDUCTION OF PARTIAL DI?ERENTIAL EQUATIONS SOME HISTORICAL NOTICES TO THE CHAPTERS V AND VI REFERENCES IN GENERAL, WE DENOTE OPEN SUBSETS IN RN BY THE SYMBOL ? BY THE SYMBOL M WE INDICATE THE TOPOLOGICAL CLOSURE AND BY ? M THE OPEN KERNEL OF A SET M ?RN IN THE SEQUEL, WE SHALL USE THE FOLLOWING LINEAR SPACES OF FUNCTIONS: C0(?) CONTINUOUS FUNCTIONS ON ? CK(?) K-TIMES CONTINUOUSLY DI?ERENTIABLE FUNCTIONS ON ? CK 0 (?) K-TIMES CONTINUOUSLY DI?ERENTIABLE FUNCTIONS F ON ?WITH THE COMPACT SUPPORT SUPP F = {X ??: F(X) ?= 0} ?? CK(?) K-TIMES CONTINUOUSLY DI?ERENTIABLE FUNCTIONS ON ?, WHOSE DERIVATIVES UP TO THE ORDER K CAN BE CONTINUOUSLY EXTENDED ONTO THE CLOSURE ? CK 0 (???) K-TIMES CONTINUOUSLY DI?ERENTIABLE FUNCTIONS F ON ?, WHOSE DERIVATIVES UP TO THE ORDER K CAN BE EXTENDED ONTO THE CLOSURE ?CONTINUOUSLY WITH THE PROPERTY SUPP F ???? C? ?(?, K) SPACE OF FUNCTIONS AS ABOVE WITH VALUES IN K = RN OR K = C FINALLY, WE UTILIZE THE NOTATIONS ?U GRADIENT (UX1, , UXN) OF A FUNCTION U = U(X1, , XN) ? C1(RN) , X > F (K)(0) K! XK IN THE FOLLOWING PROPOSITION, WE INTRODUCE A ’MOLLI?ER’ WHICH ENABLES US TO SMOOTH FUNCTIONS PROPOSITION 1 WE CONSIDER THE FOLLOWING FUNCTION TO EACH ? > 0, NAMELY K?(Z) := ? = PROOF: 1 THE EXPONENTIAL FUNCTION IS POSITIVE, AND THE STATEMENT IS OBVIOUS 2 WE SUBSTITUTE Z = ??X WITH DZ = ?? N DX AND CALCULATE RN K?(Z) DZ = ? DZ = DT ? ? N = 1 3 WE UTILIZE THE SUBSTITUTION FROM PART 2 OF OUR PROOF AND OBTAIN |Z|?? K?(Z) DZ = FOR ? ?0+ , AND CONSEQUENTLY THE FUNCTIONS F?(X) CONVERGE UNIFORMLY ON THE SPACE RN TO- WARDS THE FUNCTION F(X) PROOF: ON ACCOUNT OF ITS COMPACT SUPPORT, THE FUNCTION F(X) IS UNIFORMLY CONTINUOUS ON THE SPACE RN THE NUMBER ? > 0 BEING GIVEN, WE ?ND A NUMBER ? = ?(?) > 0 SUCH THAT X, Y ?RN, |X ?Y| ?? =? |F(X) ?F(Y)| ?? SINCE F IS BOUNDED, WE ?ND A QUANTITY ?0 = ?0(?) > 0 SATISFYING 2 SUP Y?RN |F(Y)| |Y?X|?? K?(Y ?X) DY ?? FOR ALL 0 < ? < ?0 WE NOTE THAT |F?(X) ?F(X)| = RN K?(Y ?X) F(Y) DY ?F(X) RN K?(Y ?X) DY ? |Y?X|?? K?(Y ?X) {F(Y) ?F(X)} DY + |Y?X|?? K?(Y ?X) {F(Y) ?F(X)} DY , 0(RN) HOLDS TRUE DUE TO PROPOSITION 2, THE FAMILY OF FUNCTIONS D?F?(X) CONVERGES UNIFORMLY ON THE SPACE RN TOWARDS D?F(X) - FOR ALL |?| ?K - WHEN ? ?0+ HOLDS TRUE NOW WE CHOOSE THE RADIUS R > 0 SUCH THAT SUPP F ?BR IS VALID TAKING THE NUMBER ? > 0 AS ?XED, WE CONSIDER THE POWER SERIES K?(Z) = ? J , WHICH CONVERGES UNIFORMLY IN B2R THEREFORE, EACH NUMBER ? > 0 POSSESSES AN INDEX N0 = N0(?, R) SUCH THAT THE POLYNOMIAL P?,R(Z) := N ? J I DI?ERENTIATION AND INTEGRATION ON MANIFOLDS IS SUBJECT TO THE FOLLOWING ESTIMATE: SUP |Z|?2R |K?(Z) ?P?,R(Z)| ?? WITH THE EXPRESSION F?,R(X) := RN P?,R(Y ?X)F(Y) DY WE OBTAIN A POLYNOMIAL IN THE VARIABLES X1, , XN - FOR EACH ? > 0 FURTHER- MORE, WE DEDUCE D? F?,R(X) = RN P?,R(Y ?X)D?F(Y) DY FOR ALL X ?RN, |?| ?K NOW WE ARRIVE AT THE SUBSEQUENT ESTIMATE FOR ALL |?| ?K AND |X| ?R, NAMELY |D?F?(X) ?D? F?,R(X)| = |Y|?R K?(Y ?X) ?P?,R(Y ?X) D?F(Y) DY ? |Y|?R |K?(Y ?X) ?P?,R(Y ?X)||D?F(Y)| DY ?? |Y|?R |D?F(Y)| DY THEREFORE, THE POLYNOMIALS D? F?,R(X) CONVERGE UNIFORMLY ON BR TOWARDS THE DERIVATIVES D?F(X) CHOOSING THE NULL-SEQUENCE ? = M WITH M = 1, 2, , WE OBTAIN AN APPROXIMATING SEQUENCE OF POLYNOMIALS PM,R(X) := F FOR ALL |?| ?K AND ALL COMPACT SUBSETS C ?? Q E D THEOREM 1 ABOVE PROVIDES A UNIFORM APPROXIMATION BY POLYNOMIALS IN THE IN- TERIOR OF THE DOMAIN FOR THE RESPECTIVE FUNCTION CONTINUOUS FUNCTIONS DE?NED ON COMPACT SETS CAN BE UNIFORMLY APPROXIMATED UP TO THE BOUNDARY OF THE DOMAIN HERE WE NEED THE FOLLOWING THEOREM 2 (TIETZE’S EXTENSION THEOREM) LET C ?RN DENOTE A COMPACT SET AND F(X) ?C0(C, C) A CONTINUOUS FUNCTION DE?NED ON C THEN WE HAVE A CONTINUOUS EXTENSION OF F ONTO THE WHOLE SPACE RN WHICH MEANS: THERE EXISTS A FUNCTION G(X) ?C0(RN, C) SATISFYING F(X) = G(X) FOR ALL POINTS X ?C I DI?ERENTIATION AND INTEGRATION ON MANIFOLDS PROOF: 1 WE TAKE X ?RN AND DE?NE THE FUNCTION D(X) := MIN Y?C |Y ?X| , WHICH MEASURES THE DISTANCE OF THE POINT X TO THE SET C SINCE C IS COM- PACT, WE ?ND TO EACH POINT X ?RN A POINT Y ?C SATISFYING |Y ?X| = D(X) WHEN X1, X2 ?RN ARE CHOSEN, WE INFER THE FOLLOWING INEQUALITY FOR Y2 ?C WITH |Y2 ?X2| = D(X2), NAMELY D(X1) ?D(X2) = INF Y?C |X1 ?Y|) ?|X2 ?Y2| ?|X1 ?Y2| ?|X2 ?Y2| ?|X1 ?X2| INTERCHANGING THE POINTS X1 AND X2, WE OBTAIN AN ANALOGOUS INEQUALITY AND INFER |D(X1) ?D(X2)| ?|X1 ?X2| FOR ALL POINTS X1, X2 ?RN IN PARTICULAR, THE DISTANCE D : RN ?R REPRESENTS A CONTINUOUS FUNCTION 2 FOR X /?C AND A ?RN, WE CONSIDER THE FUNCTION ?(X, A) := MAX 2 ?|X ?A| D(X) , AS WELL AS ?(X, A) = 2?K? X, A(K) > ?K(X) F A(K) ?F(Z) ? K:|A(K)?X|?2D(X) ?K(X) F A(K) ?F(Z) ? SUP A?C : |A?X|?2D(X) |F(A) ?F(Z)| ? SUP A?C : |A?Z|?2D(X)+|X?Z| |F(A) ?F(Z)| ? SUP A?C : |A?Z|?3|X?Z| |F(A) ?F(Z)| SINCE THE FUNCTION F : C ?C IS UNIFORMLY CONTINUOUS, WE INFER LIM X?Z X??C H(X) = F(Z) FOR Z ??C AND X /?C Q E D T , IF T > ?µ(X) > FOR ALL X ?RN ?µ(X) ARE DIRECTLY INFERRED FROM THE CONSTRUCTION ABOVE Q E D DE?NITION 1 WE NAME THE FUNCTIONS ?1, ?2, , ?M FROM THEOREM 4 A PAR- TITION OF UNITY SUBORDINATE TO THE OPEN COVERING {OX}X?K OF THE COMPACT SET K §2 PARAMETER-INVARIANT INTEGRALS AND DI?ERENTIAL FORMS IN THE BASIC LECTURES OF ANALYSIS THE FOLLOWING FUNDAMENTAL RESULT IS ESTAB- LISHED THEOREM 1 (TRANSFORMATION FORMULA FOR MULTIPLE INTEGRALS) LET ?, ? ?RN DENOTE TWO OPEN SETS, WHERE WE TAKE N ?N FURTHERMORE, LET Y = (Y1(X1, , XN), , YN(X1, , XN)) : ??? DENOTE A BIJECTIVE MAPPING OF THE CLASS C1(?, RN) SATISFYING JY(X) := DET ?YI(X) ?XJ I,J=1,,N ?= ?SM (S) ?TM ?S1 (S) ?TM ?SM (S) > FOR ALL S ?T; 2 X(S) = X T(S) FOR ALL S ?T WE SAY THAT X ORIGINATES FROM X BY AN ORIENTATION-PRESERVING REPARAMETRIZA- TION THE EQUIVALENCE CLASS [X] CONSISTING OF ALL THOSE PARAMETRIC REPRESENTA- TIONS WHICH ARE EQUIVALENT TO X IS NAMED AN OPEN, ORIENTED, M-DIMENSIONAL, REGULAR SURFACE OF THE CLASS CK IN RN WE NAME A SURFACE EMBEDDED INTO THE SPACE RN IF ADDITIONALLY THE MAPPING X : T ?RN IS INJECTIVE EXAMPLE 1 (CURVES IN RN) ON THE INTERVAL T = (A, B) ?R WE CONSIDER THE MAPPING X = X(T) = X1(T), , XN(T) ?C1(T, RN), T ?T SATISFYING |X?(T)| = {X? 1(T)}2 + + {X?N(T)}2 > FOR ALL T ?T THEN THE INTEGRAL L(X) = B A |X?(T)| DT DETERMINES THE ARC LENGTH OF THE CURVE X = X(T) I DI?ERENTIATION AND INTEGRATION ON MANIFOLDS EXAMPLE 2 (CLASSICAL SURFACES IN R3) WHEN T ?R2 DENOTES AN OPEN PARAMETER DOMAIN, WE CONSIDER THE GAUSSIAN SURFACE REPRESENTATION X(U, V) = X(U, V), Y(U, V), Z(U, V) : T ??R3 ?C1(T, R3) THE VECTOR IN THE DIRECTION OF THE NORMAL TO THE SURFACE IS GIVEN BY XU ?XV = ?(Y, Z) ?(U, V), ?(Z, X) ?(U, V), ?(X, Y) ?(U, V) = (YUZV ?ZUYV, ZUXV ?XUZV, XUYV ?XVYU) THE UNIT NORMAL VECTOR TO THE SURFACE X IS DE?NED BY THE FORMULA N(U, V) := XU ?XV |XU ?XV| , AND WE NOTE THAT |N(U, V)| = 1, N(U, V) XU(U, V) = N(U, V) XV(U, V) = SUCH THAT A(X) = T |XU|2|XV|2 ?(XU XV)2 DUDV FOLLOWS EXAMPLE 3 (HYPERSURFACES IN RN) LET X : T ?RN DENOTE A REGULAR SURFACE - DE?NED ON THE PARAMETER DOMAIN T ?RN?1 THE (N ?1) VECTORS XT1, , XTN?1 ARE LINEAR INDEPENDENT FOR ALL T ?T ; AND THEY SPAN THE TANGENTIAL SPACE TO THE SURFACE AT THE POINT X(T) ? RN NOW WE SHALL CONSTRUCT THE UNIT NORMAL VECTOR ?(T) ?RN THEREFORE, WE REQUIRE |?(T)| = AND ?(T) XTK(T) = AS WELL AS DET XT1(T), , XTN?1(T), ?(T) > ?IDI FOR ALL ?1, , ?N ?R NOW WE INTRODUCE THE UNIT NORMAL VECTOR ?(T) = ?1(T), , ?N(T) = (DJ(T))2 DT1 DTN?1 CORRESPONDINGLY, WE DE?NE THE SUBMATRICES OF THE MATRIX B THEN WE HAVE THE IDENTITY DET (AT ?B) = 1?I1<<IM?N DET AI1IM DET BI1IM PROOF: WE ?X A AND SHOW THAT THE IDENTITY ABOVE HOLDS TRUE FOR ALL MATRICES B 1 WHEN WE CONSIDER THE UNIT VECTORS E1, , EN AS COLUMNS IN RN, THE FORMULA ABOVE HOLDS TRUE FOR ALL B = (EJ1, , EJM) WITH J1, , JM ?{1, , N} AT ?RST 2 WHEN THE FORMULA ABOVE HOLDS TRUE FOR THE MATRIX B = (B1, , BM), THIS REMAINS TRUE FOR THE MATRIX B? = (B1, , ?BI, , BM) 3 WHEN WE HAVE OUR FORMULA FOR THE MATRICES B? = (B1, , B? I, , BM) AND B?? = (B1, , B?? I , , BM), THIS REMAINS TRUE FOR THE MATRIX B = (B1, , B? I + B?? I , , BM) Q E D COROLLARY: GIVEN THE N × M-MATRIX A, WE HAVE THE IDENTITY DET (AT ?A) = 1?I1<<IM?N (DET AI1IM)2 WE WRITE THE METRIC TENSOR INTO THE FORM GIJ(T) I,J=1,,M = ?X(T)T ??X(T) WITH THE FUNCTIONAL MATRIX ?X(T) = XT1(T), XTM(T) , AND WE DEDUCE G(T) = DET GIJ(T) I,J=1,,M = 1?I1<<IM?N ?(XI1, , XIM) ?(T1, , TM) (T) THEREFORE, THE SURFACE ELEMENT SATIS?ES D?(T) = G(T) DT1 DTM = 1?I1<<IM?N ?(XI1, , XIM) ?(T1, , TM) (T) DT1 DTM DE?NITION 2 THE SURFACE AREA OF AN OPEN, ORIENTED, M-DIMENSIONAL, REGULAR C1-SURFACE IN RN WITH THE PARAMETRIC REPRESENTATION X(T) : T ?RN IS GIVEN BY THE IMPROPER RIEMANNIAN INTEGRAL I DI?ERENTIATION AND INTEGRATION ON MANIFOLDS A(X) := T 1?I1<<IM?N ?(XI1, , XIM) ?(T1, , TM) AI1IM X(T) ?(XI1, , XIM) ?(T1, , TM) DT1 DTM = ?(X) THEREFORE, ? IS A MAPPING WHICH IS DE?NED ON THE EQUIVALENCE CLASSES OF THE ORIENTED SURFACES [X] WITH X ?F 5 AN ORIENTATION-REVERSING PARAMETRIC TRANSFORMATION T = T(S) WITH J(S) < 0, S ?T INDUCES THE CHANGE OF SIGN: ?( X) = ??(X) DE?NITION 4 A 0-FORM OF THE CLASS CK(O) IS SIMPLY A FUNCTION F(X) ?CK(O) AND MORE PRECISELY ? = F(X), X ?O WHEN 1 ?M ?N IS ?XED, WE NAME ?M := DXI1 ? ?DXIM, 1 ?I1, , IM ?N A BASIC M-FORM I DI?ERENTIATION AND INTEGRATION ON MANIFOLDS DE?NITION 5 LET ?, ?1, ?2 REPRESENT THREE M-FORMS OF THE CLASS C0(O) AND CHOOSE C ?R THEN WE DE?NE THE DI?ERENTIAL FORMS C? AND ?1 + ?2 BY THE PRESCRIPTION (C?)(X) := C?(X) FOR ALL X ?F AND (?1 + ?2)(X) := ?1(X) + ?2(X) FOR ALL X ?F RESPECTIVELY THE M-DIMENSIONAL DI?ERENTIAL FORMS CONSTITUTE A VECTOR SPACE WITH THE NULL- ELEMENT O(X) = 4 IN PARTICULAR, WHEN THE TWO INDICES IJ1 AND IJ2 COINCIDE, WE DEDUCE DXI1 ? ?DXIL = 0 THEREFORE, EACH M-FORM IN RN WITH THE DEGREE M > N VANISHES IDENTICALLY 5 AN L-FORM ?1 AND AN M-FORM ?2 ARE SUBJECT TO THE COMMUTATOR RELATION ?1 ??2 = (?1)LM?2 ??1 THEREFORE, THE EXTERIOR PRODUCT IS NOT COMMUTATIVE 6 WE CAN REPRESENT EACH M-FORM IN THE FOLLOWING WAY: ? = 1?I1<<IM?N AI1IM(X) DXI1 ? ?DXIM THE BASIC M-FORMS DXI1 ? ?DXIM, 1 ?I1 < < IM ?N CONSTITUTE A BASIS FOR THE SPACE OF ALL DI?ERENTIAL FORMS, WITH COE?CIENT FUNCTIONS IN THE CLASS CK(O), WHERE K ?N0 HOLDS TRUE DE?NITION 7 LET THE SYMBOL ? = 1?I1<<IM?N AI1IM(X) DXI1 ? ?DXIM, X ?O DENOTE A CONTINUOUS DI?ERENTIAL FORM ON THE OPEN SET O ?RN, WITH 1 ? M ?N BEING ?XED THEN WE DE?NE THE IMPROPER RIEMANNIAN INTEGRAL OF THE DI?ERENTIAL FORM ? OVER THE SURFACE [X] ?O VIA [X] ? := T 1?I1<<IM?N AI1IM X(T) ?(XI1, , XIM) ?(T1, , TM) DT1 DTM, IF ? IS ABSOLUTELY INTEGRABLE OVER X AND CONSEQUENTLY [X] |?| := T 1?I1<<IM?N AI1IM X(T) ?(XI1, , XIM) ?(T1, , TM) DT1 DTM < +? IS SATIS?ED REMARK: WITH THE AID OF THE TRANSFORMATION FORMULA, WE SHOW THAT THESE INTEGRALS ARE INDEPENDENT OF THE CHOICE OF THE REPRESENTATIVES FOR THE SURFACE THEREFORE, WE ARE ALLOWED TO WRITE [X] |?| = X |?|, [X] ? = X ? FXI(X) DXI, X ?O WHEN ? = 1?I1<<IM?N AI1IM(X) DXI1 ? ?DXIM REPRESENTS AN M-FORM OF THE CLASS C1(O), WE DE?NE ITS EXTERIOR DERIVATIVE AS THE (M + 1)-FORM D? := 1?I1<<IM?N DAI1IM(X) ?DXI1 ? ?DXIM REMARKS: 1 WHEN ?1 AND ?2 ARE TWO M-FORMS IN RN AND ?1, ?2 ?R ARE GIVEN, WE HAVE THE IDENTITY D(?1?1 + ?2?2) = ?1D?1 + ?2D?2 THEREFORE, THE DI?ERENTIAL OPERATOR D CONSTITUTES A LINEAR OPERATOR 2 WHEN ? DENOTES AN L-FORM AND ? AN M-FORM OF THE CLASS C1(O), WE INFER THE PRODUCT RULE D(? ??) = (D?) ?? + (?1)M? ?D? WE SHALL PROVE ONLY THE LAST STATEMENT HERE IT SU?CES TO CONSIDER THE SITUATION ? = F(X)?M, ? = G(X)?L , WHERE ?M AND ?L ARE BASIC FORMS OF THE ORDER M AND L, RESPECTIVELY NOW WE DEDUCE ?AI ?XI (X) # DX1 ? ?DXN = DIV A(X) DX1 ? ?DXN 0(H ?S), WE OBTAIN H D? = X D? = ??I ?YJ (Y) DYJ, I = 1, , N AND ?? := 1?I1<<IM?N AI1IM ?(Y) D?I1 ? ?D?IM , WE OBTAIN THE TRANSFORMED M-FORM ?? WITH RESPECT TO THE MAPPING ? REMARKS: 1 WHEN ?1, ?2 ARE TWO M-FORMS AND ?1, ?2 ?R ARE GIVEN, WE INFER THE IDENTITY (?1?1 + ?2?2)? = ?1(?1)? + ?2(?2)? 2 WHEN ? REPRESENTS AN L-FORM AND ? AN M-FORM, WE HAVE THE RULE (? ??)? = ?? ??? THE FOLLOWING RESULT IS IMPORTANT FOR THE EVALUATION OF INTEGRALS FOR DI?ERENTIAL FORMS OVER SURFACES THEOREM 1 (PULL-BACK OF DI?ERENTIAL FORMS) LET ? DENOTE A CONTINUOUS M-FORM IN THE OPEN SET O ?RN ON THE OPEN SET T ?RM WE DE?NE A SURFACE X BY THE PARAMETRIC REPRESENTATION X = ?(Y) : T ??O ?C1(T ) WITH ?(T ) ??O FINALLY, WE DE?NE THE SURFACE Y (T) = (T1, , TM), T ?T AND NOTE THAT X(T) = ? ?Y (T), T ?T THEN THE FOLLOWING IDENTITY HOLDS TRUE: X ? = Y ?? ? ? ??IM ?YJM D?JM = ? ? I1,,IM AI1IM ?(Y) D?I1 ? ?D?IM ? ? Y=?(Z) , AND CONSEQUENTLY ???? = (??)? HERE WE PERFORM OUR SUMMATION OVER THE INDICES I1, , IM ?{1, , N}, J1, , JM ?{1, , L?}, AND K1, , KM ?{1, , L??} Q E D WITH THE LATERAL LENGTHS 2R THIS OBJECT HAS THE UPPER BOUNDING SIDE SR(?) := Y ?RM+1 : Y1 = ?1, YJ ?(?J ?R, ?J + R) FOR J = 2, , M + WE COMPREHEND HR(?) AND SR(?) AS SURFACES IN RM+1: HR(?) : Y (T1, , TM+1) = (?1 + T1, , ?M+1 + TM+1) WITH ?R < T1 < 0, |TJ| < R, J = 2, , M + I DI?ERENTIATION AND INTEGRATION ON MANIFOLDS THE CONSIDERATIONS IN PART 1 ) REVEAL H(L) (D?)?(K) = S(L) ??(K) FOR ALL K, L ?N ?N THE TRANSITION TO THE LIMIT K ??IMPLIES H(L) (D?)? = S(L) ?? ON ACCOUNT OF $ F |D?| < +?, THE LIMIT PROCEDURE L ??YIELDS F D? = H (D?)? = S ?? = S ? THIS IS EXACTLY THE IDENTITY STATED ABOVE Q E D NOW WE INTRODUCE THE FUNDAMENTAL NOTION OF A DI?ERENTIABLE MANIFOLD DE?NITION 1 LET US ?X THE DIMENSIONS 1 ?M ?N AS WELL AS THE SET M ?RN WE NAME M AN M-DIMENSIONAL CK-MANIFOLD, IF EACH POINT ? ?M POSSESSES AN ELEMENT ? ?RM AND OPEN NEIGHBORHOODS U ?RN OF ? ?U AND V ?RM OF ? ?V AS WELL AS AN EMBEDDED REGULAR SURFACE X = ?(Y) : V ??U ?CK(V ) SUCH THAT ? = ?(?) AND ?(V ) = M ?U IS CORRECT; HERE WE HAVE CHOSEN K ?N ADEQUATELY WE CALL (?, V ) A CHART OF THE MANIFOLD ALL CHARTS TOGETHER A := (??, V?) : ? ?J CONSTITUTE AN ATLAS OF THE MANIFOLD WHEN ?J : VJ ?UJ ?M WITH J = 1, ? ? AND AN OPEN NEIGHBORHOOD U ?RN OF ? ?U WITH THE FOLLOWING PROPERTIES: ?(?) = ?, ? SR(?) = ? M ?U, ? HR(?) = M ?U THE SET OF REGULAR BOUNDARY POINTS WILL BE DENOTED BY THE SYMBOL ?M DE?NITION 3 FOR THE BOUNDED MANIFOLD M FROM DE?NITION 2, WE DE?NE THE SET OF SINGULAR BOUNDARY POINTS ?M ACCORDING TO ?M := ? M \ ?M IN THE CASE ?M = ?, WE OBTAIN A COMPACT MANIFOLD WITH REGULAR BOUNDARY IF THE CONDITION ?M = ?IS FUL?LLED ADDITIONALLY, WE SPEAK OF A CLOSED MANIFOLD M ?M ?M ?M = ? ?M = ? ?M = ? ? ? ? : ? ? PPPP Q WITH THE PARAMETER 0 ?? ?1, WE CONSIDER THE MATRICES M(?) := ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? (1 ??)?Y1(?) + ?N ?Y2(?) ?YM+1(?) N N N?M ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? , % M(?) := ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? (1 ??)?Y1(?) + ?N DET % M(0)?1 ?M(0) = DET ?(??1 ??) ? > ?K(X) = FOR ALL X ?SUPP ? WE DE?NE ?L(X) = 0(M ??M, [0, 1]) IS NAMED EXHAUSTING THE MANIFOLD, WHEN EACH COMPACT SET K ?M ??M POSSESSES AN INDEX K0 = K0(K) ?N SUCH THAT ?K(X) = X = ?(?, ?) X := SUP |?? ?|2 : ? ?TM(X), |?| = NOW WE ARRIVE AT OUR CENTRAL RESULT, NAMELY