On the Differentiability of Real, Complex and Quaternion Functions
|
|
- Jewel Fitzgerald
- 5 years ago
- Views:
Transcription
1 Bulletin of TICMI Vol. 8, No., 204, On the Differentibility of Rel, Complex nd Quternion Funtions Omr Dzgnidze A. Rzmdze Mthemtil Institute of Iv. Jvkhishvili Tbilisi Stte University, 6 Tmrshvili St., 077, Tbilisi, Georgi (Reeived Deember 26, 203; Revised Mrh 25, 204; Aepted June 6, 204) In the pper, the neessry nd suffiient onditions re estblished for the ontinuity of funtions of severl vribles, for the differentibility of funtions of severl rel vribles, nd for the C n -differentibility of funtions of severl omplex vribles. Also, the rules re given for lulting H-derivtives of quternion funtions, nd the neessry nd suffiient onditions re obtined for the H-differentibility of quternion funtions. Keywords: Seprtely strong (ngulr) prtil ontinuity, nonintense ngulr prtil ontinuity, ngulr (strong) prtil derivtive, ngulr (strong) grdient, nonintense ngulr prtil derivtive, differentibility, totl differentil, vrition in the Hrdy (Arzel) sense, indefinite integrl, bsolutely ontinuity, Lebesgue s intense point, definite integrl with prmeter, C n -differentibility, Hrtog s min theorem, H-derivtive AMS Subjet Clssifition: 32A20, 30G35, 30A05, 30B0. Introdution A funtion of mny vribles will not hve the ontinuity or differentibility property only beuse it hs the sme property with respet to eh independent vrible. Funtions with this drwbk t individul points hve been known sine the lte 9th entury, nd on the mssive set sine the 20th entury. Nmely, the following sttement is vlid. Sttement A ([, pp ]): There exists the funtion of two vribles whih is disontinuous t lmost every point of the unit squre nd t every point of tht squre ontinuous with respet to every vrible. Note tht this Tolstov s funtion does not posses lmost everywhere even the property of ontinuity on the whole (see [2]; [3, pp ]). These nd nlogous problems re studied e.g. in Z. Piotrowski [4]. Here the problem onsists in finding out whether there exists or does not exist ny notion of funtion with respet to n independent vrible nd whether the fulfillment of this notion for ll independent vribles will be the neessry nd suffiient ondition for the ontinuity nd differentibility of the funtion itself. In this pper, the disussion onentrtes on this problem. Emil: odzgni@rmi.ge ISSN: print 204 Tbilisi University Press
2 94 Bulletin of TICMI In formulting the min results, we use the following nottion: x = (x,..., x n ), = (,..., x0 n), x( k ) = (x,..., x k, k, x k+,..., x n ).. The onditions for the ontinuity. The funtion f is lled strong prtil ontinuous with respet to the vrible x k t the point, if the equlity [f(x) f(x( k ))] = 0 (.) x is fulfilled nd f is lled seprtely strong prtil ontinuous t the point, if f with respet to every vrible is strongly prtil ontinuous t, i.e. equlity (.) is fulfilled for ll k =, 2,..., n. Theorem. ([5]; [2]; [3, pp ]): For the ontinuity of the funtion f t the point, it is neessry nd suffiient tht it possesses seprtely strong prtil ontinuity t. 2. The expression f(x) f(x( k )) for x j j j x k k, j k, depending on the vribles x,..., x n, is lled n ngulr prtil inrement of the funtion f t the point with respet to the vrible x k, orresponding to the olletion = (,..., k, k+,..., n ) of positive onstnts. The ngulr prtil ontinuity of the funtion of t the point with respet to the vrible x k mens the fulfillment of the equlity x k k [f(x) f(x( k ))] = 0 (.2) x j j j x k k j k for every olletion = (,..., k, k+,..., n ) of positive onstnts. The funtion f is lled seprtely ngulr prtil ontinuous t the point, if with respet to every vrible the funtion f possesses the property of ngulr prtil ontinuity t the point, i.e. if for ll k =,..., n nd for every olletion = (,..., k, k+,..., n ) of positive onstnts, equlity (.2) holds. Theorem.2 ([5]; [2] ; [3, pp ]): For the ontinuity of the funtion f t the point, the neessry nd suffiient ondition is the seprtely ngulr prtil ontinuity t. 3. If in the definition of the ngulr prtil ontinuity we put j = for ll j k, then we hve the nonintense ngulr prtil ontinuity t the point of the funtion f with respet to the vrible x k. Theorem.3 ([3, pp ]) : For the ontinuity of the funtion f t the point, the neessry nd suffiient ondition is the seprtely nonintense ngulr prtil ontinuity of the funtion f t the point.
3 2. Angulr prtil derivtive nd ngulr grdient Vol. 8, No., The existene of finite prtil derivtives of ll orders, i.e. ordinry grdients of the rel funtion f t the point does not imply the differentibility of the funtion f t the sme point. Even the funtion, possessing finite grdient t the point, my be disontinuous t. Suh, for exmple, re t the point (0, 0) the most of the funtions of two vribles indited in Piotrowski s work [4]. It is remrkble tht this ft n be relized t ll points of set, whose plne mesure is rbitrrily nerly to totl mesure. Sttement B ([, 4]): For every number µ < there exists the funtion F, defined on the squre Q = {(x, y) R 2 ; 0 x 2, 0 y }, possessing t ll points of the Q finite prtil derivtives of ll orders, nd t the sme time F is disontinuous on ertin set E Q of plne mesure µ 2. We sy tht the funtion F hs t the point n ngulr prtil derivtive with respet to the vrible x k, symbolilly f x k ( ), if for every olletion = (,..., k, k+,..., n ) of positive n onstnts there exists n independent of the finite it f x k ( ) = f(x) f(x( k )) x k k x k. (2.) x j j j x k k k j k The existene of f x k ( ) implies existene of the prtil derivtive f x k ( ), nd the equlity f x k ( ) = f x k ( ). To show this, we hve to put in (2.) x j = j for ll j k. The existene of the ngulr prtil derivtive does not, in generl, follows from existene of the ordinry prtil derivtive. If f x k ( ) is finite, then the funtion f with respet to the vrible x k hs the property of ngulr prtil ontinuity t the point. If there exist finite f x k ( ), k =,..., n, then we ll f the funtion possessing n ngulr grdient t the point nd write nggrd f( ) = (f x ( ),..., f x n ( )). Theorem 2. ([5]; [6]; [3, pp ]): For the funtion f to be differentible t the point, it is neessry nd suffiient tht nggrd f( ) is finite. The totl differentil df( ) of the differentible t the point funtion f dmits the following representtion n df( ) = f x k ( ) dx k. k= Theorem 2.2 ([6]; [3, p. 65]): For the funtion f to be differentible t the point, it is neessry nd suffiient tht the nonintense ngulr prtil deriv-
4 96 Bulletin of TICMI tives D xk f( ) = re finite for ll k =,..., n. f(x) f(x( k )) x k k x k x j j x k k k j k Corollry 2.3 ([6]; [3, p. 65]) : of ll f x k ( ), nd the equlity The finiteness of ll D xk f( ) implies finiteness f x k ( ) = D xk f( ), k =,..., n, n df( ) = D xk f( ) dx k. k= 3. Exmples on the differentibility Using Theorem 2.2, we n estblish the differentibility s well s nondifferentibility of onrete funtions. On the differentibility we investigte some ppering frequently funtions. Proposition 3. ([6]; [3, p. 66]): Suppose the numbers α j re positive, j =,..., n. Then the ondition α + α α n > is neessry nd suffiient for the everywhere ontinuous funtion φ(x,..., x n ) = x α x 2 α2 x n α n to be differentible t the point = (0,..., 0). In prtiulr, the funtion γ(x,..., x n ) = ( x x n ) α is differentible t the point if nd only if α > /n. If α + α α n, then ll D xk φ( ) re devoid of existene. Proposition 3.2 ([3, p. 67]): the funtion Suppose the numbers β j >, j =,..., n. Then n x j βj for ll rtionl x j Φ(x,..., x n ) = j= 0 t the remining points (3.) is differentible t the point = (0,..., 0), dφ( ) = 0 nd disontinuous t ll the remining points (x,..., x n ) (0,..., 0).
5 Proposition 3.3 ([3, p. 68]): Vol. 8, No., ( n q x j ) for ll rtionl x j Ψ(x,..., x n ) = j= 0 t the remining points possesses the sme properties s the funtion (3.). Proposition 3.4 ([3, p. 68]): (3.2) The orresponding to the number α > 0 funtion ( n ) +α x 2 2 j for ll rtionl x j ω(x,..., x n ) = j= 0 t the remining points possesses ll properties of funtions (3.) nd (3.2). Proposition 3.5 ([3, p. 69]): g(x, x 2 ) = The funtion { x x 2 sin x x 2 for x x for x x 2 = 0 is differentible t the point = (0, 0), nd its grdient grd g(x, x 2 ) is indeterminte in the puntured neighborhood of the point. Proposition 3.6 ([3, pp ]) : ψ(x, x 2 ) = possesses the following properties: The orresponding to the number q > funtion { x 2 x 2 The funtion x 2 +x 2 2 for x 2 + x2 2 > 0, 0 for x = 0 = x 2 ) ψ(x, x 2 ) is ontinuous everywhere; 2) grd ψ(x, x 2 ) is finite everywhere; 3) ψ(x, x 2 ) is not differentible t the point (0, 0); 4) grd ψ(x, x 2 ) is not ontinuous t the point (0, 0). 4. A strong prtil derivtive nd strong grdient We sy tht the funtion f possesses t the point strong prtil derivtive with respet to the vrible x k, symbolilly f [x k ] (x0 ), if there exists finite it f [x f(x) f(x( k ] (x0 k ) = )) x x k. k We sy tht the funtion f hs t the point strong grdient, symbolilly strgrd f( ), if for every k =,..., n there exist finite f [x k] (x0 ), nd we write strgrd f( ) = (f [x k ] (x0 ),..., f [x n ] (x0 )).
6 98 Bulletin of TICMI If there exists strgrd f( ), then there likewise exists nggrd f( ), nd equlities strgrd f( ) = nggrd f( ) = grd f( ) hold. Consequently, we hve Theorem 4. ([5]; [6]; [3, p. 77]) : The existene of finite strgrd f( ) implies existene of totl differentil df( ) nd strgrd f( ) = nggrd f( ) = grd f( ). If the grd f(x) is ontinuous t the point, then we hve the equlity strgrd f( ) = grd f( ) ([3, p. 75]). Besides, the existene of the finite strgrd f( ) does not, in generl, imply the ontinuity of grd f(x) t the point. For, the grdient of the funtion g(x, x 2 ) from the Proposition 3.5 is not ontinuous t (0, 0), but strgrd g(0, 0) = (0, 0). In ddition to this, we hve Theorem 4.2 ([3, p. 76]): There exists n bsolutely ontinuous funtion of two vribles whih hs lmost everywhere both finite strong nd disontinuous grdients. Remrk : We should begin the proof of Theorem 4.2 from [3] with the following. There exists mesurble set e [0, ] suh tht the sets e (α, β) nd ([0, ] \ e) (α, β) hve positive mesures for ll subintervls (α, β) [0, ] ([7, p. 50] or [8, p. 49]). By α(x) we denote the hrteristi funtion of the set e. Anlogously, we obtin the funtion β(y). Proposition 4.3 ([5]; [6]; [3, p. 77]) : The finiteness of nggrd f( ) or, wht is the sme, the existene of df( ) does not imply the existene of strgrd f( ). The funtion λ(x, x 2 ) = x x 2 2/3 is differentible t the point = (0.0) (see Proposition 4.3), but strgrd λ( ) does not exist. Afterwrds, G.G. Onini estblished tht the existene of finite strong grdient is essentilly stronger property thn the differentibility. Theorem 4.4 ([]; [2]) : f : [0, ] n R suh tht: For rbitrry n 2 there exists ontinuous funtion. f is differentible lmost everywhere, 2. f devoid of finite strong grdient lmost everywhere. The following theorem is improvement of Theorem 4.4. Theorem 4.5 ([0, Theorem 4]): For rbitrry n 2 there exists ontinuous funtion f : [0, ] n R tht is differentible lmost everywhere, but everywhere devoid of finite strong grdient. As is known, funtions of bounded vritionss in the Hrdy or Arzel sense possess the differentibility property lmost everywhere, i.e. hve finite ngulr grdients lmost everywhere. As to the existene of strong grdient, funtions of these lsses behve differently. Theorem 4.6 ([9]; [0]): Every funtion f : [0, ] n R of bounded vrition in the Hrdy sense hs finite strong grdient lmost everywhere.
7 Vol. 8, No., Theorem 4.7 ([0, Theorem 3]): For rbitrry n 2 there exists ontinuous funtion f : [0, ] n R of bounded vrition in the Arzel sense tht everywhere devoid of finite the strong grdient. 5. Clssifition of funtions by vrious grdients Theorem 5. ([3, p. 80]): A lss with ontinuous t the point grdients of funtions is ontined stritly in lss with finite t the point strong grdients of funtions, nd the ltter is ontined stritly in lss of funtions with finite t the point ngulr grdients. This lss oinides with the lss of differentible t funtions. Remrk : The notions of ngulr nd strong grdients were generlized by Leri Bntsuri, who introdued the notion of grdient with respet to the bsis nd estblished, in prtiulr, the reltionship between the differentiblity nd the existene of the grdient whih he hs introdued ([3], [4]). 6. Differentibility of n indefinite integrl nd of n bsolutely ontinuous funtions Let the funtion of two vribles f be summble on the retngle Q = {(x, y) R 2 : x b, y d}, f L(Q). Consider for the funtion f the indefinite double integrl F (x, y) = x y f(t, τ) dt dτ. (6.) The following problems re quite nturl. I. Does the indefinite double integrl (6.), hve or hve no totl differentil lmost everywhere? II. If F hs totl differentil, then t wht points nd how the set of suh points is onneted with the funtion f? The nswer to problem I will be given here, nd problem II will be onsidered in Setion 8. Theorem 6. ([5, Theorem 6.7]; [5]; [3, pp ]): Indefinite integrl (6.) hs totl differentil t lmost ll points (x, y) Q for every funtion f L(Q). Theorem 6.2 ([5]; [5]; [3, p. 04]): At every point (, y 0 ) Q of differentibility of the indefinite integrl (6.) with f L(Q) we hve h 0 k 0 h + k x0 +h y0 +k y 0 f(t, τ) dt dτ = 0. (6.2) In prtiulr, equlity (6.2) is fulfilled t lmost ll points (, y 0 ) Q. Theorem 6.3 ([5]; [3, p. 05]): Let the indefinite integrl (6.) for f L(Q) hve in the neighborhood of the point (, y 0 ) Q finite F x, F y nd F x,y. Then for the funtion F to be differentible t the point (, y 0 ), it is neessry nd suffiient
8 00 Bulletin of TICMI tht h 0 k 0 h k h + k F x,y( + θ h, y 0 + θ 2 k) = 0, 0 < θ, θ 2 <. If, in ddition, F x,y = f in the neighborhood of the point (, y 0 ), then for the differentibility of the funtion F t (, y 0 ) it is neessry nd suffiient tht h 0 k 0 h k h + k f( + θ h, y 0 + θ 2 k) = 0, 0 < θ, θ 2 <. Theorem 6.4 ([5]; [5]; [3, pp ]): Every bsolutely ontinuous on the Q funtion hs totl differentil lmost everywhere on the Q. Its prtil nd mixed prtil derivtives re summble on the Q funtions. Note tht Theorem 6.4 is not true for seprtely bsolutely ontinuous funtions! 7. The finiteness of strong grdient of n indefinite integrl nd n bsolutely ontinuous funtion Theorem 7. ([5, Theorem 6.6]; [3, p. 09]): Let the funtion f be summble on the retngle Q = {(x, y) R 2 : x b, y d}. Then the orresponding indefinite integrl F (x, y) = possesses the following properties: x y f(t, τ) dt dτ ) for lmost every [, b] nd for every y 0 [, d] the F [x] (, y 0 ) is finite, nd F [x] (x, y) = y0 f(, τ) dτ; 2) for every [, b] nd for lmost every y 0 [, d] the F [y] (, y 0 ) is finite, nd F [y] (x, y) = x0 f(t, y 0 ) dt; 3) t lmost every point (, y 0 ) Q the strgrd F (, y 0 ) is finite. Theorem 7.2 ([5, Theorem 6.8]; [3, pp. 2]): For every summble on the retngle Q = {(x, y) R 2 : x b, y d} funtion f the following sttements re vlid: ) there exists mesurble set e [, b] with e = b, suh tht t every point (, y 0 ) with e nd y [, d] the integrl y 0 f(, τ) dτ is finite,
9 Vol. 8, No., nd h 0 h y y 0 x0 +h y0 f(t, τ) dt dτ = y0 f(, τ) dτ; (7.) 2) there exists mesurble set e 2 [, d] with e 2 = d, suh tht t every point (, y 0 ) with [, b] nd y 0 e 2 the integrl f(t, y 0) dt is finite, nd y0 +k x0 x0 f(t, τ) dt dτ = f(t, y 0 ) dt; (7.2) k 0 k y x 0 3) equlities (7.) nd (7.2) re fulfilled simultneously t the points (, y 0 ) E, where E = e e 2, E = Q. To formulte this nd the subsequent theorems in short, we introdue the following mesurble sets: A) E = m( ), E = Q, e where the mesurble set e [, b] with e = b is dopted from sttement ) of Theorem 7.2, nd the vertil losed intervl m( ) is defined by the equlity B) E 2 = y 0 e 2 n(y 0 ), m( ) = {(, y) : y d}; E 2 = Q, where the mesurble set e 2 [, d] with e 2 = d is dopted from sttement 2) of Theorem 7.2, nd the horizontl losed intervl n(y 0 ) is defined by the equlity n(y 0 ) = {(x, y 0 ) : x b}. Now Theorem 7.2 n be rephrsed s follows. Theorem 7.3 ([3, p. 2]): For every funtion f L(Q), equlities (7.) nd (7.2) tke ple t the points (, y 0 ) E nd (, y 0 ) E 2, respetively. Equlities (7.) nd (7.2) re fulfilled simultneously t the points (, y 0 ) E 3, where E 3 = E E 2, E 3 = Q. Theorem 7.4 ([5]; [3, pp. 2 3]): sttements tke ple: For every funtion f L(Q) the following ) t the points (, y 0 ) E the equlity holds; (h,k) (0,0) h x0 +h y0 +k y 0 f(t, τ) dt dτ = 0 (7.3)
10 02 Bulletin of TICMI 2) t the points (, y 0 ) E 2 the equlity (h,k) (0,0) k x0+h y0+k y 0 f(t, τ) dt dτ = 0 (7.4) is vlid; 3) equlities (7.3) nd (7.4) re fulfilled simultneously t the points (, y 0 ) E 3, where E 3 = E E 2, E 3 = Q ; 4) t the points (, y 0 ) E 3 the equlity holds. h + k (h,k) (0,0) hk x0+h y0+k y 0 f(t, τ) dt dτ = 0 (7.5) Remrk ([3, p. 3]): If S(x, y) L(Q) is Sk s funtion, then the expression hk x+h y+k x y S(t, τ) dt dτ (7.6) hs the strong supper it + t every point (x, y) Q. At the sme time, Theorem 7.4 shows tht tending of expression (7.6) to + is subordinte to equlities (7.3) nd (7.4) t the points (, y 0 ) E E 2, i.e. hk x0+h y0+k y 0 ( S(t, τ) dt dτ = O mx(h, k) Theorem 7.5 ([3, pp. 3 4]): To every bsolutely ontinuous on the retngle Q funtion Φ there orresponds triple of funtions φ L(Q), g L([, b]) nd h L([, d]), suh tht the following sttements tke ple: ) for lmost every [, b] nd for every y 0 [, d] there exists the finite Φ [x] (, y 0 ), nd ). Φ [x] (, y 0 ) = y0 φ(, y) dy + g( ); 2) for every [, b] nd for lmost every y 0 [, d] there exists the finite Φ [y] (, y 0 ), nd Φ [y] (, y 0 ) = x0 φ(x, y 0 ) dx + h(y 0 ); 3) t lmost every point (, y 0 ) Q the strgrdφ(, y 0 ), Φ x,y(, y 0 ) nd Φ y,x(, y 0 ) re finite, nd Φ x,y(, y 0 ) = φ(, y 0 ) = Φ y,x(, y 0 ). An n 2-dimensionl nlogue of sttement 3) of Theorem 7. is
11 Vol. 8, No., Theorem 7.6 ([6]; [7]; [3, p. ]): For every n 2 nd f L(0, ) n the indefinite integrl of f, t lmost every point is differentible, moreover, hs finite strong grdient. 8. Lebesgue s intense points nd finiteness t these points of strong grdient of n indefinite integrl Definition 8. ([5]; [3, p. 5]): Let the funtion f belong to the spe L p (Q) for some p. The point (, y 0 ) Q is lled jointly Lebesgue s intense point (of p-th degree) of the funtion f, symbolilly (, y 0 ) int L p x,y(f), if the following two onditions re fulfilled: (h,k) (0,0) h (h,k) (0,0) k x0 +h y0 +k y 0 y0 +k x0 +h y0 f(x, y) dy x0 f(x, y) dy p f(, y) dy dx = 0, (8.) p f(x, y 0 ) dx dy = 0. (8.2) When equlity (8.) is fulfilled, then the point (, y 0 ) is lled Lebesgue s intense point with respet to the vrible x (of p-th degree) of the funtion f, symbolilly (, y 0 ) int L p x(f). When equlity (8.2) is fulfilled, then the point (, y 0 ) is lled Lebesgue s intense point with respet to the vrible y (of p-th degree) of the funtion f, symbolilly (, y 0 ) int L p y(f). Theorem 8.2 ([3, p. 5]): Let the funtion f belong to the spe L p (Q) for some p. The following sttements tke ple: ) there exists mesurble set e [, b] with e = b, suh tht the set of ll points (, y 0 ) with e nd y 0 [, d] forms the set int L p x(f), int L p x(f) = Q ; 2) there exists mesurble set e 2 [, d] with e 2 = d, suh tht the set of ll points (, y 0 ) with [, b] nd y 0 e 2 forms the set int Lp y(f), int L p y(f) = Q ; 3) the set of ll points (, y 0 ) with e nd y 0 e 2 forms the set int L p x,y(f), int L p x,y(f) = Q. Let the funtion f L(Q). Then the orrespond- Theorem 8.3 ([5]; [3, p. 8]): ing indefinite integrl F (x, y) = possesses the following properties: x y f(t, τ) dt dτ ) t every point (, y 0 ) int L x (f) the F [x] (, y 0 ) is finite nd y0 F [x] (, y 0 ) = f(, τ) dτ,
12 04 Bulletin of TICMI or wht is the sme, h 0 h k 0 x0 +h y0 +k f(t, τ) dt dτ = y0 f(, τ) dτ; 2) t every point (, y 0 ) int L y (f) the F [y] (, y 0 ) is finite nd or wht is the sme, y0 +k h 0 k y k 0 0 x0 F [y] (, y 0 ) = f(t, y 0 ) dt, x0 +h f(t, τ) dt dτ = x0 f(t, y 0 ) dt; 3) t every point (, y 0 ) int L x,y (f) the strgrd F (, y 0 ) is finite, in prtiulr, there exists df (, y 0 ). Theorem 8.4 ([5]; [3, p. 9]): following sttements re vlid: For every funtion f L p (Q) with p the ) t every point (, y 0 ) int L p x(f) the equlity (h,k) (0,0) h x0 +h y0 +k holds; 2) t every point (, y 0 ) int L p y(f) we hve y0 +k (h,k) (0,0) k y 0 y 0 x0 +h p f(x, y) dy dx = 0 (8.3) p f(x, y) dx dy = 0; (8.4) 3) t every point (, y 0 ) int L p x,y(f) equlities (8.3) nd (8.4) re fulfilled simultneously. 9. A generliztion of lssil theorems on derivtives on n indefinite integrl nd of the definite integrl with prmeter We men the following Lebesgue s nd Ch. J. de l Vlle e Poussin s theorems. Theorem L: Let ψ be summble funtion on losed intervl [, b], nd suppose tht Ψ(x) = Then for lmost ll x [, b] the equlity x ψ(t) dt. Ψ (x) = ψ(x) (9.)
13 is fulfilled. Vol. 8, No., Theorem VP: Let funtion f(x, y) be summble with respet to x on losed intervl [, b] for every fixed y from losed intervl [, d]. Consider finite on [, d] funtion definite integrl with prmeter y Φ(y) = b f(x, y) dx. Suppose tht the following onditions re fulfilled: (A) f(x, y) is funtion bsolutely ontinuous with respet to y on [, d] for every fixed x [, b]; (B) prtil derivtive f y with respet to y is summble funtion on losed retngle Q = {(x, y) R 2 : x b, y d}. Then for lmost ll y [, d]. Φ (y) = b f y(x, y) dx (9.2) We hve the following generliztion of Theorem L nd Theorem VP. Theorem 9. ([8]; [3, pp ]): Let ssumptions (A) nd (B) of Theorem VP be fulfilled. Then the funtion n indefinite integrl with prmeter y possesses the following properties: F (x, y) = x f(t, y) dt (9.3) (i) there exists e [, b] suh tht e = b, F [x] (, y 0 ) is finite for (, y 0 ) e [, d], nd F [x] (, y 0 ) = f(, y 0 ); (9.4) (ii) there exists e 2 [, d] suh tht e 2 = d, F [y] (, y 0 ) is finite for (, y 0 ) [, b] e 2, nd x0 F [y] (, y 0 ) = f y(t, y 0 ) dt; (9.5) (iii) the strgrd F (, y 0 ) is finite t lmost ll points (, y 0 ) Q, in prtiulr, there exists the totl differentil df (, y 0 ). If in equlity (9.3) we put x = b, then equlity (9.5) for = b tkes the form of equlity (9.2) beuse the derivtive of the funtion of one vrible is, in ft, its strong prtil derivtive with respet to the sme vrible, if we onsider this funtion s the funtion of two vribles, onstnt with respet to the seond vrible. Thus equlity (9.5) is the generliztion of equlity (9.2).
14 06 Bulletin of TICMI Equlity (9.) is obtined nlogously from equlity (9.4) if the funtion f in equlity (9.3) is ssumed to be independent of the vrible y. 0. A riterion of C n -differentibility Theorem 0. ([9]): nd only if the ondition A funtion f is C n -differentible t point z C n, if f x k (z) + if ŷ k (z) = 0 or, equivlently D xk f(z) + idŷk f(z) = 0 holds for ll k =,..., n, where z = (z,..., z n ) nd z k = x k + y k. Hrtog s Min Theorem([9, p. 7]). A funtion f holomorphi (nlyti) with respet to eh vrible in n open set G C n is C n -holomorphi (C n -nlyti) in G.. On the H-differentibility Definition. ([20]) : A quternion funtion f(z), z = + x i + x 2 i 2 + x 3 i 3, defined on some neighborhood of point z 0 = 0 + x0 i + 2 i i 3, is lled H- differentible t z 0 if there exists two sequenes of quternions A k (z 0 ) nd B k (z 0 ) suh tht A k (z 0 )B k (z 0 ) is finite nd tht the inrement f(z 0 + h) f(z 0 ) of the k funtion f n be represented s f(z 0 + h) f(z 0 ) = k A k (z 0 )B k (z 0 ) + ω(z 0, h), where ω(z 0, h) = 0. h 0 h In this se, the quternion k A k (z 0 )B k (z 0 ) is lled the H-derivtive of the funtion f t the point z 0 nd is denoted f (z 0 ). Thus f (z 0 ) = k A k (z 0 )B k (z 0 ). (.) The uniqueness of the H-derivtive follows from the ft tht the right-hnd prt of (.), if it exists, is just the prtil derivtive f (z 0 ) of f t z 0 with respet to its rel vrible (see [6], equlity (2)). *For the history of differentibility of quternion funtions see [20] nd [2], p. 385.
15 Vol. 8, No., The bsi elementry qunternion funtions z n, e z, os z, sin z re H- differentible nd fulfilled the following equlities (z n ) = nz n, (e z ) = e z, (os z) = sin z, (sin z) = os z. The rules for lulting H-derivtives re identil to those derived in stndrd lulus ourse: (f) (z) = f (z), (f) (z) = f (z), (f + φ) (z) = f (z)+φ (z), (fφ) (z) = f (z)φ(z)+f(z)φ (z), (/φ) (z) = /φ(z) φ (z) /φ(z), (f /φ) (z) == f (z) /φ(z) f(z) /φ(z) φ (z) /φ(z), (/φ f) (z) = /φ(z) φ (z) /φ(z) f(z) + /φ(z) f (z). Right now we formulte reltionship between H-differentibility of quternion funtion f(z) = u 0 (z) + u (z)i + u 2 (z)i 2 + u 3 (z)i 3 of quternion vrible z = + x i + x 2 i 2 + x 3 i 3 nd the existene of the differentil df(z) (with respet to rel vribles, x, x 2, x 3 ). Sine the prtil ngulr derivtives re the derivtives with respet to rel vribles (see the Setion 2), the ondition of differentibility for rel, omplex nd quternion funtions re expressed in the sme form. It then follows tht for the differentibility of quternion funtion f t point z = +x i +x 2 i 2 +x 3 i 3, neessry nd suffiient ondition is the existene finite prtil ngulr derivtives f x k = (u 0 ) x k +i (u 2 ) x k +i 2 (u 2 ) x k +i 3 (u 3 ) x k, k = 0,, 2, 3. Moreover, when f is differentible t z, the following equlities hold for its differentil df(z): df(z) = f (z)d + f x (z)dx + f x 2 (z)dx 2 + f x 3 (z)dx 3, df(z) = du 0 (z) + i du (z) + i 2 du 2 (z) + i 3 du 3 (z). Theorem.2 ([22]): If quternion funtion f is H-differentible t point z = + x i + x 2 i 2 + x 3 i 3, then f is differentible t the sme point z nd its prtil ngulr derivtives f (z), f x (z), f x 2 (z), f x 3 (z) n be expressed in terms of the H-derivtive f (z) = A k (z)b k (z) s follows: k f (z) = k A k (z)b k (z) = f (z), (.2) f x (z) = k A k (z)i B k (z), (.3) f x 2 (z) = k A k (z)i 2 B k (z), (.4) f x 3 (z) = k A k (z)i 3 B k (z). (.5) Moreover, we hve df(z) = k A k (z) dz B k (z). (.6) Remrk : Eqution (.6) n be interpreted s follows. As in the lssil se, the differentil df(z) of n H-differentible funtion f is liner with respet to the differentil dz of the independent vrible z.
16 08 Bulletin of TICMI Theorem.3 ([22]): If quternion funtion f is differentible t point z nd its prtil ngulr derivtives f (z), f x (z), f x 2 (z) nd f x 3 (z) n be expressed in the forms (.2) (.5) for some quternions A k (z) nd B k (z), then f is H- differentible t the point z nd f (z) = k A k (z)b k (z). We n ombine Theorems.2 nd.3 to obtin the following theorem Theorem.4 ([22]): The existene of the differentil df(z) of quternion funtion f nd its representbility in the form df(z) = k A k (z) dz B k (z) (.7) is equivlent to the existene of the derivtive f (z) nd its representbility in the form f (z) = k A k (z)b k (z). Corollry.5 ([22]): When x 2 = 0 = x 3 nd u 2 = 0 = u 3, then one hs omplex funtion f(z) = u(z) + iv(z) of omplex vrible z = x + iy. In this se, Eq. (.7) hs the form df(z) = (z)dz = (z)dx + i(z)dy, where (z) = k A k(z)b k (z), from whih we obtin the equlities f x (z) = (z) nd f ŷ (z) = i(z). Thus, we hve f x (z) + if ŷ (z) = 0. (.8) Note tht Eq.(.8) is neessry nd suffiient ondition for the omplex funtion f to be C -differentible t the point z (see [9], Theorem 3., when n = ). Moreover, we hve obtined the well known equlities f (z) = f x (z) nd f (z) = if ŷ (z) for the derivtive f (z). Corollry.6 ([22]): For quternion z = + x i + x 2 i 2 + x 3 i 3, we hve dz n = z n dz + z n 2 dz z + z n 3 dz z zdz z n 2 + dz z n for ll n = 0,, 2,.... Corollry.7 ([22]): For the prtil derivtives of the funtions f n (z) = z n, n = 0,, 2,..., with respet to rel vribles x k, k = 0,, 2, 3, we hve (z n ) x k = z n i k + z n 2 i k z + + z i k z n 2 + i k z n. Referenes [] G.P. Tolstov, On prtil derivtives (Russin), Izvestiy Akd. Nuk SSSR. Ser. Mt., 3 (949), ; trnsltion in Amer. Mth. So. (952), [2] O.P. Dzgnidze, Seprtely ontinuous funtions in new sense re ontinuous, Rel Anl. Exhnge, 24, 2 (998/99), [3] O. Dzgnidze, Some new results on the ontinuity nd differentibility of funtions of severl rel vribles, Pro. A. Rzmdze Mth. Inst., 34 (2004), -38 [4] Z. Piotrowski, The genesis of seprte versus joint ontinuity, Ttr Mt. Mth. Publ., 8 (996), 3-26
17 Vol. 8, No., [5] O.P. Dzgnidze, On the differentibility of funtions of two vribles nd of indefinite double integrls, Pro. A. Rzmdze Mth. Inst., 06 (993), 7-48 [6] O. Dzgnidze, A neessry nd suffiient ondition for differentibility funtions of severl vribles, Pro. A. Rzmdze Mth. Inst., 23 (2000), [7] Yu.S. Očn, Colletion of Problems nd Theorems on the Theory of Funtions of Rel Vrible, Prosveshheniye, Mosow, 965 [8] Yu.S. Očn, Colletion of Problems on Mthemtil Anlysis (generl theory of sets nd funtions), Prosveshheniye, Mosow, 98 [9] L.D. Bntsuri, G.G. Onini, On the differentil properties of funtions of bounded vrition in Hrdy sense, Pro. A. Rzmdze Mth. Inst., 39 (2005), [0] L.D. Bntsuri, G.G. Onini, On differentil properties of funtions of bounded vrition, Anl. Mth., 38, (202), -7 [] G. Onini, On the reltion between onditions of the differentibility nd existene of the strong grdient, Pro. A. Rzmdze Mth. Inst., 32 (2003), 5-52 [2] G.G.Onini, On the inter-reltion between differentibility onditions nd the existene of strong grdient (Russin), Mt. Zmetki, 77, (2005), 93-98; trnsltion in Mth. Notes, 77, -2 (2005), [3] L. Bntsuri, On the reltion between the differentibility ondition nd the ondition of the existene of generlized grdient, Bull. Georgin Ad. Si., 7, 2 (2005), [4] L. Bntsuri, On the reltionship between the onditions of differentibility nd existene of generlized grdient, Book of Abstrts of the III Interntionl Conferene of the Georgin Methemtil Union (Btumi, 202, September 2-9), [5] O. Dzgnidze, Totl differentil of the indefinite Lebesgue integrl, Pro. A. Rzmdze Mth. Inst., 4 (997), [6] O. Dzgnidze, G. Onini, On one nlogue of Lebesgue theorem on the differentition of indefinite integrl for funtions of severl vribles, Pro. A. Rzmdze Mth. Inst., 32 (2003), [7] O. Dzgnidze, G. Onini, On one nlogue of Lebesgue theorem on the differentition of indefinite integrl for funtions of severl vribles, Pro. A. Rzmdze Mth. Inst., 33 (2003), -5 [8] O. Dzgnidze, The Lebesgue nd de l Vllée Poussin s theorems on derivtion of n integrl, Ttr Mt. Mth. Publ., 35 (2007), 07-3 [9] O. Dzgnidze, A riterion of joint C-differentibility nd new proof of Hrtogs min theorem, J. Appl. Anl., 3, (2007), 3-7 [20] O. Dzgnidze, On the differentibility of quternion funtions, Tbil. Mth. J., 5 (202), -5. ArXiv: V[mt.CV] 26 Mr, 202 [2] G.E. Shilov, Mthemtil Anlysis: Funtions of Severl Rel Vribles, prts 2. Nuk, Hed Editoril Offie for Physil nd Mthemtil Literture, Mosow, 972. [22] O. Dzgnidze, Neessry nd suffiient onditions for H-differentibility of quternion funtions, Georgin Mth. J. (submitted).
T b a(f) [f ] +. P b a(f) = Conclude that if f is in AC then it is the difference of two monotone absolutely continuous functions.
Rel Vribles, Fll 2014 Problem set 5 Solution suggestions Exerise 1. Let f be bsolutely ontinuous on [, b] Show tht nd T b (f) P b (f) f (x) dx [f ] +. Conlude tht if f is in AC then it is the differene
More informationLecture 1 - Introduction and Basic Facts about PDEs
* 18.15 - Introdution to PDEs, Fll 004 Prof. Gigliol Stffilni Leture 1 - Introdution nd Bsi Fts bout PDEs The Content of the Course Definition of Prtil Differentil Eqution (PDE) Liner PDEs VVVVVVVVVVVVVVVVVVVV
More informationPart 4. Integration (with Proofs)
Prt 4. Integrtion (with Proofs) 4.1 Definition Definition A prtition P of [, b] is finite set of points {x 0, x 1,..., x n } with = x 0 < x 1
More informationLine Integrals and Entire Functions
Line Integrls nd Entire Funtions Defining n Integrl for omplex Vlued Funtions In the following setions, our min gol is to show tht every entire funtion n be represented s n everywhere onvergent power series
More informationarxiv: v1 [math.ca] 21 Aug 2018
rxiv:1808.07159v1 [mth.ca] 1 Aug 018 Clulus on Dul Rel Numbers Keqin Liu Deprtment of Mthemtis The University of British Columbi Vnouver, BC Cnd, V6T 1Z Augest, 018 Abstrt We present the bsi theory of
More informationAP Calculus AB Unit 4 Assessment
Clss: Dte: 0-04 AP Clulus AB Unit 4 Assessment Multiple Choie Identify the hoie tht best ompletes the sttement or nswers the question. A lultor my NOT be used on this prt of the exm. (6 minutes). The slope
More informationf (x)dx = f(b) f(a). a b f (x)dx is the limit of sums
Green s Theorem If f is funtion of one vrible x with derivtive f x) or df dx to the Fundmentl Theorem of lulus, nd [, b] is given intervl then, ording This is not trivil result, onsidering tht b b f x)dx
More informationCo-ordinated s-convex Function in the First Sense with Some Hadamard-Type Inequalities
Int. J. Contemp. Mth. Sienes, Vol. 3, 008, no. 3, 557-567 Co-ordinted s-convex Funtion in the First Sense with Some Hdmrd-Type Inequlities Mohmmd Alomri nd Mslin Drus Shool o Mthemtil Sienes Fulty o Siene
More informationThe Riemann-Stieltjes Integral
Chpter 6 The Riemnn-Stieltjes Integrl 6.1. Definition nd Eistene of the Integrl Definition 6.1. Let, b R nd < b. ( A prtition P of intervl [, b] is finite set of points P = { 0, 1,..., n } suh tht = 0
More informationMATH34032: Green s Functions, Integral Equations and the Calculus of Variations 1. 1 [(y ) 2 + yy + y 2 ] dx,
MATH3403: Green s Funtions, Integrl Equtions nd the Clulus of Vritions 1 Exmples 5 Qu.1 Show tht the extreml funtion of the funtionl I[y] = 1 0 [(y ) + yy + y ] dx, where y(0) = 0 nd y(1) = 1, is y(x)
More informationINTEGRATION. 1 Integrals of Complex Valued functions of a REAL variable
INTEGRATION NOTE: These notes re supposed to supplement Chpter 4 of the online textbook. 1 Integrls of Complex Vlued funtions of REAL vrible If I is n intervl in R (for exmple I = [, b] or I = (, b)) nd
More information(a) A partition P of [a, b] is a finite subset of [a, b] containing a and b. If Q is another partition and P Q, then Q is a refinement of P.
Chpter 7: The Riemnn Integrl When the derivtive is introdued, it is not hrd to see tht the it of the differene quotient should be equl to the slope of the tngent line, or when the horizontl xis is time
More informationMore Properties of the Riemann Integral
More Properties of the Riemnn Integrl Jmes K. Peterson Deprtment of Biologil Sienes nd Deprtment of Mthemtil Sienes Clemson University Februry 15, 2018 Outline More Riemnn Integrl Properties The Fundmentl
More informationSolutions to Assignment 1
MTHE 237 Fll 2015 Solutions to Assignment 1 Problem 1 Find the order of the differentil eqution: t d3 y dt 3 +t2 y = os(t. Is the differentil eqution liner? Is the eqution homogeneous? b Repet the bove
More informationLecture Summaries for Multivariable Integral Calculus M52B
These leture summries my lso be viewed online by liking the L ion t the top right of ny leture sreen. Leture Summries for Multivrible Integrl Clulus M52B Chpter nd setion numbers refer to the 6th edition.
More informationON THE C-INTEGRAL BENEDETTO BONGIORNO
ON THE C-INTEGRAL BENEDETTO BONGIORNO Let F : [, b] R be differentible function nd let f be its derivtive. The problem of recovering F from f is clled problem of primitives. In 1912, the problem of primitives
More informationMA10207B: ANALYSIS SECOND SEMESTER OUTLINE NOTES
MA10207B: ANALYSIS SECOND SEMESTER OUTLINE NOTES CHARLIE COLLIER UNIVERSITY OF BATH These notes hve been typeset by Chrlie Collier nd re bsed on the leture notes by Adrin Hill nd Thoms Cottrell. These
More informationON THE INEQUALITY OF THE DIFFERENCE OF TWO INTEGRAL MEANS AND APPLICATIONS FOR PDFs
ON THE INEQUALITY OF THE DIFFERENCE OF TWO INTEGRAL MEANS AND APPLICATIONS FOR PDFs A.I. KECHRINIOTIS AND N.D. ASSIMAKIS Deprtment of Eletronis Tehnologil Edutionl Institute of Lmi, Greee EMil: {kehrin,
More informationSection 3.6. Definite Integrals
The Clulus of Funtions of Severl Vribles Setion.6 efinite Integrls We will first define the definite integrl for funtion f : R R nd lter indite how the definition my be extended to funtions of three or
More informationElectromagnetism Notes, NYU Spring 2018
Eletromgnetism Notes, NYU Spring 208 April 2, 208 Ation formultion of EM. Free field desription Let us first onsider the free EM field, i.e. in the bsene of ny hrges or urrents. To tret this s mehnil system
More informationMATH 409 Advanced Calculus I Lecture 22: Improper Riemann integrals.
MATH 409 Advned Clulus I Leture 22: Improper Riemnn integrls. Improper Riemnn integrl If funtion f : [,b] R is integrble on [,b], then the funtion F(x) = x f(t)dt is well defined nd ontinuous on [,b].
More informationNew Expansion and Infinite Series
Interntionl Mthemticl Forum, Vol. 9, 204, no. 22, 06-073 HIKARI Ltd, www.m-hikri.com http://dx.doi.org/0.2988/imf.204.4502 New Expnsion nd Infinite Series Diyun Zhng College of Computer Nnjing University
More informationHyers-Ulam stability of Pielou logistic difference equation
vilble online t wwwisr-publitionsom/jns J Nonliner Si ppl, 0 (207, 35 322 Reserh rtile Journl Homepge: wwwtjnsom - wwwisr-publitionsom/jns Hyers-Ulm stbility of Pielou logisti differene eqution Soon-Mo
More informationGreen s Theorem. (2x e y ) da. (2x e y ) dx dy. x 2 xe y. (1 e y ) dy. y=1. = y e y. y=0. = 2 e
Green s Theorem. Let be the boundry of the unit squre, y, oriented ounterlokwise, nd let F be the vetor field F, y e y +, 2 y. Find F d r. Solution. Let s write P, y e y + nd Q, y 2 y, so tht F P, Q. Let
More informationSection 4.4. Green s Theorem
The Clulus of Funtions of Severl Vriles Setion 4.4 Green s Theorem Green s theorem is n exmple from fmily of theorems whih onnet line integrls (nd their higher-dimensionl nlogues) with the definite integrls
More informationNote 16. Stokes theorem Differential Geometry, 2005
Note 16. Stokes theorem ifferentil Geometry, 2005 Stokes theorem is the centrl result in the theory of integrtion on mnifolds. It gives the reltion between exterior differentition (see Note 14) nd integrtion
More informationThe Regulated and Riemann Integrals
Chpter 1 The Regulted nd Riemnn Integrls 1.1 Introduction We will consider severl different pproches to defining the definite integrl f(x) dx of function f(x). These definitions will ll ssign the sme vlue
More informationINTEGRATION. 1 Integrals of Complex Valued functions of a REAL variable
INTEGRATION 1 Integrls of Complex Vlued funtions of REAL vrible If I is n intervl in R (for exmple I = [, b] or I = (, b)) nd h : I C writing h = u + iv where u, v : I C, we n extend ll lulus 1 onepts
More information1.9 C 2 inner variations
46 CHAPTER 1. INDIRECT METHODS 1.9 C 2 inner vritions So fr, we hve restricted ttention to liner vritions. These re vritions of the form vx; ǫ = ux + ǫφx where φ is in some liner perturbtion clss P, for
More informationTutorial Worksheet. 1. Find all solutions to the linear system by following the given steps. x + 2y + 3z = 2 2x + 3y + z = 4.
Mth 5 Tutoril Week 1 - Jnury 1 1 Nme Setion Tutoril Worksheet 1. Find ll solutions to the liner system by following the given steps x + y + z = x + y + z = 4. y + z = Step 1. Write down the rgumented mtrix
More informationarxiv: v1 [math.ca] 11 Jul 2011
rxiv:1107.1996v1 [mth.ca] 11 Jul 2011 Existence nd computtion of Riemnn Stieltjes integrls through Riemnn integrls July, 2011 Rodrigo López Pouso Deprtmento de Análise Mtemátic Fcultde de Mtemátics, Universidde
More informationMath 113 Fall Final Exam Review. 2. Applications of Integration Chapter 6 including sections and section 6.8
Mth 3 Fll 0 The scope of the finl exm will include: Finl Exm Review. Integrls Chpter 5 including sections 5. 5.7, 5.0. Applictions of Integrtion Chpter 6 including sections 6. 6.5 nd section 6.8 3. Infinite
More informationMAT 403 NOTES 4. f + f =
MAT 403 NOTES 4 1. Fundmentl Theorem o Clulus We will proo more generl version o the FTC thn the textook. But just like the textook, we strt with the ollowing proposition. Let R[, ] e the set o Riemnn
More information6.1 Definition of the Riemann Integral
6 The Riemnn Integrl 6. Deinition o the Riemnn Integrl Deinition 6.. Given n intervl [, b] with < b, prtition P o [, b] is inite set o points {x, x,..., x n } [, b], lled grid points, suh tht x =, x n
More information7.2 Riemann Integrable Functions
7.2 Riemnn Integrble Functions Theorem 1. If f : [, b] R is step function, then f R[, b]. Theorem 2. If f : [, b] R is continuous on [, b], then f R[, b]. Theorem 3. If f : [, b] R is bounded nd continuous
More informationDIFFERENCE BETWEEN TWO RIEMANN-STIELTJES INTEGRAL MEANS
Krgujev Journl of Mthemtis Volume 38() (204), Pges 35 49. DIFFERENCE BETWEEN TWO RIEMANN-STIELTJES INTEGRAL MEANS MOHAMMAD W. ALOMARI Abstrt. In this pper, severl bouns for the ifferene between two Riemn-
More informationThe study of dual integral equations with generalized Legendre functions
J. Mth. Anl. Appl. 34 (5) 75 733 www.elsevier.om/lote/jm The study of dul integrl equtions with generlized Legendre funtions B.M. Singh, J. Rokne,R.S.Dhliwl Deprtment of Mthemtis, The University of Clgry,
More informationMath 32B Discussion Session Week 8 Notes February 28 and March 2, f(b) f(a) = f (t)dt (1)
Green s Theorem Mth 3B isussion Session Week 8 Notes Februry 8 nd Mrh, 7 Very shortly fter you lerned how to integrte single-vrible funtions, you lerned the Fundmentl Theorem of lulus the wy most integrtion
More informationPhysics 116C Solution of inhomogeneous ordinary differential equations using Green s functions
Physics 6C Solution of inhomogeneous ordinry differentil equtions using Green s functions Peter Young November 5, 29 Homogeneous Equtions We hve studied, especilly in long HW problem, second order liner
More informationSOLUTIONS TO MATH38181 EXTREME VALUES EXAM
SOLUTIONS TO MATH388 EXTREME VALUES EXAM Solutions to Question If there re norming onstnts n >, b n nd nondegenerte G suh tht the df of normlized version of M n onverges to G, i.e. ( ) Mn b n Pr x F n
More informationCHAPTER V INTEGRATION, AVERAGE BEHAVIOR A = πr 2.
CHAPTER V INTEGRATION, AVERAGE BEHAVIOR A πr 2. In this hpter we will derive the formul A πr 2 for the re of irle of rdius r. As mtter of ft, we will first hve to settle on extly wht is the definition
More informationIntegrals Depending on a Parameter
Universidd Crlos III de Mdrid Clulus II Mrin Delgdo Téllez de Ceped Unit 3 Integrls Depending on Prmeter Definition 3.1. Let f : [,b] [,d] R, if for eh fixed t [,d] the funtion f(x,t) is integrble over
More informationMATH Final Review
MATH 1591 - Finl Review November 20, 2005 1 Evlution of Limits 1. the ε δ definition of limit. 2. properties of limits. 3. how to use the diret substitution to find limit. 4. how to use the dividing out
More informationUNIFORM CONVERGENCE MA 403: REAL ANALYSIS, INSTRUCTOR: B. V. LIMAYE
UNIFORM CONVERGENCE MA 403: REAL ANALYSIS, INSTRUCTOR: B. V. LIMAYE 1. Pointwise Convergence of Sequence Let E be set nd Y be metric spce. Consider functions f n : E Y for n = 1, 2,.... We sy tht the sequence
More informationLine Integrals. Partitioning the Curve. Estimating the Mass
Line Integrls Suppose we hve curve in the xy plne nd ssocite density δ(p ) = δ(x, y) t ech point on the curve. urves, of course, do not hve density or mss, but it my sometimes be convenient or useful to
More informationAdvanced Calculus: MATH 410 Uniform Convergence of Functions Professor David Levermore 11 December 2015
Advnced Clculus: MATH 410 Uniform Convergence of Functions Professor Dvid Levermore 11 December 2015 12. Sequences of Functions We now explore two notions of wht it mens for sequence of functions {f n
More informationMATH , Calculus 2, Fall 2018
MATH 36-2, 36-3 Clculus 2, Fll 28 The FUNdmentl Theorem of Clculus Sections 5.4 nd 5.5 This worksheet focuses on the most importnt theorem in clculus. In fct, the Fundmentl Theorem of Clculus (FTC is rgubly
More informationSECTION A STUDENT MATERIAL. Part 1. What and Why.?
SECTION A STUDENT MATERIAL Prt Wht nd Wh.? Student Mteril Prt Prolem n > 0 n > 0 Is the onverse true? Prolem If n is even then n is even. If n is even then n is even. Wht nd Wh? Eploring Pure Mths Are
More informationLearning Objectives of Module 2 (Algebra and Calculus) Notes:
67 Lerning Ojetives of Module (Alger nd Clulus) Notes:. Lerning units re grouped under three res ( Foundtion Knowledge, Alger nd Clulus ) nd Further Lerning Unit.. Relted lerning ojetives re grouped under
More informationHomework 4. (1) If f R[a, b], show that f 3 R[a, b]. If f + (x) = max{f(x), 0}, is f + R[a, b]? Justify your answer.
Homework 4 (1) If f R[, b], show tht f 3 R[, b]. If f + (x) = mx{f(x), 0}, is f + R[, b]? Justify your nswer. (2) Let f be continuous function on [, b] tht is strictly positive except finitely mny points
More informationAP Calculus BC Chapter 8: Integration Techniques, L Hopital s Rule and Improper Integrals
AP Clulus BC Chpter 8: Integrtion Tehniques, L Hopitl s Rule nd Improper Integrls 8. Bsi Integrtion Rules In this setion we will review vrious integrtion strtegies. Strtegies: I. Seprte the integrnd into
More information63. Representation of functions as power series Consider a power series. ( 1) n x 2n for all 1 < x < 1
3 9. SEQUENCES AND SERIES 63. Representtion of functions s power series Consider power series x 2 + x 4 x 6 + x 8 + = ( ) n x 2n It is geometric series with q = x 2 nd therefore it converges for ll q =
More informationW. We shall do so one by one, starting with I 1, and we shall do it greedily, trying
Vitli covers 1 Definition. A Vitli cover of set E R is set V of closed intervls with positive length so tht, for every δ > 0 nd every x E, there is some I V with λ(i ) < δ nd x I. 2 Lemm (Vitli covering)
More informationIntegrals along Curves.
Integrls long Curves. 1. Pth integrls. Let : [, b] R n be continuous function nd let be the imge ([, b]) of. We refer to both nd s curve. If we need to distinguish between the two we cll the function the
More informationEntrance Exam, Real Analysis September 1, 2009 Solve exactly 6 out of the 8 problems. Compute the following and justify your computation: lim
1. Let n be positive integers. ntrnce xm, Rel Anlysis September 1, 29 Solve exctly 6 out of the 8 problems. Sketch the grph of the function f(x): f(x) = lim e x2n. Compute the following nd justify your
More informationCzechoslovak Mathematical Journal, 55 (130) (2005), , Abbotsford. 1. Introduction
Czechoslovk Mthemticl Journl, 55 (130) (2005), 933 940 ESTIMATES OF THE REMAINDER IN TAYLOR S THEOREM USING THE HENSTOCK-KURZWEIL INTEGRAL, Abbotsford (Received Jnury 22, 2003) Abstrct. When rel-vlued
More informationLECTURE. INTEGRATION AND ANTIDERIVATIVE.
ANALYSIS FOR HIGH SCHOOL TEACHERS LECTURE. INTEGRATION AND ANTIDERIVATIVE. ROTHSCHILD CAESARIA COURSE, 2015/6 1. Integrtion Historiclly, it ws the problem of computing res nd volumes, tht triggered development
More informationLecture 1. Functional series. Pointwise and uniform convergence.
1 Introduction. Lecture 1. Functionl series. Pointwise nd uniform convergence. In this course we study mongst other things Fourier series. The Fourier series for periodic function f(x) with period 2π is
More informationExam 2, Mathematics 4701, Section ETY6 6:05 pm 7:40 pm, March 31, 2016, IH-1105 Instructor: Attila Máté 1
Exm, Mthemtics 471, Section ETY6 6:5 pm 7:4 pm, Mrch 1, 16, IH-115 Instructor: Attil Máté 1 17 copies 1. ) Stte the usul sufficient condition for the fixed-point itertion to converge when solving the eqution
More informationMath Advanced Calculus II
Mth 452 - Advnced Clculus II Line Integrls nd Green s Theorem The min gol of this chpter is to prove Stoke s theorem, which is the multivrible version of the fundmentl theorem of clculus. We will be focused
More information1 PYTHAGORAS THEOREM 1. Given a right angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.
1 PYTHAGORAS THEOREM 1 1 Pythgors Theorem In this setion we will present geometri proof of the fmous theorem of Pythgors. Given right ngled tringle, the squre of the hypotenuse is equl to the sum of the
More informationI. INTEGRAL THEOREMS. A. Introduction
1 U Deprtment of Physics 301A Mechnics - I. INTEGRAL THEOREM A. Introduction The integrl theorems of mthemticl physics ll hve their origin in the ordinry fundmentl theorem of clculus, i.e. xb x df dx dx
More informationLine Integrals. Chapter Definition
hpter 2 Line Integrls 2.1 Definition When we re integrting function of one vrible, we integrte long n intervl on one of the xes. We now generlize this ide by integrting long ny curve in the xy-plne. It
More informationIntroduction to the Calculus of Variations
Introduction to the Clculus of Vritions Jim Fischer Mrch 20, 1999 Abstrct This is self-contined pper which introduces fundmentl problem in the clculus of vritions, the problem of finding extreme vlues
More informationSections 5.3: Antiderivatives and the Fundamental Theorem of Calculus Theory:
Setions 5.3: Antierivtives n the Funmentl Theorem of Clulus Theory: Definition. Assume tht y = f(x) is ontinuous funtion on n intervl I. We ll funtion F (x), x I, to be n ntierivtive of f(x) if F (x) =
More informationPartial Derivatives. Limits. For a single variable function f (x), the limit lim
Limits Prtil Derivtives For single vrible function f (x), the limit lim x f (x) exists only if the right-hnd side limit equls to the left-hnd side limit, i.e., lim f (x) = lim f (x). x x + For two vribles
More informationf (z) dz = 0 f(z) dz = 2πj f(z 0 ) Generalized Cauchy Integral Formula (For pole with any order) (n 1)! f (n 1) (z 0 ) f (n) (z 0 ) M n!
uhy s Theorems I Ang M.S. Otober 26, 212 Augustin-Louis uhy 1789 1857 Referenes Murry R. Spiegel omplex V ribles with introdution to onf orml mpping nd its pplitions Dennis G. Zill, P. D. Shnhn A F irst
More informationUniversity of Sioux Falls. MAT204/205 Calculus I/II
University of Sioux Flls MAT204/205 Clulus I/II Conepts ddressed: Clulus Textook: Thoms Clulus, 11 th ed., Weir, Hss, Giordno 1. Use stndrd differentition nd integrtion tehniques. Differentition tehniques
More informationChapter 22. The Fundamental Theorem of Calculus
Version of 24.2.4 Chpter 22 The Fundmentl Theorem of Clculus In this chpter I ddress one of the most importnt properties of the Lebesgue integrl. Given n integrble function f : [,b] R, we cn form its indefinite
More informationNotes on length and conformal metrics
Notes on length nd conforml metrics We recll how to mesure the Eucliden distnce of n rc in the plne. Let α : [, b] R 2 be smooth (C ) rc. Tht is α(t) (x(t), y(t)) where x(t) nd y(t) re smooth rel vlued
More informationHermite-Hadamard inequality for geometrically quasiconvex functions on co-ordinates
Int. J. Nonliner Anl. Appl. 8 27 No. 47-6 ISSN: 28-6822 eletroni http://dx.doi.org/.2275/ijn.26.483 Hermite-Hdmrd ineulity for geometrilly usionvex funtions on o-ordintes Ali Brni Ftemeh Mlmir Deprtment
More informationReview of Riemann Integral
1 Review of Riemnn Integrl In this chpter we review the definition of Riemnn integrl of bounded function f : [, b] R, nd point out its limittions so s to be convinced of the necessity of more generl integrl.
More informationQUADRATIC EQUATION. Contents
QUADRATIC EQUATION Contents Topi Pge No. Theory 0-04 Exerise - 05-09 Exerise - 09-3 Exerise - 3 4-5 Exerise - 4 6 Answer Key 7-8 Syllus Qudrti equtions with rel oeffiients, reltions etween roots nd oeffiients,
More informationTHE EXISTENCE-UNIQUENESS THEOREM FOR FIRST-ORDER DIFFERENTIAL EQUATIONS.
THE EXISTENCE-UNIQUENESS THEOREM FOR FIRST-ORDER DIFFERENTIAL EQUATIONS RADON ROSBOROUGH https://intuitiveexplntionscom/picrd-lindelof-theorem/ This document is proof of the existence-uniqueness theorem
More informationMATH 409 Advanced Calculus I Lecture 19: Riemann sums. Properties of integrals.
MATH 409 Advnced Clculus I Lecture 19: Riemnn sums. Properties of integrls. Drboux sums Let P = {x 0,x 1,...,x n } be prtition of n intervl [,b], where x 0 = < x 1 < < x n = b. Let f : [,b] R be bounded
More informationElectromagnetic-Power-based Modal Classification, Modal Expansion, and Modal Decomposition for Perfect Electric Conductors
LIAN: EM-BASED MODAL CLASSIFICATION EXANSION AND DECOMOSITION FOR EC 1 Eletromgneti-ower-bsed Modl Clssifition Modl Expnsion nd Modl Deomposition for erfet Eletri Condutors Renzun Lin Abstrt Trditionlly
More information(4.1) D r v(t) ω(t, v(t))
1.4. Differentil inequlities. Let D r denote the right hnd derivtive of function. If ω(t, u) is sclr function of the sclrs t, u in some open connected set Ω, we sy tht function v(t), t < b, is solution
More informationReview of Calculus, cont d
Jim Lmbers MAT 460 Fll Semester 2009-10 Lecture 3 Notes These notes correspond to Section 1.1 in the text. Review of Clculus, cont d Riemnn Sums nd the Definite Integrl There re mny cses in which some
More informationAn Overview of the Theory of Distributions
An Overview of the Theory of Distributions Mtt Guthrie Adpted from Hlperin [1] 1 Chpter 1 Introduction For the rnge of time between the introduction of the opertionl clculus t the end of the 19th century
More informationEuler-Maclaurin Summation Formula 1
Jnury 9, Euler-Mclurin Summtion Formul Suppose tht f nd its derivtive re continuous functions on the closed intervl [, b]. Let ψ(x) {x}, where {x} x [x] is the frctionl prt of x. Lemm : If < b nd, b Z,
More information1.3 The Lemma of DuBois-Reymond
28 CHAPTER 1. INDIRECT METHODS 1.3 The Lemm of DuBois-Reymond We needed extr regulrity to integrte by prts nd obtin the Euler- Lgrnge eqution. The following result shows tht, t lest sometimes, the extr
More information1 The Riemann Integral
The Riemnn Integrl. An exmple leding to the notion of integrl (res) We know how to find (i.e. define) the re of rectngle (bse height), tringle ( (sum of res of tringles). But how do we find/define n re
More informationMath 115 ( ) Yum-Tong Siu 1. Lagrange Multipliers and Variational Problems with Constraints. F (x,y,y )dx
Mth 5 2006-2007) Yum-Tong Siu Lgrnge Multipliers nd Vritionl Problems with Constrints Integrl Constrints. Consider the vritionl problem of finding the extremls for the functionl J[y] = F x,y,y )dx with
More information( ) { } [ ] { } [ ) { } ( ] { }
Mth 65 Prelulus Review Properties of Inequlities 1. > nd > >. > + > +. > nd > 0 > 4. > nd < 0 < Asolute Vlue, if 0, if < 0 Properties of Asolute Vlue > 0 1. < < > or
More information1. On some properties of definite integrals. We prove
This short collection of notes is intended to complement the textbook Anlisi Mtemtic 2 by Crl Mdern, published by Città Studi Editore, [M]. We refer to [M] for nottion nd the logicl stremline of the rguments.
More informationCalculus Cheat Sheet. Integrals Definitions. where F( x ) is an anti-derivative of f ( x ). Fundamental Theorem of Calculus. dx = f x dx g x dx
Clulus Chet Sheet Integrls Definitions Definite Integrl: Suppose f ( ) is ontinuous Anti-Derivtive : An nti-derivtive of f ( ) on [, ]. Divide [, ] into n suintervls of is funtion, F( ), suh tht F = f.
More informationarxiv: v1 [math.ca] 7 Mar 2012
rxiv:1203.1462v1 [mth.ca] 7 Mr 2012 A simple proof of the Fundmentl Theorem of Clculus for the Lebesgue integrl Mrch, 2012 Rodrigo López Pouso Deprtmento de Análise Mtemátic Fcultde de Mtemátics, Universidde
More informationu(t)dt + i a f(t)dt f(t) dt b f(t) dt (2) With this preliminary step in place, we are ready to define integration on a general curve in C.
Lecture 4 Complex Integrtion MATH-GA 2451.001 Complex Vriles 1 Construction 1.1 Integrting complex function over curve in C A nturl wy to construct the integrl of complex function over curve in the complex
More informationMA 124 January 18, Derivatives are. Integrals are.
MA 124 Jnury 18, 2018 Prof PB s one-minute introduction to clculus Derivtives re. Integrls re. In Clculus 1, we lern limits, derivtives, some pplictions of derivtives, indefinite integrls, definite integrls,
More informationDong-Myung Lee, Jeong-Gon Lee, and Ming-Gen Cui. 1. introduction
J. Kore So. Mth. Edu. Ser. B: Pure Appl. Mth. ISSN 16-0657 Volume 11, Number My 004), Pges 133 138 REPRESENTATION OF SOLUTIONS OF FREDHOLM EQUATIONS IN W Ω) OF REPRODUCING KERNELS Dong-Myung Lee, Jeong-Gon
More informationAP CALCULUS Test #6: Unit #6 Basic Integration and Applications
AP CALCULUS Test #6: Unit #6 Bsi Integrtion nd Applitions A GRAPHING CALCULATOR IS REQUIRED FOR SOME PROBLEMS OR PARTS OF PROBLEMS IN THIS PART OF THE EXAMINATION. () The ext numeril vlue of the orret
More informationFinal Exam Review. [Top Bottom]dx =
Finl Exm Review Are Between Curves See 7.1 exmples 1, 2, 4, 5 nd exerises 1-33 (odd) The re of the region bounded by the urves y = f(x), y = g(x), nd the lines x = nd x = b, where f nd g re ontinuous nd
More informationRiemann is the Mann! (But Lebesgue may besgue to differ.)
Riemnn is the Mnn! (But Lebesgue my besgue to differ.) Leo Livshits My 2, 2008 1 For finite intervls in R We hve seen in clss tht every continuous function f : [, b] R hs the property tht for every ɛ >
More information10 Vector Integral Calculus
Vector Integrl lculus Vector integrl clculus extends integrls s known from clculus to integrls over curves ("line integrls"), surfces ("surfce integrls") nd solids ("volume integrls"). These integrls hve
More information1.1. Linear Constant Coefficient Equations. Remark: A differential equation is an equation
1 1.1. Liner Constnt Coefficient Equtions Section Objective(s): Overview of Differentil Equtions. Liner Differentil Equtions. Solving Liner Differentil Equtions. The Initil Vlue Problem. 1.1.1. Overview
More informationSome estimates on the Hermite-Hadamard inequality through quasi-convex functions
Annls of University of Criov, Mth. Comp. Sci. Ser. Volume 3, 7, Pges 8 87 ISSN: 13-693 Some estimtes on the Hermite-Hdmrd inequlity through qusi-convex functions Dniel Alexndru Ion Abstrct. In this pper
More informationThe Bochner Integral and the Weak Property (N)
Int. Journl of Mth. Anlysis, Vol. 8, 2014, no. 19, 901-906 HIKARI Ltd, www.m-hikri.com http://dx.doi.org/10.12988/ijm.2014.4367 The Bochner Integrl nd the Wek Property (N) Besnik Bush Memetj University
More informationFUNCTIONS OF α-slow INCREASE
Bulletin of Mthemticl Anlysis nd Applictions ISSN: 1821-1291, URL: http://www.bmth.org Volume 4 Issue 1 (2012), Pges 226-230. FUNCTIONS OF α-slow INCREASE (COMMUNICATED BY HÜSEYIN BOR) YILUN SHANG Abstrct.
More informationLIAPUNOV-TYPE INTEGRAL INEQUALITIES FOR CERTAIN HIGHER-ORDER DIFFERENTIAL EQUATIONS
Eletroni Journl of Differentil Equtions, Vol. 9(9, No. 8, pp. 1 14. ISSN: 17-6691. URL: http://ejde.mth.txstte.edu or http://ejde.mth.unt.edu ftp ejde.mth.txstte.edu LIAPUNOV-TYPE INTEGRAL INEQUALITIES
More informationA BRIEF INTRODUCTION TO UNIFORM CONVERGENCE. In the study of Fourier series, several questions arise naturally, such as: c n e int
A BRIEF INTRODUCTION TO UNIFORM CONVERGENCE HANS RINGSTRÖM. Questions nd exmples In the study of Fourier series, severl questions rise nturlly, such s: () (2) re there conditions on c n, n Z, which ensure
More information