Theorem on the differentiation of a composite function with a vector argument

Transcription

1 Poceedings of the Estonian Academy of Sciences doi:.376/poc..3. Avaiae onine at Theoem on the diffeentiation of a composite function with a vecto agument Vadim Kapain and Üe Kotta Institute of Cyenetics at Tainn Univesity of Technoogy Akadeemia tee 68 Tainn Estonia; kotta@cc.ioc.ee Received 9 Noveme 9 evised 5 Januay accepted 8 Januay Astact. The pape povides a theoem on the diffeentiation of a composite function with a vecto agument. The theoem shows how the patia deivative of the tota deivative of the composite function can e expessed though the tota deivative of the patia deivative of the composite function. The poof of the theoem is ased on Mishkov s fomua which is the geneaization of the we-known Faà di Buno s fomua fo a composite function with a vecto agument. Key wods: diffeentia cacuus patia deivative tota deivative composite function.. INTRODUCTION The theoem poved in this pape was equied as an intemediate esut in soving the poem of the tansfomation of the noninea conto system descied y state equations into the oseve fom and finding the necessay conditions fo the possiiity of such tansfomation []. The deduction of the necessay conditions invoves fequent appication of the diffeentiation of the composite functions with espect to time agument and taking the patia deivatives of the diffeentiated composite function with espect to one of the vaiaes o its deivatives. The goa of this pape is to pesent and pove a fomua commutation ue which aows changing the ode of taking the tota highe-ode deivatives of the composite function and thei patia deivatives with espect to one of the vaiaes o its deivative. Since this esut may e usefu in the soution of othe noninea conto poems we popose it as a sepaate contiution. Fo exampe poay the main esut povided in the pape can e appied fo oseve design in [4]. The main too fo poving the theoem commutation ue is Mishkov s theoem [3] which povides the expicit fomua fo the nth deivative of a composite function with a vecto agument. Mishkov s fomua is a staightfowad geneaization of the we-known Faà di Buno s fomua [] which gives an expicit equation fo the nth-ode deivative of the composite function with a scaa agument. Coesponding autho vkapain@cc.ioc.ee

2 96 Poceedings of the Estonian Academy of Sciences MAIN RESULT The foowing theoem shows how the patia deivative of the tota deivative of the composite function can e expessed though the tota deivative of the patia deivative of this function. The composite function with the vecto agument with an aitay nume of components is consideed. Theoem. Assume that f ξ tξ t...ξ t is a composite function fo which deivatives up to ode a + ae defined; then f ξ tξ t...ξ t a+ f ξ a = Ca+ ξ tξ t...ξ t t ξ t whee =... Ca+ is the inomia coefficient and a ae nonnegative integes. Poof. In the poof we omit the vaiae t of ξ i t i.e. use instead of ξ i t a shote notation ξ i which aows the uky fomuas to e witten in a moe compact fom. Accoding to Mishkov s fomua [3] the a+th deivative of the composite function with a vecto agument can e computed y the fomua f ξ ξ...ξ a+ = a+ a+ i! k i a +! a+ q i j! k f ξ p ξ p ξ p a+ j= ξ i qi j j whee the espective sums ae taken ove a nonnegative intege soutions of the Diophantine equations as foows: k + k + + a + k a+ = a + q i + q i + + q i = k i 3 i fo i =...a + and p j and k on the ight-hand side of satisfy the eations p j = q j + q j + + q a+ j j =... k = p + p + + p = k + k + + k a+. 4 In taking the patia deivative of sum with espect to ξ a ony addends of sum with q a wi matte. Denote y h and g the pats of sum coesponding to q a and q a = espectivey; then f ξ ξ...ξ a+ = h + g. 5 Note that it is possie to state that h equas the expession in the ight-hand side of whee in addition to the estictions expessed y 3 and 4 the condition q a has to e satisfied. Note aso that if q a then k a. We pove the fomua sepaatey fo the cases a > and a. Fist conside the case when a >. Since k a and q a in ode to satisfy and 3 the foowing must hod k a = k i = < i a + i a q a = q a j = j =... j q i j = < i a + i a j =... As a esut unde the condition q a one can ewite as foows: and in 3 now i =... k + k + + k = 7 6

3 V. Kapain and Ü. Kotta: Diffeentiation of a composite function 97 Using 6 and changing the notations taking p j = p j fo j =... j p = p and k = k equations 4 may e ewitten as p j = q j + q j + + q j j =... k = p + p + + p = k + k + + k. 8 Note aso that unde conditions 6 a+ i! k i = a! i! k i a+ q i j! = q i j! and a+ ξ i qi j j = ξ a ξ i qi j j. Taking into account the aove equations and the fact that the patia deivative of g in 5 with espect to equas we otain in new vaiaes p j and k: ξ a f ξ ξ...ξ a+ ξ a = a! a +! i! k i q i j! k+ f ξ p ξ p ξ p + ξ p + + ξ p ξ i qi j j. 9 Note that in 9 a the patia deivatives with espect to ξ j ae of ode p j except with espect to ξ when the ode of the patia deivative is p +. In ode to unify the odes denote f := f ξ. We aso mutipy the ight-hand side of equation 9 y!! to otain f ξ ξ...ξ a+ ξ a = C a+ i! k i! q i j! k f ξ p ξ p ξ p ξ i qi j j. It is easy to oseve now that accoding to Mishkov s fomua the sum on the ight-hand side of togethe with the conditions 3 fo i =... 7 and 8 is the th-ode tota deivative of the function f. Consequenty f ξ ξ...ξ a+ ξ a = Ca+ f = Ca+ ξ f ξ ξ...ξ. Second conside the case a. Since k a in ode to satisfy and 3 the foowing must hod: k i = < i a + q i j = < i a + j =...

4 98 Poceedings of the Estonian Academy of Sciences Theefoe it is possie to ewite condition as k + + a k a + ak a + a + k a+ + + k = 3 and in 3 now i =... Again in ode to unify the notation in 3 one can take k i = k i fo i =... i a and k a = k a. This aows 3 to e ewitten as foows: k + k + + k = 4 and 3 as i a q i + q i + + q i = k i i =... i a q a + q a + + q a = k a +. Since q a we can denote q a := q a and the emaining q s as q i j := q i j. Theey 5 can e ewitten in unified notation as i 5 q i + q i + + q i = k i 6 fo i =... Changing notations taking p j = p j fo j =... j p = p and k = k equations 4 may e ewitten as p j = q j + q j + + q j j =... k = p + p + + p = k + k + + k. 7 Taking into account and using vaiaes k i and q i j we have a+ a+ i! k i = a! = ξ ξ i qi j j q i! k i a+ ξ a qa q i j! = q a + ξ a qa + q i j! ξ a+ qa+ ξ q j= j 8 ξ i qi j j. Futhemoe on the asis of 8 and the fact that the patia deivative of g in 5 with espect to ξ a equas we otain in new vaiaes p j and k f ξ ξ...ξ a+ ξ a = a! a +! i! k i q i j! k+ f ξ p ξ p ξ p + ξ p + + ξ p ξ i qi j j.

5 V. Kapain and Ü. Kotta: Diffeentiation of a composite function 99 Like in case a > we denote f = f ξ and mutipy the ight-hand side of the equaity given aove y! otain f ξ ξ...ξ a+ ξ a = C a+ i! k i! q i j! k f ξ p ξ p ξ p! to ξ i qi j j. 9 Again it is not difficut to oseve that accoding to Mishkov s fomua the sum on the ight-hand side of equation 9 togethe with the conditions 4 6 and 7 is the th-ode tota deivative of the function f. Consequenty hods again and this competes the poof. Some usefu cooaies of the theoem ae given eow. Cooay. Unde the assumptions of Theoem f ξ tξ t...ξ t m+n f ξ tξ t...ξ t m n = ξ t ξ t whee m and n ae nonnegative integes. Cooay. Unde the assumptions of Theoem f ξ tξ t...ξ t n f ξ tξ t...ξ t n = ξ t ξ t whee n is a nonnegative intege. 3. EXAMPLE The exampe in this section iustates the statement of Theoem. Conside the composite function f xtyt and assume that we need to take the patia deivative with espect to ÿt of the 3d-ode tota deivative of the function. Diect computations yied f xtyt 3 ÿt = 3 f xtyt yt ẏt + 3 f xtyt ẋt. xt yt On the othe hand taking the patia deivative of f xtyt with espect to yt and the tota deivative of the otained esut one gets f xtyt = f xtyt yt yt ẏt + f xtyt ẋt. xt yt Mutipying oth sides of the aove equaity y C3 we have f xtyt C3 = 3 f xtyt yt yt ẏt + 3 f xtyt ẋt. xt yt It is not difficut to check that f xtyt 3 f xtyt = C3. ÿt yt

6 Poceedings of the Estonian Academy of Sciences CONCLUSIONS The pape shows how to commute the opeations of taking highe-ode tota and patia deivatives of composite functions with vecto aguments. The fomua povided in the pape may e appicae not ony in diffeentia cacuus. As aeady mentioned in the intoduction the theoem was a usefu too in deiving sovaiity conditions of a cetain poem in noninea conto theoy. With high poaiity it may e usefu in deaing with othe noninea conto poems whee the deivatives of the composite functions with a vecto agument often show up. ACKNOWLEDGEMENT This wok was patiay suppoted y the Estonian Science Foundation though gant No. 69. REFERENCES. Johnson W. P. The cuious histoy of Faa di Buno s Fomua. Ame. Math. Monthy Kapain V. and Kotta Ü. Necessay and sufficient conditions in tems of diffeentia-foms fo ineaization of the state equations up to input-output injections. UKACC Intenationa Confeence on CONTROL 7 Septeme Coventy UK sumitted fo puication. 3. Mishkov R. L. Geneaization of the fomua of Faa di Buno fo a composite function with a vecto agument. Intenat. J. Math. Math. Sci Mishkov R. L. Noninea oseve design y educed geneaized oseve canonica fom. Intenat. J. Conto Teoeem vektoagumendiga iitfunktsiooni difeentseeimisest Vadim Kapain ja Üe Kotta On tõestatud teoeem vektoagumendiga iitfunktsiooni difeentseeimise kohta. Teoeemis esitatud vaem näita kuidas iitfunktsiooni täistuetise osatuetist saa väjendada tema osatuetise täistuetise kaudu. Teoeemi tõestus põhine Mishkovi vaemi mis omakoda kujuta endast tuntud Faà di Buno vaemi üdistust vektoagumendiga iitfunktsiooni jaoks. Näide iusteei teoeetiist tuemust.