Using Maple to Evaluate the Partial Derivatives of Two-Variables Functions

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1 Available Online at wwwijcmccom International Journal of Comuter Science and Mobile Comuting A Monthly Journal of Comuter Science and Information Technology ISSN X IJCSMC, Vol, Iue 6, June 013, g5 3 RESEARCH ARTICLE Uing Male to Evaluate the Partial Derivative of Two-Variable Function Chii-Huei Yu Deartment of Management and Information, Nan Jeon Intitute of Technology, Tainan City, Taiwan chiihuei@mailnjtcedutw Abtract Thi tudy ue the mathematical oftware Male for the auxiliary tool to evaluate the artial derivative of two tye of two-variable function We can obtain the infinite erie form of any order artial derivative of thee two tye of two-variable function by uing differentiation term by term At the ame time, we rovide two examle of two-variable function to determine their artial derivative ractically The reearch method adoted in thi tudy involved finding olution through manual calculation and verifying thee olution by uing Male Thi tye of reearch method not only allow the dicovery of calculation error, but alo hel modify the original direction of thinking from manual and Male calculation Therefore, Male rovide inight and guidance regarding roblem-olving method Keyword artial derivative; two-variable function; infinite erie form; differentiation term by term; Male I INTRODUCTION The comuter algebra ytem (CAS) ha been widely emloyed in mathematical and cientific tudie The raid comutation and the viually aealing grahical interface of the rogram render creative reearch oible Male oee ignificance among mathematical calculation ytem and can be conidered a leading tool in the CAS field The ueriority of Male lie in it imle intruction and eae of ue, which enable beginner to learn the oerating technique in a hort eriod In addition, through the numerical and ymbolic comutation erformed by Male, the logic of thinking can be converted into a erie of intruction The comutation reult of Male can be ued to modify our reviou thinking direction, thereby forming direct and contructive feedback that can aid in imroving undertanding of roblem and cultivating reearch interet Inquiring through an online uort ytem rovided by Male or browing the Male webite (wwwmaleoftcom) can facilitate further undertanding of Male and might rovide unexected inight A for the intruction and oeration of Male, we can refer to [1]-[7] In calculu and engineering mathematic coure, the tudy of evaluating the artial derivative of multivariable function i an imortant iue For examle, Lalace equation, wave equation, a well a ome other imortant hyical equation are involved the artial derivative of multivariable function Therefore, whether in hyic, engineering or other cience, the evaluation and numerical calculation of the artial derivative are imortant and can refer to [8]-[11] In thi aer, we mainly tudy the evaluation of the artial derivative of the following two tye of two-variable function f y co[( n 1)( r co[ n( co( cx + d ) + r (1) 013, IJCSMC All Right Reerved 5

2 Chii-Huei Yu, International Journal of Comuter Science and Mobile Comuting Vol Iue 6, June- 013, g 5-3 g y in[( n 1)( r in[ n( co( cx + d ) + r (), where r,, c, d are real number, n i an integer We can obtain the infinite erie form of any order artial derivative of thee two tye of two-variable function by uing differentiation term by term; thee are the major reult in thi aer (ie, Theorem 1 and ), and hence greatly reduce the difficulty of determining higher order artial derivative value of thee two tye of two-variable function A for the related tudy of evaluating artial derivative of two-variable function can refer to [1]-[16] On the other hand, we rooe two examle of two-variable function to determine their artial derivative ractically The reearch method adoted in thi tudy involved finding olution through manual calculation and verifying thee olution by uing Male Thi tye of reearch method not only allow the dicovery of calculation error, but alo hel modify the original direction of thinking from manual and Male calculation For thi reaon, Male rovide inight and guidance regarding roblem-olving method II MAIN RESULTS Firtly, we introduce ome notation and formula ued in thi tudy Notation (i) Let z a + ib be a comlex number, where i 1, a, b are real number We denote a the real art of z by Re( z ), and b the imaginary art of z by Im( z ) (ii)suoe t i any real number, and m i any oitive integer Define ( t) m t ( t 1) ( t m + 1) ( t ) 0 1 (iii)aume q, are non-negative integer For the two-variable function f, the -time artial derivative with reect to x, and then q -time artial derivative with reect to y i a ( q + ) -th order artial derivative of f, and denoted by Formula (i)euler' formula θ e i coθ + i inθ, where θ i any real number (ii)demoivre' formula n (coθ + i inθ ) co nθ + i in nθ, where n i any integer, θ i any real number, and (iii)geometric erie 1 Suoe z i a comlex number, and z < 1 Then 1 z k k z 0 Next, we introduce an imortant theorem ued in thi tudy Differentiation term by term theorem ([17]) If, for all non-negative integer k, the function g k : ( a, b) R atify the following three condition:(i) there exit a oint x0 ( a, b) uch that gk ( x 0 ) i convergent, (ii) all function g k (x ) are differentiable on oen k 0 013, IJCSMC All Right Reerved 6

3 Chii-Huei Yu, International Journal of Comuter Science and Mobile Comuting Vol Iue 6, June- 013, g 5-3 interval ( a, b), (iii) k 0 d dx g k ( x ) i uniformly convergent on ( a, b) Then g k ( x ) i uniformly convergent k 0 d and differentiable on ( a, b) Moreover, it derivative g k ( x) dx k 0 k 0 Before deriving the major reult in thi aer, we need a lemma d dx g k ( x) Lemma Suoe z i a comlex number, n i any integer, r i a real number, r 0 and z r Then zz n r 1 k + n z k 0 r if z < r (3) r k z k 1+ n k 0 if z > r (4) Proof If z < r, then zz n r z n 1 1 r z 1 r k n 1 z z (by geometric erie) r k 0 r 1 k + n z k 0 r If z > r, then zz n r z n 1 1 z r 1 z k n 1 r z z k 0 z (by geometric erie) r k z k 1+ n k 0 The following i the firt reult in thi tudy, we determine the infinite erie form of any order artial derivative of two-variable function (1) r,, c, d are real number, n,, q are non-negative integer If the domain of the two- Theorem 1 Suoe variable function f i {( x, y ) R y exit, y r, y 0} y co[( n 1)( r co[ n( co( cx + d ) + r Cae(i) If y < r, then the ( q + ) -th order artial derivative of f c k 0 + n ) ) q k q y co r + n )( cx + d ) + (5) Cae(ii) If y > r, then 013, IJCSMC All Right Reerved 7

4 Chii-Huei Yu, International Journal of Comuter Science and Mobile Comuting Vol Iue 6, June- 013, g 5-3 k k q c r 1 + n) ( k ) q y co cx + d ) + k 0 (6) Proof In Lemma, we take ( cx d ) z y e i + Cae(i) If y < r, then by (3) of Lemma, we obtain i( d ) n [ y e ] 1 i( cx + d ) k + n y e i( d ) [ ] y e r k 0 r n in( d ) y e [ y co( cx + d) r] + iy in( cx + d ) 1 + n ) i( k + n )( cx + d ) y e k 0 r (by Euler' formula and DeMoivre' formula) in( cx + d ) e {[ y co( cx + d ) r] iy in( cx + d )} [ y co( cx + d ) r] + [ y in( 1 k i + n )( cx + d ) y e k 0 r {co[ n( + i in[ n( } {[ y co( cx + d) r] iy in( cx + d )} co( cx + d ) + r 1 k i( k + n )( cx + d ) y e k 0 r (7) By the equality of real art of both ide of (7), we have f y co[( n 1)( r co[ n( co( cx + d ) + r 1 k y co[ + n )( k 0 r Therefore, by differentiation term by term, differentiating -time artial derivative with reect to x, and then q -time artial derivative with reect to y on both ide of (8), we obtain the ( q + ) -th order artial derivative of f (8) c k 0 + n ) ) q k q y co r + n )( cx + d ) + Cae(ii) If y > r, then by (4) of Lemma, we obtain By the equality of real art of both ide of (9), we obtain f i( d ) n [ y e ] k i( cx + d ) k 1+ n r y e i( d ) [ ] (9) y e r k 0 y co[( n 1)( r co[ n( co( cx + d ) + r k ( k 1) r y co[ (10) k 0 Alo, by differentiation term by term, differentiating -time artial derivative with reect to x, and then q - time artial derivative with reect to y on both ide of (10), we have k k q c r 1 + n) ( k ) q y co cx + d ) + k 0 013, IJCSMC All Right Reerved 8

5 Chii-Huei Yu, International Journal of Comuter Science and Mobile Comuting Vol Iue 6, June- 013, g 5-3 Theorem If the aumtion are the ame a Theorem 1, and the domain of the two-variable function g i {( x, y ) R y exit, y r, y 0} y in[( n 1)( r in[ n( co( cx + d ) + r Cae(i) If y < r, then the ( q + ) -th order artial derivative of g c k 0 + n ) ) q k q y in r + n )( cx + d ) + (11) Cae(ii) If y > r, then k c + r 1 n) k 0 k q ) q y in cx + d ) + (1) Proof Cae(i) If y < r, then by the equality of imaginary art of both ide of (7), we obtain g y in[( n 1)( cx + d)] r in[ n( 1 k y in[ + n)( co( cx + d ) + r k 0 r Therefore, by differentiation term by term, differentiating -time artial derivative with reect to x, and then q -time artial derivative with reect to y on both ide of (13), we obtain the ( q + ) -th order artial derivative of g (13) c k 0 + n ) ) q k q y in r + n )( cx + d ) + Cae(ii) If y > r, then by the equality of imaginary art of both ide of (9), g y in[( n 1)( cx + d)] r in[ n( k ( k 1) r y in[( k 1 + n )( (14) co( cx + d ) + r k 0 Alo, by differentiation term by term, differentiating -time artial derivative with reect to x, and then q - time artial derivative with reect to y on both ide of (14), we obtain the ( q + ) -th order artial derivative of g k c + r 1 n) k 0 k q ) q y in cx + d ) + III EXAMPLES In the following, we rooe two examle of two-variable function to determine their any order artial derivative, and ome of their higher order artial derivative value ractically On the other hand, we ue Male to calculate the aroximation of thee higher order artial derivative value and their infinite erie form for verifying our anwer Examle 1 If the domain of the two-variable function 013, IJCSMC All Right Reerved 9

6 Chii-Huei Yu, International Journal of Comuter Science and Mobile Comuting Vol Iue 6, June- 013, g i {( x, y ) R y 7, y 0} f 3 y co[ 4( x 3)] 7 co[ 5( x 3)] 6 3 y 14 y co( x 3) + 49 (15) Cae(i) If y 3 < 7 and y 0, then by (5) of Theorem 1, we obtain any ( q + ) f -th order artial derivative of k 0 + 5) (3k ) q 3k q y co 7 + 5)( x 3) + (16) 1 1 Thu, we can evaluate the 7-th order artial derivative value of f y ) at,, 3 7 f 1 1, k ) (3k ) k 4 in 10 ) (17) Next, we ue Male to verify the correctne of (17) >f:(x,->(y^3*co(4*(*x-3))-7*co(5*(*x-3)))/(y^6-14*y^3*co(*x-3)+49); >evalf(d[1$3,$4](f)(1/,1/3),14); >evalf(-8*um((k+5)^3*roduct(3*k-j,j03)*(1/3)^(3*k-4)/7^(k+1)*in(-*k-10),k0infinit,14); Cae(ii) If y 3 > 7, then by (6) of Theorem 1, we obtain k 3k 3 q 7 + 4) ( 3k 3) q y co + 4)( x 3) + k 0 (18) Hence, we can evaluate the 9-th order artial derivative value of f y ) at ( 1, ), 9 f (1,) 5 4 k 16 7 ( k k ) ( 3k 3k 8 3) 5 co 4) (19) We alo ue Male to verify the correctne of (19) >evalf(d[1$4,$5](f)(1,),0); >evalf(16*um(7^k*(-k+4)^4*roduct(-3*k-3-j,j04)*^(-3*k-8)*co(k-4),k0infinit,0); Examle If the domain of the two-variable function 013, IJCSMC All Right Reerved 30

7 Chii-Huei Yu, International Journal of Comuter Science and Mobile Comuting Vol Iue 6, June- 013, g 5-3 g y ) 5 i {( x, y ) R y 4, y 0} 5 y in[ 5(4 x + )] + 4 in[ 6(4 x + )] 10 5 y + 8 y co( 4 x + ) + 16 (0) Cae(i) If y 5 < 4 and y 0, then by (11) of Theorem, we obtain any ( q + ) of g -th order artial derivative 4 k 0 + 6) (5k ) q 5k q y in ( 4) + 6)( 4 x + ) + (1) Therefore, we can determine the 8-th order artial derivative value of g y ) at 1 1, 4 5, 8 g 4 4 1, k ) (5k ) 4 ( 4) 1 5 5k 4 in + 6) () In the following, we ue Male to verify the correctne of () >g:(x,->(y^5*in(5*(4*x+))+4*in(6*(4*x+)))/(y^10+8*y^5*co(4*x+)+16); >evalf(d[1$4,$4](g)(-1/4,1/5),14); >evalf(-56*um((k+6)^4*roduct(5*k-j,j03)*(1/5)^(5*k-4)/(-4)^(k+1)*in(k+6),k0infinit,14); Cae(ii) If y 5 > 4, then by (1) of Theorem, we obtain + g k 5k 5 q 4 + ( 4) 5) ( 5k 5) q y in + 5)(4 x + ) + k 0 3 Hence, we can evaluate the 9-th order artial derivative value of g y ) at, 3, 4 (3) 9 g 3, k 64 ( 4) ( k k ) ( 5k 5k 11 5) 6 ( 3) co 5) (4) Uing Male to verify the correctne of (4) a follow: >evalf(d[1$3,$6](g)(-3/4,-3),8); >evalf(-64*um((-4)^k*(-k+5)^3*roduct(-5*k-5-j,j05)*(-3)^(-5*k-11)*co(k-5),k0infinit,8); , IJCSMC All Right Reerved 31

8 Chii-Huei Yu, International Journal of Comuter Science and Mobile Comuting Vol Iue 6, June- 013, g 5-3 IV CONCLUSIONS From the above dicuion, we know the differentiation term by term lay a ignificant role in the theoretical inference of thi tudy In fact, the alication of thi theorem i extenive, and can be ued to eaily olve many difficult roblem; we endeavor to conduct further tudie on related alication On the other hand, Male alo lay a vital aitive role in roblem-olving In the future, we will extend the reearch toic to other calculu and engineering mathematic roblem and olve thee roblem by uing Male Thee reult will be ued a teaching material for Male on education and reearch to enhance the connotation of calculu and engineering mathematic REFERENCES [1] D Richard, Advanced Mathematical Method with Male, New York: Cambridge Univerity Pre, 00 [] C Tocci and S G Adam, Alied Male for Engineer and Scientit, Boton: Artech Houe, 1996 [3] M L Abell and J P Braelton, Male by Examle, 3rd ed, New York: Elevier Academic Pre, 005 [4] F Garvan, The Male Book, London: Chaman & Hall/CRC, 001 [5] R J Stroeker and J F Kaahoek, Dicovering Mathematic with Male : An Interactive Exloration for Mathematician, Engineer and Econometrician, Bael: Birkhauer Verlag, 1999 [6] C T J Dodon and E A Gonzalez, Exeriment in Mathematic Uing Male, New York: Sringer- Verlag, 1995 [7] J S Roberton, Engineering Mathematic with Male, New York: McGraw-Hill, 1996 [8] L E Fraenkel, Formulae for high derivative of comoite function, Mathematical Proceeding of the Cambridge Philoohical Society, vol 83, , 1978 [9] C H, Bichof, G Corli, and A Griewank, Structured econd and higher-order derivative through univariate Taylor erie, Otimization Method and Software, vol, 11-3, 1993 [10] D N Richard, An efficient method for the numerical evaluation of artial derivative of arbitrary order, ACM Tranaction on Mathematical Software, vol 18(), , 199 [11] T-W, Ma, Higher chain formula roved by combinatoric, The Electronic Journal of Combinatoric, vol 16, #N1, 009 [1] C-H Yu, Partial derivative of ome tye of two-variable function, Pure and Alied Mathematic Journal, vol (), 56-61, Aril 013 [13] C-H Yu, Evaluating artial derivative of two-variable function by uing Male, 6th IEEE/International Conference on Advanced Infocomm Technology, National United Univerity, Taiwan, July 013, in re [14] C-H Yu, Alication of Male: taking the artial differential roblem of ome tye of two-variable function a an examle, Proceeding of International Conference on e-learning, Tamkang Univerity, Taiwan, , 013 [15] C-H Yu, Alication of Male on the artial differential roblem of four tye of two-variable function, Proceeding of The International Conference on Advanced Information Technologie, Chaoyang Univerity of Technology, Taiwan, no87, 013 [16] C-H, Yu, Alication of Male: taking the artial differential roblem of two-variable function a an examle, Proceeding of 013 Buine Innovation and Develoment Symoium, Mingdao Univerity, Taiwan, B , March 013 [17] T M Aotol, Mathematical Analyi, nd ed, Boton: Addion-Weley, 30, , IJCSMC All Right Reerved 3