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1 DonnishJounals Donnish Jounal of Educational Reseach and Reviews. Vol 1(1) pp Novembe, Copyight 014 Donnish Jounals Oiginal Reseach Pape Vecto Analysis Using MAXIMA Savaş Tuylu Compute Science, Nigeian Tukish Nile Univesity, Nigeia. Accepted 3d Novembe, 014. The methods of calculus lie at the heat of the physical sciences and engineeing. Maxima can help you make faste pogess, if you ae just leaning calculus. The examples in this eseach pape will offe an oppotunity to see some Maxima tools in the context of simple examples, but you will likely be thinking about much hade poblems you want to solve as you see these tools used hee. This eseach pape includes Vecto Analysis. Keywods: Vecto Addition, Vecto Subtaction, Dot Poduct of Vectos, The Coss poduct, Gadient, The Divegence, Cul, Laplacian Opeato, Maxima. 1. INTRODUCTION Maxima is a system fo the manipulation of symbolic and numeical expessions, including diffeentiation, integation, Taylo seies, Laplace tansfoms, odinay diffeential equations, systems of linea equations, polynomials, sets, lists, vectos, matices, tensos, and moe. Maxima yields high pecision numeic esults by using exact factions, abitay pecision integes, and vaiable pecision floating point numbes. Maxima can plot functions and data in two and thee dimensions. Maxima souce code can be compiled on many compute opeating systems, including Windows, Linux, and MacOS X. The souce code fo all systems and pecompiled binaies fo Windows and Linux ae available at the SouceFoge file manage. Maxima is a descendant of Macsyma, the legenday compute algeba system developed in the late 1960s at the Massachusetts Institute of Technology. It is the only system based on that effot still publicly available and with an active use community, thanks to its open souce natue. Macsyma was evolutionay in its day, and many late systems, such as Maple and Mathematica, wee inspied by it. The Maxima banch of Macsyma was maintained by William Schelte fom 198 until he passed away in 001. In 1998 he obtained pemission to elease the souce code unde the GNU Geneal Public License (GPL). It was his effots and skills that made the suvival of Maxima possible. Vecto Analysis povides Vecto Addition and Subtaction, Dot Poduct of Vectos, The Coss poduct, Gadient, The Divegence, Cul, Laplacian Opeato, a basic intoduction to Vecto, some functions and vaiables of vecto opeatos using Maxima, definitions of Maxima s functions and vaiables of vecto opeatos, solving functions and vaiables of vecto opeatos examples using Maxima. MAXIMA has been constantly updated and used by eseache and enginees as well as by students.. VECTOR ANALYSIS Vectos, epesented in all thee coodinate systems, i.e. catesian, cylindical and spheical, must be used in electomagnetics. The vectos ae denoted by boldface symbols in ode to distinguish vectos fom scalas..1. Geneal A vecto field is a function having, at each instant in time, a magnitude at evey point in space and a diection. Vecto quantities have both magnitude and diection. They ae epesented gaphically by aows as shown in Figue.1. The length and aowhead show the magnitude and diection of a vecto, espectively. Coesponding Autho: savastyludf@hotmail.com

2 S a v a ş T u y l u D o n n. J. E d u. R e s. R e v. Figue.1: A vecto.. Vecto Analysis Using Maxima The vecto algeba and calculus pogams ae stoed in the package vect of MAXIMA Vesion that povides to combine and simplify symbolic expessions of dot poducts, coss poducts, gadient, divegence, cul, and laplacian opeatos of vectos. The command load(vect) is used fo vecto analysis. The following opeatos ae defined by the vect package: Table. Command vecto1. vecto vecto1 x vecto gad expession n cul expession laplacian expession Desciption the dot poduct of vecto1 and vecto the coss poduct of vecto1 and vecto the gadient of its agument, expession the cul of its agument, expession the laplacian of its agument, expession Figue.3. Vecto addition and Subtaction Vecto addition and subtaction using Maxima Example.3.1: The vectos A and B ae given in catesian coodinate system as A kx ly (.3.1), B ly k (.3.). Calculate the vecto addition A + B and vecto subtaction A - B. Examples.4. Dot Poduct of Vectos The dot poduct of two vectos is the poduct of one vecto which is aligned with anothe vecto theefoe it is obtained by the poduct of thei magnitudes multiplied by the cosine of thei angle (0 ) between vectos as A. B A B cos( ) (.4.1). AB The esult is scala. The dot poduct of two vectos in catesian coodinates.3. Vecto Addition and Subtaction The vecto sum of A and B is defined in elation to the gaphic sketch of the vectos, as shown in Fig..3. The ule is to put the initial point of one vecto to the end point of anothe vecto. A A x A y A (.4.), B B x B y B (.4.3), is AB A B A B A B (.4.4).. x x y y

3 S a v a ş T u y l u D o n n. J. E d u. R e s. R e v. 3 Same as above, the dot poduct of two vectos in spheical coodinates A A A A (.4.5), B B B B (.4.6), is AB A B A B A B (.4.7).. Dot poduct using Maxima Example.4.1: Conside two vectos in catesian coodinates A A x A y A B B x B y B Calculate the dot poduct of A and B. The coss poduct of two vectos in catesian coodinates A A x A y A (.5.), B B x B y B (.5.3), AB x A A A B B B ( A B A B ) x ( A B A B ) y ( A B A B ) (.5.4). y y x x x y y x Coss poduct using Maxima Example.5.1 Conside two vectos in catesian coodinates A A x A y A, and B B x B y B, calculate the coss poduct of A and B..5. The Coss Poduct The coss poduct of two vectos is obtained by the poduct of thei magnitudes multiplied by the sine of thei angle (0 ) between vectos as given in Fig..5.1, AxB A B sin( AB ) c (.5.1), whee c is the unit vecto which is nomal to the plane detemined by these vectos.the diection is obtained by using the ight hand ule..6. Gadient The gadient is the ate of change of a scala function f. It is a vecto in the diection of the geatest incease of function f. It is a slope. The f points towads the lage values of given function. If thee ae thee vaiables, the gadient f in catesian, cylindical, and spheical coodinate system, espectively, can be shown as f f f f x y x y (.6.1), f f f f (.6.), f 1 f 1 f f sin (.6.3). Gadient using MAXIMA Figue.5.1: The Coss poduct Example.6.1 Conside the scala function f, calculate the gadient f in catesian, cylindical, and spheical coodinate system.

4 S a v a ş T u y l u D o n n. J. E d u. R e s. R e v. 4 Divegence using Maxima Example.7.1 Calculate the divegence of vecto A in catesian, cylindical, and spheical coodinate systems.7. The Divegence The divegence is an opeato which shows the magnitude of the vecto field's souce at the given point. The divegence is scala and beas a similaity to the deivative of a function. The divegence of a vecto A in catesian, cylindical, and spheical coodinate system, espectively, can be shown as A A x y A A x y (.7.1), 1 ( A) 1 A A A (.7.), 1 ( A ) 1 ( A sin ) 1 A A (.7.3). sin sin.8. Cul The cul is a vecto opeato which descibes the otation of a vecto. The diection of the cul is the axis of the otation. The cul of a vecto A in catesian, cylindical, and spheical coodinate system, espectively, can be shown as:

5 S a v a ş T u y l u D o n n. J. E d u. R e s. R e v. 5 A A y A A x A y Ax xa ( ) x ( ) y ( ) x y y x x y A A A (.8.1), 1 1 1A 1 ( ) A A A A A xa ( ) ( ) ( ) A A A (.8.), 1 ( Asin ) A 1 1 ( A ) A 1 ( A) A xa ( ) ( ) ( ) sin sin Cul using Maxima Example.8.1 (.8.3). Calculate the cul of A in catesian, cylindical, and spheical coodinate system..9. Laplacian Opeato The laplacian opeato is a second ode diffeential opeato, applied Nabla opeato. It is the sum of all second patial deivatives in catesian coodinates. The laplacian of vecto A in catesian, cylindical, and spheical coodinate system, espectively, can be shown as f f f x y f f 1 f 1 f f f ( ) (.9.1), (.9.), f ( f ) f (sin f ) sin sin Laplacian using Maxima Example.9.1 (.9.3). Conside the scala function f, calculate the laplacian f in catesian, cylindical, and spheical coodinate system.

6 S a v a ş T u y l u D o n n. J. E d u. R e s. R e v. 6 CONCLUSION Reseach pape can apply each and evey pat of Vectos, help application of the physical sciences and engineeing, make faste pogess, and help to undestand Vectos. The pape paticulaity helps to undestand pats of Calculus and is going to extend to othe pats of the Calculus. ACKNOWLEDGEMENTS I would like to thank the Maxima developes, Nigeian Tukish Nile Univesity, and Pof. Niyai ARI fo thei fiendly help. REFERENCES [1] Niyai ARI, Lectue notes, Univesity of Technology, Zuich, Switeland. [] Niyai ARI, Symbolic computation of electomagnetics with Maxima (013). [3]