Vector Calculus and Di erential Forms with Applications to ...

1 Vector Calculus and DifferentialForms with Applications toElectromagnetismSean RobersonMay7,2015P R E F A C EThis paper is written as a final project for a course in Vector analysis,taught at Texas A&M University - San Antonio in the spring of2015as an independent study in mathematics, physics, engineering, and the sciencesusually go through a sequence of three Calculus courses before go-ing on to differential equations, real analysis, and linear algebra . Inthe third course, traditionally reserved for multivariable Calculus , stu-dents usually learn how to differentiate functions of several variableand integrate over general domains in space. Very rarely, as was mycase, will professors have time to cover the important integral theo-rems using Vector functions: Green s Theorem, Stokes Theorem, some universities, such as UCSD and Cornell, honors students areable to take an accelerated Calculus sequence using the textVector Cal-culus, Linear algebra , and Differential Formsby John Hamal Hubbardand Barbara Burke Hubbard.

2 Here, students learn multivariable cal-culus using linear algebra and real analysis, and then they generalizefamiliar integral theorems using the language of differential paper was written over the course of one semester, where themajority of the book was details, such as orientation of manifolds, topology, and thefoundation of the integral were skipped to save length. The papershould still be readable by a student with at least three semesters ofcalculus, one course in linear algebra , and one course in real analysis- all at the undergraduate level. Many of the fundamental theoremsdo not have their proofs, for some would be lengthy or require manydetails from real idea of this paper is to introduce differential Forms - an objectthat allows Maxwell s equations of electromagnetism to be conciselyrepresented in two lines as opposed to four.

3 The development of thisidea takes time, but it is worth it in the long Zachary Roberson2A C K N O W L E D G E M E N T SMany individuals supported me in writing this paper. First, I wouldlike to acknowledge my professor Donald F. Myers for teaching thisclass in a way that tapped into my interests and offered me a chal-lenge. He has also provided me with numerous resources that ex-tended my knowledge of mathematics and my methods of problemsolving. I would also like to thank Dr. John G. Romo for being ableto have this course offered. It served as an excellent extension of myknowledge from real analysis taken the semester before, as well aslinear algebra and differential writing this paper, I have received numerous amounts of sup-port from my coworkers, each of whom are writing tutors. I wouldlike to extend my thanks to Brandi Anderson, Stephanie Gil, CarlosLopez, Coral Rosario, Teresa Ruiz, Lisa Sanna, and Sheridan Santensfor their encouragement in motivating me to finish this project, allwhile still tending to my responsibilities at work.

4 I also would liketo thank Chris Tingwald and Dr. Katherine T. Bridgman for theirsupport as importantly, I could not have been motivated to finish thispaper without the undying support of my family and friends, espe-cially my mother, Denise Ginther, for giving me the motivation tofinish college and pursue a all your help, I would not have finished this paper, norwould it have been a quality U N D A M E N T A L D I F F E R E N T I A L O P E R A T O R SIn Vector analysis, it is of interest to examine the rate of flow of var-ious quantities such as heat, electricity, and fluids. The concept ofderivative can be extended to vectors in three ways, each using somesort of partial derivative. Each differential operator uses the symbol to operate on a function of several variables. This symbol is treatedas the following Vector : := x1 x2 xn gradientThe first differential operator is the gradient, denoted by for gradient operates on aC1functionf:Rn Rand returns avector of first derivatives.

5 For example, iff(x,y,z) =2x+3xy 4yz2,then f= 2+3y3x 4z2 8yz There are scalar product and quotient rules for gradients, withgrad(f g) =fgradg+ggradfandgrad(fg)=ggradf fgradgg2,similar to the product and quotient rules familiar in single divergenceThe next operator is the divergence, and it returns a sum of partialderivatives of a Vector functionF:Rn Rn. Using the nabla nota-tion, divergence can be denoted by F, or as divF. By definition,divF:= 1 j n Fj div is a differential operator, there are product and quotientrules for differentiation. The product rule states that for a scalar func-tionfand a Vector functionG,div(fG) =G gradf+ , the quotient rule statesdiv(Gf)=fdivG G differential operator can be built from the divergence op-erator. TheLaplacianis a differential operator that returns a sum ofsecond derivatives. It it defined as 2:= ,or, when acting on a functionf, 2f=div gradf= f= 1 j n 2f Laplacian appears in many partial differential equations, suchas thewave equation 2u t2=c2 2u,thediffusion equation u t=k 2u,and theLaplace equation 2u=0.

6 It is important to note that theLaplacian operator acts on spatial variables such asx,y,zand not thetime variablet(Strauss,14). curlThe last fundamental differential operator exists only inR3. The curlreturns a Vector of derivatives according to the rule F. That is,curlF:= F3 y F2 z F1 z F3 x F2 x F1 y .The curl of a product of a scalar and Vector function is given bycurl(fG=fcurlG+gradf G,and the curl of the quotient of a Vector function with a scalar func-tion iscurl(Gf)=fcurlG G (gradf) also have two important results regarding the f is differentiable. Thencurl gradf= definition,curl gradf= 2f y z 2f z y 2f x z 2f z x 2f x y 2f y x =0since mixed partials a differentiable Vector function curlF= definition,div curlF=( x y z)T F3 y F2 z F3 x F1 z F2 x F1 y = 2F3 x y 2F2 x z+ 2F1 y z 2F3 y x+ 2F2 z x 2F1 z y=0,since, as above, mixed partials curlThere are many other product rules for Vector derivatives.)

7 Griffithsgives the following three product rules utilizing both the inner andcross products:1. grad(F G) = (F curlG) + (G curlF) + (F )G+ (G )F2. div(F G) =G curlF F curlG3. curl(F G) = (G )F (F )G+F(divG) G(divF)72C L A S S I C A L T H E O R E M S A N D R E S U L T SAs with any branch of mathematics, Vector Calculus has its own set ofclassic results. The theorems here are presented in the order in whichthey appear in James Stewart sCalculus: Early fundamental theorem for line integralsThe Fundamental Theorem for Line Integrals, also called the Gradi-ent Theorem by David Griffiths gives a way to evaluate special lineintegrals ( ). In order to understand the theorem, we must firstdefine aconservative Vector fieldand apotential (Conservative Vector Field, Potential Function).Avector field is conservative if it is the gradient of some differentiablefunction.

8 That is,Fis conservative iffF= a function fis called a potential function these potential functions amounts to solving an exact dif-ferential equation, defined below, along with the definition of an (Exact Differential, Exact Differential Equation).Letz=f(x,y)be a differentiable function. The symbol dz denotes thedifferential of f , given as f xdx+ f expression M dx+N dyis an exact differential if M= f xand N= f y- that is, it correspondsto the differential of a function. The associated differential equationM dx+N dy=0is exact if the left side is an exact these exact differential equations uses the equality of mixedderivatives. This establishes the following exactness condition; a simi-lar expression will be used in the computations for Green s (Condition for Exactness).A necessary and sufficientcondition for M dx+N dy to an exact differential on some rectangle fundamental theorem for line integrals M y= N x,provided that M and N are differentiable on proof for the necessity is given in Zill, while the proof of thesufficiency outlines the method of solution to an exact differentialequation ( ).

9 To solve such an equation, one of the functionsMorNis selected and then integrated with respect to their attached variable - that is, integrateMwith respect tox, andNwith respecttoy. Afterwards, differentiate with respect to the remaining variableand then solve forf(x,y). This procedure is similar to finding a po-tential function for a conservative Vector (Conservative Criterion).IfF=P(x,y)i+Q(x,y)jisconserva tive, then P y= Q converse holds on an open, simply-connected region - a regionwhose boundary curve does not intersect itself and is , to find a potential function for the Vector fieldFdescribedabove, it suffices to solve the exact differential equationP dx+Q dy= are now ready to state the Fundamental Theorem of Line (Fundamental Theorem of Line Integrals, GradientTheorem).Let C be a smooth curve parameterized byr(t),a t f is differentiable with a continuous gradient on Cgradf dr=f(r(b)) f(r(a)).

10 From the Gradient Theorem follows a property of integrals takenover a closed (Path Independence Theorem).The integral CF drispath independent on a domain D iff CF dr=0for each closed pathC green s green s theoremThe following result is due to George Green. It relates the line integralof a function over a closed curved to an associated double integraltaken over the interior of the (Green s Theorem).Suppose C is a curve with positiveorientation (that is, oriented in the counterclockwise direction). Supposealso that C is smooth and closed. If P and Q are of classC1on someopen set contained in C, then CP dx+Q dy= C Q x P yd immediately follows from the theorem that if the integrand onthe left is an exact differential, the integral is may also write Green s Theorem in two Vector Forms : one withthe curl and the other with the ( Vector Forms of Green s Theorem).