A Practical Introduction to Differential Forms Alexia E. Schulz

1 A Practical Introduction toDifferential FormsAlexia E. SchulzandWilliam C. SchulzAugust 12, 2013 Transgalactic Publishing CompanyFlagstaff, Vienna, Cosmopolisiic 2012 by Alexia E. Schulz and William C. SchulzContents1 Introduction and Basic Introduction .. Some Conventions .. Some Formulas to Recall .. Coordinate systems .. The Algebra of Differential Forms .. The Operatord.. Orientation .. Differential Forms and Vectors .. grad, curl and div .. The Poincar e Lemma and it s Converse .. Boundaries .. Integrals of Forms .. Variable Changes .. Surface integrals .. The Generalized Stokes Theorem .. Curvilinear Coordinates I: preliminary formulas .. Curvilinear Coordinates II: the calculations .. Surfaces and Manifolds .. The Dualizing Operator .. The Codifferential .. The Laplacian .. Maxwell s Equations in 3-space .. Indefinite Inner Product Spaces.

2 Maxwell s equations in Space-Time .. Energy Flow and Energy Density .. 712 Mathematical Introduction .. Manifolds Mappings and Pullbacks .. Proofs of Stokes Theorem .. Proofs of the Converse of the Poincare Lemma .. Permutations .. The operator .. 76iiiivCONTENTSC hapter 1 Introduction and BasicApplications12 CHAPTER 1. Introduction AND BASIC INTRODUCTIONT hese notes began life as an Introduction to differential Forms for amathematicalphysics class and they still retain some of that flavor. Thus the material isintroduced in a rather formal manner and the mathematical complexities areput off to later sections. We have tried to write so that those whoseprimaryinterest is in the applications of differential Forms can avoid the theoreticalmaterial provided they are willing to accept the formulas that are derived inthe mathematical sections, which are clearly marked as such. Those who wishmay read the mathematical sections as they occur, or later, or indeed may putthem off to a more convenient time, perhaps in a future life, without loss to thecontinuity of the applied thread.

3 Anyway, such is my hope. But we want to alsoemphasize that those who wish will find all the mathematical details available, ata level of rigor usual to the better mathematical physics books. The treatmentis mostly local, and what little manifold theory is needed is quietly developedas we go. We have tried to introduce abstract material in circumstances whereit is useful to do so and we have also tried to avoid introducing a lot of abstractmathematical material all at one two areas most completely addressed in these notes, besides the foun-dational material, are coordinate changes and Maxwell s equations since wefeel that these illustrate the power of differential Forms quite treatMaxwell s equations in both three and four dimensions in separate sections. Wewill also look at a few other has been carefully chosen to be consistent with standardtensornotation to facilitate comparison with such treatments, and to facilitate learningbasic differential treatment of Maxwell s equations requires the derivation of thepotentialequations.

4 Although not strictly necessary, we have introduced the codifferential and the Laplace operator d + dsince this is the natural route using modernmathematics. For example we point out that the condition of Lorenzcan beexpressed instantly and easily in terms of the codifferntial in four as long as we have it available we can look at a couple of other applicationsof the Laplace operator on justified criticism of these notes might be that many things are donetwice, which is not efficient. We have sacrificed efficiency for convenience to thereader who may wish to deal with only one particular thing, and so would like arelatively complete treatment in the section without having to read five , many formulas are repeated at the beginning of sections where theyare used, rather than referred to in previous sections. The increase in paper israther small, and for those getting it electronically there is no wasteat all. Itis difficult for a mathematician to resist the call of generality but sinceone ofus is a physicist the brakes have been applied, and we hope that the product isa reasonable compromise between the siren song of mathematics and the needsof Practical SOME Some ConventionsHere we will introduce some conventions that will be used throughout thesenotes.

5 The letterAwill be used for a region of 2-dimensional space, for examplethe unit disk consisting of points whose distance from the origin is lessthan orequal to 1. It s boundary would be the unit circle consisting of pointswhosedistance from the origin is exactly 1. We will use the symbol to indicate theboundary. Thus ifAis the unit diskA={x R2| |x| 1}then the boundaryofAis A={x R2| |x|= 1}which is the unit circle. Notice carefully thedifference between the terms DISK and CIRCLE. (DISK and CIRCLEare oftenconfused in common speech.)The letterMwill be used for a (solid) region of 3 dimensional space, forexample the unit ball,M={x R3| |x| 1}whose boundary is the unitsphere M={x R3| |x|= 1}. (The terms BALL and SPHERE are oftenconfused in common speech, particularly in cases like a beach ball or abasketballsince they are filled with air.)The letterSwill be used for a (2 dimensional) surface in three dimensionalspace, for example the upper half of the unit sphere.

6 The boundary of thisSwould be a circle in thex, we do not wish to specify dimension, we will use the letterK. The use ofKindicates that the formula will work in any dimension, and this usually meansanydimension, not just 1, 2 or 3 dimensional space. Naturally Kmeans theboundary ball and sphere have analogs in every dimension. It is customarytorefer to the ball inRnis then-ball and its boundary as the (n 1) example, the unit disk is the 2-ball and its boundary, the unit circle, isthe 1-sphere. Note that them-sphere lives inRm+1. It is called them-spherebecause it requiresmvariables to describe it, like latitude and longitude on useful to know are the terms open and closed. This is a tricky topo-logical concept, so we will treat it only it includesits boundary. Thus the unit disk and unit ball are closed. If we remove theboundary KfromKthe resulting setK is calledopen. Thus for the unit ballinR3we haveM={x R3| |x| 1}closed 3-ballM ={x R3| |x|<1}open 3-ball M={x R3| |x|= 1}2-sphereWe want to give a real world example here but remember it must be inex-act since real world objects are granular (atomic) in constitution,so can onlyapproximate the perfect mathematical objects.

7 Some people prefer to eat theclosed peach (with fuzzy skin), some people prefer the open peach(fuzzy skinremoved, peach ) and the boundary of the peach, peach, is the fuzzy this will help you remember. Deeper knowledge of these matters canbe found in the wonderful book [2] and also [4].4 CHAPTER 1. Introduction AND BASIC APPLICATIONSFor functions we will use a slightly augmented variant of the physics conven-tion. When we writef:S Rwe mean a function whose input is a pointp Sand whose output is a real number. This is theoretically useful but not suitablefor calculation. When we wish to calculate, we need to introduce coordinates. Ifwe are dealing with the upper half of the unit sphere (set of points inR3whosedistance from the origin is exactly one and for whichz 0) then we might writef(x, y) if we choose to represent points in thex, ycoordinate system. Notice,and this is an important point, that the coordinatextakes as inputp Sandoutputs a real number, it sxcoordinate.

8 Hence the coordinatesxandyarefunctions just likef. IfSis the upper half of the unit sphere inR3thenxandyare not really good coordinates. It would be be better to use longitudeand colatitude for my coordinates and then we would writef( , ).1 Noteuse of the same letterfno matter what the coordinate system, because thefrepresents a quantity in physics, whereas in math it represents a functional re-lationship and we would not use the same letter for different coordinates. Notealso thatf(.5, .5) is ambiguous in physics unless you have already specified thecoordinate system. Not so with the math , we will almost always use the lettersf, g, hfor functions onA, M, S, these will occur in coordinate form , for examplef(x, y, z) for a Some Formulas to RecallYou are all familiar with thedx, dy, dzwhich occur in the derivative notationdydxand the integral notationZMf(x, y)dxdyZMf(x, y, z)dxdydzand you recall the Green, divergence and Stokes theorems, whichI list here forconvenience:Green s theoremZ Af(x, y)dx+g(x, y)dy=ZA g x f ydxdyThe divergence theorem or Gauss s theoremZ Mf(x, y, z)dydz+g(x, y, z)dzdx+h(x, y, z)dxdy=ZM f x+ g y+ h zdxdydz1 BEWARE.

9 Is longitude in physics but colatitude in mathematics. is colatitude inphysics but longitude in SOME FORMULAS TO RECALL5 The Classical Stokes theoremZ Sf(x, y, z)dx+g(x, y, z)dy+h(x, y, z)dz=ZM h y g z dydz+ f z h x dzdx+ g x f y dxdyYou might be more familiar with the last two in the vector formsZ Mv dS=ZMdivvdVandZ Sv d =ZScurlv dSThere are some conventions on integrals that we will mention now. Infor-mer times when integrating over a three dimensional object we wouldwriteR R RMdivvdVThis is now completely antiquated, and we will not do the other hand, there is a convention that when integrating around curvesor surfaces that have no boundary we put a small circle on the integral, so thatwe writeI Mv dSforZ Mv dSSince this is favored by the physics community we will mostly use it. Notice thatif a geometric object is the boundary of something, then it itself hasno boundary,and so we will use the circled integral almost exclusively with our purposes we will define a differential form to be an object likef(x, y)dxf(x, y, z)dydzf(x, y, z)dxdydzwhich we find as integrands in the written out Forms of the Green, divergenceand Stokes theorem above.

10 If is a sum of such objects it turns out that thethree theorems collapse to one mighty theorem, calledthe generalized Stokestheorem, which is valid for all dimensions:I S =ZSd To use this theorem and for other purposes it is only necessary the algebra that thedx, dy, dzsatisfy which is almost the same asordinary algebra with one important the rule for the operatordwhich is almost these are learned differential Forms can be manipulated easily and withconfidence. It is also useful to learn how various things that happen in vectoranalysis can be mimicked by differential Forms , and we will do this, naively atfirst and then in much more 1. Introduction AND BASIC APPLICATIONSIf you are concerned about what differential Forms ARE, the answer is alittle tricky and we am going to put it off for the moment. Later we willdiscuss the surprisingly dull answer to this question. Incidentally, the difficultyin explaining what they really are is one reason they have not become morecommon in elementary textbooks despite their extreme to give a tiny hint of the geometrical interpretation of differential two form measures the density of lines of force of a field, as introduced byJames Faraday a century and a half ago.