Introduction to di erential forms

1 Introduction to differential formsDonu ArapuraMay 6, 2016 The calculus of differential forms give an alternative to vector calculus whichis ultimately simpler and more flexible. Unfortunately it is rarely encounteredat the undergraduate level. However, the last few times I taught undergraduateadvanced calculus I decided I would do it this way. So I wrote up this briefsupplement which explains how to work with them, and what they are good for,but the approach is kept informal. In particular, multlinear algebra is kept toa minimum, and I don t define manifolds or anything like that. By the time Igot to this topic, I had covered a certain amount of standard material, which isbriefly summarized at the end of these thanks to Jo ao Carvalho, John Crow, Mat u s Goljer and Josh Hill forcatching some 1- forms .

2 Exactness inR2.. Parametric curves .. Line integrals .. Work .. Green s theorem for a rectangle .. Exercise Set 1 .. Wedge product .. 2- forms .. Exactness inR3and conservation of energy .. Derivative of a 2-form and divergence .. Poincar e s lemma for 2- forms .. Exercise Set 2.. 163 Surface Parameterized Surfaces .. Surface Integrals .. Surface Integrals (continued) .. Length and Area .. Exercise Set 3 .. 274 Stokes Green and Stokes .. Proof of Stokes theorem .. Cauchy s theorem* .. Exercise Set 4.

3 355 Gauss Triple integrals .. Gauss theorem .. Proof for the cube .. Gravitational Flux .. Laplace s equation* .. Exercise Set 5 .. 426 Beyond3dimensions* Beyond 3D .. Maxwell s equations inR4.. 477 Further reading49A Review of multivariable Differential Calculus .. Integral Calculus .. differential 1-form (or simply a differential or a 1-form) on an open subset ofR2is an expressionF(x,y)dx+G(x,y)dywhereF,GareR- valued functions onthe open set. A very important example of a differential is given as follows: Iff(x,y) isC1R-valued function on an open setU, then its total differential (orexterior derivative) isdf= f xdx+ f ydyIt is a differential a similar fashion, a differential 1-form on an open subset ofR3is anexpressionF(x,y,z)dx+G(x,y,z)dy+H(x,y, z)dzwhereF,G,HareR-valuedfunctions on the open set.

4 Iff(x,y,z) is aC1function on this set, then its totaldifferential isdf= f xdx+ f ydy+ f zdz2At this stage, it is worth pointing out that a differential form is very similarto a vector field. In fact, we can set up a correspondence:Fi+Gj+Hk Fdx+Gdy+Hdzwherei,j,kare the standard unit vectors along thex,y,zaxes. Under this setup, the gradient fcorresponds todf. Thus it might seem that all we are doingis writing the previous concepts in a funny notation. However, the notation isvery suggestive and ultimately quite powerful. Suppose that thatx,y,zdependon some parametert, andfdepends onx,y,z, then the chain rule saysdfdt= f xdxdt+ f ydydt+ f zdzdtThus the formula fordfcan be obtained by Exactness inR2 Suppose thatFdx+Gdyis a differential onR2withC1coefficients.

5 We willsay that it isexactif one can find aC2functionf(x,y) withdf=Fdx+GdyMost differential forms are not exact. To see why, note that the above equationis equivalent toF= f x, G= f iffexists then F y= 2f y x= 2f x y= G xBut this equation would fail for most examples such asydx. We will call adifferentialclosedif F yand G xare equal. So we have just shown that if adifferential is to be exact, then it had better be is a very important concept. You ve probably already encounteredit in the context of differential equations. Given an equationdydx=F(x,y)G(x,y)we can rewrite it asFdx Gdy= 0 IfFdx Gdyis exact and equal to say,df, then the curvesf(x,y) =cgivesolutions to this concepts arise in physics.

6 For example given a vector fieldF=F1i+F2jrepresenting a force, one would like find a functionP(x,y) calledthe potential energy, such thatF= P. The force is calledconservative(seesection ) if it has a potential energy function. In terms of differential forms ,Fis conservative precisely whenF1dx+F2dyis Parametric curvesBefore discussing line integrals, we have to say a few words about parametriccurves. A parametric curve in the plane is vector valued functionC: [a,b] other words, we letxandydepend on some parametertrunning fromatob. It is not just a set of points, but the trajectory of particle travelling along thecurve.

7 To begin with, we will assume thatCisC1. Then we can define the thevelocity or tangent vectorv= (dxdt,dydt). We want to assume that the particletravels without stopping,v6= 0. Thenvgives a direction toC, which we alsorefer to as itsorientation. IfCis given byx=f(t), y=g(t),a t bthenx=f( u), y=g( u), b u awill be called C. This represents the same set of points, but traveled in theopposite thatCis given depending on some parametert,x=f(t), y=g(t)and thattdepends in turn on a new parametert=h(u) such thatdtdu6= we can get a new parametric curveC x=f(h(u)), y=g(h(u))It the derivativedtduis everywhere positive, we want to view the oriented curvesCandC as the equivalent.

8 If this derivative is everywhere negative, then CandC are equivalent. For example, the curvesC:x= cos , y= sin ,0 2 C :x= sint, y= cost,0 t 2 represent going once around the unit circle counterclockwise and clockwise re-spectively. SoC should be equivalent to C. We can see this rigorously bymaking a change of variable = /2 will be convient to allow piecewiseC1curves. We can treat these as unionsofC1curves, where one starts where the previous one ends. We can talk aboutparametrized curves inR3in pretty much the same Line integralsNow comes the real question. Given a differentialFdx+Gdy, when is it exact?Or equivalently, how can we tell whether a force is conservative or not?

9 Checkingthat it s closed is easy, and as we ve seen, if a differential is not closed, thenit can t be exact. The amazing thing is that the converse statement is often(although not always) true:4 THEOREM (x,y)dx+G(x,y)dyis a closed form on all ofR2withC1coefficients, then it is prove this, we would need solve the equationdf=Fdx+Gdy. In otherwords, we need to undo the effect ofdand this should clearly involve some kindof integration process. To define this, we first have to choose a parametricC1curveC. Then we define:DEFINITION CFdx+Gdy= ba[F(x(t),y(t))dxdt+G(x(t),y(t))dydt]dtI fCis piecewiseC1, then we simply add up the integrals over we ve done everything at once, it is often easier, in practice, to dothis in steps.

10 First change the variables fromxandyto expresions int, thenreplacedxbydxdtdtetc. Then integrate with respect tot. For example, if weparameterize the unit circlecbyx= cos ,y= sin , 0 2 , we see yx2+y2dx+xx2+y2dy= sin (cos ) d + cos (sin ) d =d and therefore C yx2+y2dx+xx2+y2dy= 2 0d = 2 From the chain rule, we getLEMMA CFdx+Gdy= CFdx+GdyIfCandC are equivalent, then CFdx+Gdy= C Fdx+GdyWhile we re at it, we can also define a line integral inR3. Suppose thatFdx+Gdy+Hdzis a differential form withC1coeffients. LetC: [a,b] R3be a piecewiseC1parametric curve, thenDEFINITION CFdx+Gdy+Hdz= ba[F(x(t),y(t),z(t))dxdt+G(x(t),y(t),z(t ))dydt+H(x(t),y(t),z(t))dzdt]dt5 The notion of exactness extends toR3automatically: a form is exact if itequalsdffor aC2function.