MAS 4156 Lecture Notes Differentil Forms Definitions Differentil forms re objects tht re defined on mnifolds. For this clss, the only mnifold we will put forms on is R 3. The full definition is: Definition: Let M be differentil mnifold. A p-form on M is n ntisymmetric, contrvrint p-tensor. Men nything to you? No. Here is the definition we re using in clss: Definition: A p-form on R 3 is n object of the form: 0-form: f(x, y, z) (i.e., just function) 1-form: M(x,y,z) dx + N(x,y,z) dy + P(x,y,z) dz 2-form: M(x,y,z) dydz + N(x,y,z) dzdx + P(x,y,z) dxdy 3-form: f(x,y,z) dxdydz Algebr of forms Given form α, you re llowed to multiply it be function g(x,y,z) (which will just multiply ll functions in the definition of α). Given two forms of the sme dimension, you re llowed to dd them like ny lgebric expressions. The d terms in form re ntisymmetric, mening tht if you switch two of the d terms in form, you get negtive sign. So dxdy dydx, dxdydz dydxdz dydzdx, etc. Using these rules, we see tht we don t need, sy, dzdy term, since it is just negtive dydz term. Note tht this ntisymmetry property kills ny form term with repeted d. For instnce, dxdxdy dxdxdy (by switching the first two terms), nd so it is equl to zero.
Note tht we will hve no 4-forms, 5-forms, etc, becuse we only hve three distinct d s. Definition: Let α nd β be two forms. Define the wedge product α β s the form you would get by foiling out α times β, keeping the d s in ech term in order, then using the ntisymmetry property to reduce. As n exmple: (2 dx + 3 dy + 4 dz) (5 dx + 6 dy + 7 dz 10 dxdx + 12 dxdy + 14 dxdz + 15 dydx +18 dydy + 21 dydz + 20 dzdx +24 dzdy + 28 dzdz (21 24) dydz + (20 14) dzdx +(12 15) dxdy 3 dydz + 6 dzdx 3 dxdy The wedge product of p form nd q form is p + q form. In prticulr, if you wedge together 1 nd 3, 2 nd 3, 3 nd 3, or 2 nd 2, you will get zero (becuse ll combos gives you gret thn 3 -form). The nottion dxdy is shorthnd nottion for dx dy. Sme for the others. The nottion dxdydz mens dx dy dz. Wedging with 0-form (which is function) is just multipliction by the function. Definition: The opertor d, clled the exterior derivtive, is mp from p forms to p + 1 forms. On 0-form f(x,y,z), d returns: df f x dx + f y On 1-form M dx + N dy + P dz, d returns: dy + f z dz d(m dx + N dy + P dz) dm dx + dn dy + dp dz On 2-form M dydz + N dzdx + P dxdy, d returns: d(m dydz + N dzdx + P dxdy) dm dydz + dn dzdx + dp dxdy Note tht d(x) 1 dx + 0 dy + 0 dz dx. Mkes sense, huh? With some computtion, you cn show tht for ny form ω, d 2 ω 0.
Connection between forms nd vector fields There is close connection between forms on R 3 nd vector fields on R 3. In prticulr, mke the following ssocitions: dx î dy ĵ dz ˆk dydz î dzdx ĵ dxdy ˆk With these ssocitions, ech 1-form is ssocited to vector field s is ech 2-form. Any 3-form f dxdydz is obviously ssocited to the function f(x,y,z), nd of course ny 0-form is ssocited to itself. With these ssocitions, ll of our vector opertions hs corresponding form opertion: The wedge of two 1-forms is ssocited to the curl of the two corresponding vector fields. The wedge of 1-form nd 2-form is ssocited to the dot product of the two corresponding vector fields. The exterior derivtive of 0-form is ssocited to the grdient of the corresponding function. The exterior derivtive of 1-form is ssocited to the curl of the corresponding vector field. The exterior derivtive of 2-form is ssocited to the divergence of the corresponding vector field. Integrting Forms We cn integrte forms. The object we integrte them on hs the sme dimension s the form: 0-forms re evluted t points, 1-forms re integrted on curves, 2-forms re integrted on surfces, 3-forms re integrted on volumes. Integrting 1-forms We begin with M dx + N dy + P dz C
where C is n oriented curve. As usul, we prmeterize C with vector-vlued function r(t) x(t),y(t),z(t), t b. Note tht x x(t), nd so dx x (t) dt. Sme for y nd z. Put this in, nd we get: C M dx + N dy + P dz M(r(t))x (t) dt + N(r(t))y (t) dt + P(r(t))z (t) dt M(r(t))x (t) + N(r(t))y (t) + P(r(t))z (t) dt But then we note tht this is just dot product. So letting F M,N,P, we cn write this s: C M(r(t)),N(r(t)),P(r(t)) x (t),y (t),z (t) dt F(r(t)) r (t) dt F ˆt ds So integrting 1-form on curve is the sme s finding the work done by the corresponding vector field. Integrting 2-forms Now we hve surfce S, nd we wnt to define M dydz + N dzdx + P dxdy S As lwys, we prmeterize S by some vector vlued function s(u,v) x(u,v),y(u,v),z(u,v) over some domin D. Now we hve x x(u,v), nd so dx x u du + x v dv. Similrly for y nd z. So, for instnce: M dydz M(s(u,v)) (y u du + y v dv) (z u du + z v dv) M(s(u,v))(y u z v y v z u ) dudv Doing this for every term gives: M(s(u,v))(y u z v y v z u ) + N(s(u,v))(z u x v z v x u ) + P(s(u,v))(x u y v x v y u ) dudv D Agin, we notice tht this is dot product between the vector field F M,N,P nd vector tht is the result of s u s v. So our integrl turns into: M dydz + N dzdx + P dxdy F(s(u,v)) (s u s v ) dudv S D F ˆn ds S
So integrting 2-form on surfce is the sme s finding the flux of the corresponding vector field. Integrting 3-forms If you re given V f(x,y,z) dxdydz nd it is esy to describe V in terms of x, y, nd z, then you integrte s usul. If you need different coordinte system (like sphericl, cylindricl, crdiodl, ellipsoidl,...), then you will hve x, y, nd z written in terms of three other vribles, sy u, v, nd w: x x(u,v,w) y y(u,v,w) z z(u,v,w) Now since x x(u,v,w), we hve dx x u du + x v dv + x w dw. Doing the sme for y nd z, then plugging the results into dxdydz, then doing out ll of the wedge product, you end up with: V f(x,y,z) dxdydz V f(x(u,v,w),y(u,v,w),z(u,v,w)) x u x v x w y u y v y w z u z v z w dudvdw In other words, the coordintes chnge with the determinnt of mtrix of prtil derivtives. Tht mtrix of prtils is clled the Jcobin, nd it is usully written s: (x,y,z) (u,v,w) The r in r dzdrdθ nd the ρ 2 sin φ in ρ 2 sin φ dρdφdθ comes from the Jcobin. Generlized Stokes Theorem We hve hd two mjor points here: The d opertor corresponds to our div, grd, nd curl (depending on dimension). Integrting forms is the sme s integrting the corresponding vector fields. Put these two fcts together, nd we ve got: Generlized Stokes Theorem: Let ω be p form, nd let M be mnifold (mening curve, surfce, or volume) of dimension p + 1. Let M be the boundry of M oriented ppropritely. Then: dω ω M M
This tkes ll of our mjor theorems - Divergence, Stokes, Fundmentl Theorem of Line Integrls, nd Green s Theorem - nd describes ll of them in one simple formul. All of our theorems were simply specil cses. This theorem is true even if we re not working in R 3. We cn define nd ply with forms in ny dimension. This theorem is true even if we re not working in Eucliden spce. Riemnnin geometry dels with objects tht re curved, twisted, contining holes, etc. Stokes is still true. Objects such s these come up in generl reltivity nd string theory. If M is curve, then we need to interpret M s the two end points, positive for the end point, negtive for the strt point. The integrl then mens to evlute ω t the end point, then subtrct ω t the strt point.