Jim Lambers MAT 280 Spring Semester Lecture 26 and 27 Notes

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1 Jim Lmbers MAT 280 pring emester Lecture 26 nd 27 Notes These notes correspond to ection 8.6 in Mrsden nd Tromb. ifferentil Forms To dte, we hve lerned the following theorems concerning the evlution of integrls of derivtives: The Fundmentl Theorem of lculus: b The Fundmentl Theorem of Line Integrls: Green s Theorem: tokes Theorem: b Guss ivergence Theorem: f (x) dx f(b) f() f(r(t)) r (t) dt f(r(b)) f(r()) (Q x P y ) da curl F d E P dx + Q dy div F dv F dr F d All of these theorems relte the integrl of the derivtive or grdient of function, or prtil derivtives of components of vector field, over higher-dimensionl region to the integrl or sum of the function or vector field over lower-dimensionl region. Now, we will see how the nottion of differentil forms cn be used to combine ll of these theorems into one. It is this nottion, s opposed to vectors nd opertions such s the divergence nd curl, tht llows the Fundmentl Theorem of lculus to be generlized to functions of severl vribles. A differentil form is n expression consisting of sclr-vlued function f : K R n R nd zero or more infinitesimls of the form dx 1, dx 2,..., dx n, where x 1, x 2,..., x n re the independent 1

2 vribles of f. The order of differentil form is defined to be the number of infinitesimls tht it includes. For simplicity, we set n 3 of three vribles. With tht in mind, 0-form, or differentil form of order zero, is simply sclr-vlued function f(x, y, z). A 1-form is function f(x, y, z) together with one of the expressions dx, dy or dz. A 2-form is function f(x, y, z) together with pir of distinct infinitesimls, which cn be either dx dy, dy dz or dz dx. Finlly, 3-form is n expression of the form f(x, y, z) dx dy dz. Exmple The function f(x, y, z) x 2 y + y 3 z is 0-form on R 3, while f dx (x 2 y + y 3 z) dx nd f dy (x 2 y + y 3 z) dy re both exmples of 1-form on R 3. Exmple Let f(x, y, z) 1/(x 2 + y 2 + z 2 ). Then f dx dy is 2-form on R 3 {(0, 0, 0}, while f dx dy dz is 3-form on the sme domin. Forms of the sme order cn be dded nd scled by functions, s the following exmples show. Exmple Let f(x, y, z) e x y sin z nd let g(x, y, z) (x 2 + y 2 + z 2 ) 3/2. Then f, g nd f + g re ll 0-forms on R 3, nd f + g e x y sin z + (x 2 + y 2 + z 2 ) 3/2. Tht is, ddition of 0-forms is identicl to ddition of functions. If we define ω 1 f dx nd ω 2 g dy, then ω 1 nd ω 2 re both 1-forms on R 3, nd so is ω ω 1 + ω 2, where Furthermore, if h(x, y, z) xy 2 z 3, nd ω f dx + g dy e x y sin z dx + (x 2 + y 2 + z 2 ) 3/2 dy. η 1 f dx dy, η 2 g dz dx re 2-forms on R 3, then is lso 2-form on R 3. η hη 1 + η 2 xy 2 z 3 e x y sin z dx dy + (x 2 + y 2 + z 3 ) 3/2 dz dx Exmple Let f(x, y, z) cos x, g(x, y, z) e y nd h(x, y, z) xyz 2. Then, ν 1 f dx dy dz nd ν 2 g dx dy dz re 3-forms on R 3, nd so is ν ν 1 + hν 2 (cos x + xyz 2 e y ) dx dy dz. It should be noted tht like ddition of functions, ddition of differentil forms is both commuttive, ssocitive, nd distributive. Also, there is never ny need to dd forms of different order, such s dding 0-form to 1-form. 2

3 We now define two essentil opertions on differentil forms. The first is clled the wedge product, multipliction opertion for differentil forms. Given k-form ω nd n l-form η, where 0 k + l 3, the wedge product of ω nd η, denoted by ω η, is (k + l)-form. It stisfies the following lws: 1. For ech k there is k-form 0 such tht η 0 0 η 0 for ny l-form η. 2. istributitivy: If f is 0-form, then (fω 1 + ω 2 ) η f(ω 1 η) + (ω 2 η). 3. Anticommuttivity: 4. Associtivity: ω η ( 1) kl (η ω). ω 1 (ω 2 ω 3 ) (ω 1 ω 2 ) ω 3 5. Homogeneity: If f is 0-form, then 6. If dx i is bsic 1-form, then dx i dx i If f is 0-form, then f ω fω. ω (fη) (fω) η f(ω η). Exmple Let ω f dx nd η g dy be 1-forms. Then by homogeneity, while ω η (f dx g dy) fg(dx dy) fg dx dy, η ω ( 1) 1(1) (ω η) fg dx dy. On the other hnd, if ν h dy dz is 2-form, then ν ω fh(dy dz dx) fh dy dz dx fh dy dx dz fh dx dy dz by homogeneity nd nticommuttivity, while ν η fh(dy dz dy) fh dy dz dy fh dy dy dz 0. Note tht if ny 3-form on R 3 is multiplied by k-form, where k > 0, then the result is zero, becuse there cnnot be distinct bsic 1-forms in the wedge product of such forms. 3

4 Exmple Let ω x dx y dy, nd η z dy dz x dz dx. Then ω η (x dx y dy) (z dy dz x dz dx) (x dx z dy dz) (y dy z dy dz) (x dx x dz dz) + (y dy x dz dx) xz dx dy dz yz dy dy dz x 2 dx dz dx + xy dy dz dx xz dx dy dz yz dy dy dz + x 2 dx dx dz + xy dy dz dx xz dx dy dz 0 0 xy dy dx dz (xz + xy) dx dy dz. The second opertion is differentition. Given k-form ω, where k < 3, the derivtive of ω, denoted by dω, is (k + 1)-form. It stisfies the following lws: 1. If f is 0-form, then df f x dx + f y dy + f z dz 2. Linerity: If ω 1 nd ω 2 re k-forms, then d(ω 1 + ω 2 ) dω 1 + dω 2 3. Product Rule: If ω is k-form nd η is n l-form, then d(ω η) (dω η) + ( 1) k (ω dη) 4. The second derivtive of form is zero; tht is, for ny k-form ω, d(dω) 0. We now illustrte the use of these differentition rules. Exmple Let ω x 2 y 3 z 4 dx dy be 2-form. Then, by Linerity nd the Product Rule, dω [d(x 2 y 3 z 4 ) dx dy] + ( 1) 0 [x 2 y 3 z 4 d(dx dy)] [( (x 2 y 3 z 4 ) x dx + (x 2 y 3 z 4 ) y dy + (x 2 y 3 z 4 ) z dz ) dx dy ] + [ x 2 y 3 z 4 {(d(dx) dy) + ( 1) 1 (dx d(dy)} ] [( 2xy 3 z 4 dx + 3x 2 y 2 z 4 dy + 4x 2 y 3 z 3 dz ) dx dy ] + [ x 2 y 3 z 4 {(0 dy) (dx 0)} ] 2xy 3 z 4 dx dx dy + 3x 2 y 2 z 4 dy dx dy + 4x 2 y 3 z 3 dz dx dy + 0 4x 2 y 3 z 3 dx dz dy 4x 2 y 3 z 3 dx dy dz. In generl, differentiting k-form ω, when k > 0, only requires differentiting the coefficient function with respect to the vribles tht re not mong ny bsic 1-forms tht re included in ω. In this exmple, since ω f dx dy, we obtin dω f z dz dx dy f z dx dy dz. 4

5 We now consider the kind of differentil forms tht pper in the theorems of vector clculus. Let ω f(x, y, z) be 0-form. Then, by the first lw of differentition, dω f dx, dy, dz. If is smooth curve with prmeteriztion r(t) x(t), y(t), z(t), t b, then b f(r(t)) r (t) dt b f(r(t)) x (t), y (t), z (t) dt It follows from the Fundmentl Theorem of Line Integrls tht dω ω(r(b)) ω(r()). b dω(r(t)) The boundry of,, consists of its initil point A nd terminl point B. If we define the integrl of 0-form ω over this 0-dimensionl region by ω ω(b) ω(a), which mkes sense considering tht, intuitively, the numbers 1 nd 1 serve s n pproprite outwrd unit norml vector t the terminl nd initil points, respectively, then we hve dω ω. Let ω P (x, y) dx + Q(x, y) dy be 1-form. Then dω d[p (x, y) dx] + d[q(x, y) dy] dp (x, y) dx P (x, y) d(dx) + dq(x, y) dy Q(x, y) d(dy) (P x dx + P y dy) dx 0 + (Q x dx + Q y dy) dy 0 P x dx dx + P y dy dx + Q x dx dy + Q y dy dy (Q y P x ) dx dy. It follows from Green s Theorem tht If we proceed similrly with 1-form ω ω F dx, dy, dz P (x, y, z) dx + Q(x, y, z) dy + R(x, y, z) dz, 5

6 then we obtin dω curl F dy dz, dz dx, dx dy (R y Q z ) dy dz + (P z R x ) dz dx + (Q y P x ) dx dy. Let be smooth surfce prmeterized by r(u, v) x(u, v), y(u, v), z(u, v), Then the (unnormlized) norml vector r u r v is given by We then hve curl F d r u r v x u, y u, z u x v, y v, z v (u, v). y u z v z u y v, z u x v x u z v, x u y v y u x v (y, z) (z, x) (x, y),,. (u, v) (u, v) (u, v) curl F n d curl F(r(u, v)) (r u r v ) du dv { (y, z) [R y (r(u, v)) Q z (r(u, v))] (u, v) + (z, x) [P z (r(u, v)) R x (r(u, v))] (u, v) + } (x, y) [Q x (r(u, v)) P y (r(u, v))] du dv (u, v) (R y Q z ) dy dz + (P z R x ) dz dx + (Q y P x ) dx dy If is the boundry curve of, nd is prmeterized by r(t) x(t), y(t), z(t), t b, then b F dr F(r(t)) r (t) dt b b P (r(t)), Q(r(t)), R(r(t)) x (t), y (t), z (t) dt ω(r(t)) dt ω. 6

7 It follows from tokes Theorem tht Let F P, Q, R. Let ω be the 2-form Then ω ω P dy dz + Q dz dx + R dx dy. dω dp dy dz + dq dz dx + dr dx dy [P x dx + P y dy + P z dz] dy dz + [Q x dx + Q y dy + Q z dz] dz dx + [R x dx + R y dy + R z dz] dx dy P x dx dy dz + Q y dy dz dx + R z dz dx dy P x dx dy dz Q y dy dx dz R z dx dz dy P x dx dy dz + Q y dx dy dz + R z dx dy dz div F dx dy dz. Let E be solid enclosed by smooth surfce with positive orienttion, nd let be prmeterized by r(u, v) x(u, v), y(u, v), z(u, v), (u, v). We then hve F d F n d F(r(u, v)) (r u r v ) du dv (y, z) (z, x) (x, y) P (r(u, v)), Q(r(u, v)), R(r(u, v)),, du dv (u, v) (u, v) (u, v) (y, z) x) y) P (r(u, v)) + Q(r(u, v)) (z, + R(r(u, v)) (x, du dv (u, v) (u, v) (u, v) P dy dz + Q dz dx + R dx dy ω. It follows from the ivergence Theorem tht ω E 7

8 Putting ll of these results together, we obtin the following combined theorem, tht is known s the Generl tokes Theorem: If M is n oriented k-mnifold with boundry M, nd ω is (k 1)-form defined on n open set contining M, then ω M The importnce of this unified theorem is tht, unlike the previously stted theorems of vector clculus, this theorem, through the lnguge of differentil forms, cn be generlized to functions of ny number of vribles. This is becuse opertions on differentil forms re not defined in terms of other opertions, such s the cross product, tht re limited to three vribles. For exmple, given 3-form ω f(x, y, z, w) dx dy dw, its integrl over 3-dimensionl, closed, positively oriented hypersurfce embedded in R 4 is equl to the integrl of dω over the 4-dimensionl solid E tht is enclosed by, where dω is computed using the previously stted rules for differentition nd multipliction of differentil forms. M 8

9 Prctice Problems Prctice problems from the recommended textbooks re: Mrsden/Tromb: ection 8.6, Exercises 1, 3, 5, 7, 11 9