Mth 452 - Advnced Clculus II Line Integrls nd Green s Theorem The min gol of this chpter is to prove Stoke s theorem, which is the multivrible version of the fundmentl theorem of clculus. We will be focused on the integrtion of certin kinds of functions over mnifolds in R n. These functions re clled differentil k-forms. In prticulr, Stoke s theorem sys tht the integrl of differentil k-form α over compct (k + 1)-mnifold V with nonempty boundry is the sme s the integrl of the differentil of tht k-form, which is (k + 1)-form dα, over the boundry of the mnifold, V. Tht is, dα We strt with the simplest cse, where V is curve. V V α. 1 Arc length nd pth prmetriztions efinition 1. A C 1 pth in R n is continuously differentible function : [, b] R n. The pth is sid to be smooth if (t) for ll t [, b]. efinition 2. The prtition P { t < t 1 <... < t k 1 < t k b} defines polygonl pproximtion to, with vertices (t ), (t 1 ),..., (t k ) nd length s(, P) given by s(, P) (t i ) (t i 1 ) efinition 3. The length s() of C 1 pth is defined s provided tht the limit exists. s() lim s(, P), P Theorem 1. If : [, b] R n is C 1 pth, then s() exists nd s() b (t) dt Proof. It suffices to prove tht, given ɛ >, tht there exists mesh threshold δ > such tht if meshp < δ, then b (t) dt s(, P) < ɛ. 1
Recll tht s(, P) (t i ) (t i 1 ) where 1,..., n re the component functions of. [ n ] 1 2 ( r (t i ) r (t i 1 )) 2 An ppliction of the men vlue theorem to the function r on the i th subintervl [t i 1, t i ] yields point t r i (t i 1, t i ) such tht Hence (1) becomes r1 r (t i ) r (t i 1 ) r(t r i )(t i t i 1 ). s(, P) [ n ] 1 2 ( r(t r i )) 2 (t i t i 1 ). (2) r1 Notice tht the sum in (2) looks very much like Riemnn sum. The rest of the proof involves proving tht it does indeed not differ much from one. efine n uxiliry function F : I [, b] n R by F (x 1,..., x n ) [ n r1 r(x r )) 2 ] 1 2. Notice tht F (t,..., t) (t). Since is C 1 -pth on the compct intervl I, it follows tht F is uniformly continuous on I. Hence there exists δ 1 > such tht for ech x r y r < δ 1. F (x) F (y) < ɛ 2(b ) We now wnt to compre the pproximtion s(, P) with the Riemnn sum (1) (3) R(, P) (t i ) (t i t i 1 ) F (t i,..., t i )(t i t i 1 ). Since the points t 1 i,..., tn i (3) tht ll lie in the intervl [t i 1, t i ] nd t i t i 1 < meshp, it follows from s(, P) R(, P) < F (t 1 i,..., t n i ) F (t i,..., t i ) (t i t i 1 ) ɛ 2(b ) (t i t i 1 ) ɛ 2, 2
if mesh P < δ 1. On the other hnd, by theorem from the lst chpter, there exists δ 2 > such tht b R(, P) (t) dt < ɛ 2 if mesh P < δ 2. The result follows by tking mesh P < min{δ 1, δ 2 } nd using the tringle inequlity. efinition 4. The pth α : [, b] R n is sid to be equivlent to the pth β : [c, d] R n if nd only if there exists C 1 function ϕ : [, b] [c, d] such tht ϕ([, b]) [c, d], α β ϕ, nd ϕ (t) > t [, b] Theorem 2. Suppose α : [, b] R n nd β : [c, d] R n re equivlent pths, nd tht f is continuous rel-vlued function whose domin of definition in R n contins the (common) imge of α nd β. Then b f(α(t)) α (t) dt d c f(β(t)) β (t) dt Proof. Use the chin rule nd the fct tht φ (t) > for ll t [, b]. Notice in the bove theorem, tht if we cn find unit-speed representtive for our pth, i.e. (t) 1, then the derivtive of the pth doesn t ply role the integrls of concern. The following proposition sys this is lwys possible. Proposition 1. Every smooth pth : [, b] R n is equivlent to smooth unit-speed pth ( smooth pth ˆ such tht ˆ (t) 1). Proof. Let L s() be the length of, nd define σ : [, b] [, L] by σ(t) so tht σ(t) is simply the length of ([, t]). t (u) du Then the fundmentl theorem of clculus gives us tht σ is C 1 function with σ (t) (t) >. Therefore σ hs C 1 inverse τ : [, L] [, b] with τ (s) 1 σ (τ(s)) >. This llows us to define the smooth pth ˆ : [, L] R n by ˆ τ, which is equivlent to by construction. 3
The chin rule nd the fct tht σ (t) (t) then give us tht which is wht we wnted. ˆ (s) (τ(s))τ (s) (τ(s)) σ (τ(s)) 1, 1.1 ifferentil 1-forms nd line integrls In lter sections, we ll tlk t length bout differentil k-forms, but for now, we ll focus on differentil 1-forms. efinition 5 (ifferentil - nd 1-forms). A differentil -form on set U R n is differentible mp f : U R. A (liner) differentil 1-form on the set U R n is mp ω which ssocites with ech point x U liner function ω x : R n R. Tht is, ω : U L(R n, R). Since differentil 1-form evluted t point v (v 1,..., v n ) R n is liner mp, it hs the form ω x (v) 1 (x)v 1 + + n (x)v n where the i : U R re mps. We sometimes write ω x simply s ω x 1 (x)dx 1 + + n (x)dx n, where we tke dx i to be projection to the i th coordinte, i.e. dx i (v) v i. efinition 6. We sy the differentil 1-form ω is continuous ( differentible, or C 1 ) when the coefficient functions 1,..., n re continuous (differentible, or C 1 ). Exmple 1. If f : U R is differentible function on n open set, then its differentil df x t x U is liner function on R n given by df x (v) 1 f(x)v 1 + + n f(x)v n nd hence is differentil 1-form. Remrk 1. ifferentil 1-forms which re differentils of differentible functions re clled closed 1-forms. Not ll differentil 1-forms re closed. We will give n exmple of this in bit. In the sequel, we ll define notion of differentition for differentil k-form, which will give us differentil (k + 1)-form. 4
efinition 7. Let ω be continuous differentil 1-form on U R n nd : [, b] U C 1 pth. The integrl of ω over the pth is defined s ω b ω (t) ( (t)) dt. efinition 8 (Line integrl). Given C 1 pth : [, b] R n nd n continuous functions f 1,..., f n whose domins of definition in R n ll contin the imge of, the line integrl f 1 dx 1 + + f n dx n is defined s f 1 dx 1 + + f n dx n b [ f1 ((t)) 1(t) + + f n ((t)) n ] dt. Remrk 2. Hence line integrl is simply the integrl of the differentil 1-form ppering s its integrnd. Formlly, we re just substituting dx i for i (t)dt. The next Theorem 3. If f is rel-vlued C 1 function on the open set U R n, nd : [, b] U is C 1 pth, then df f((b)) f(()). Proof. efine g : [, b] R by g f. Then g (t) f((t)) (t) by the chin rule, so df b b b b df (t) ( (t))dt [ 1 f((t)) 1(t) + + n f((t)) n(t) ] dt f((t)) (t)dt g (t)dt g(b) g() f((b)) f(()). The following importnt corollry is immedite: Corollry 1. If ω df, for some C 1 function f defined on U, nd α nd β re C 1 pths with the sme initil nd terminl points, then ω α β ω. 5
Exmple 2. The differentil 1-form defined on R 2 {} by ω ydx + xdy x 2 + y 2 is not the differentil of ny differentible function. To see this, observe tht if we define 1, 2 : [, 1] R 2 by then we hve while 1 (t) (cos πt, sin πt) nd 2 (t) (cos πt, sin πt) 1 ω 2 ω 1 1 (sin πt)( π sin πt) + (cos πt)(π cos πt) cos 2 πt + sin 2 dt π, πt ( sin πt)( π sin πt) + (cos πt)( π cos πt) cos 2 πt + sin 2 dt π. πt Hence Corollry 1 implies tht ω df for ny differentible f : R 2 {} R. NOTE: We cn use this observtion to prove tht the ngle function θ(x, y) rctn y x, which is defined on R 2 minus the nonnegtive x-xis, cnnot be extended to R 2 {}, since dθ ω. 1.2 The rc-length form efinition 9. The set C in R n is clled curve if nd only if it is the imge of one-to-one smooth pth. Any one-to-one smooth pth equivlent to is clled prmetriztion of C. efinition 1 (Orienttion for curve). If x (t) C with C curve, then T(x) (t) (t) is unit tngent vector to C t x. The continuous mp T : C R n is clled n orienttion for C, nd n oriented curve is the pir (C, ), often bbrevited to C. efinition 11. Given n oriented curve C in R n, its rclength form ds is defined for x C by ds x (v) T(x) v. Theorem 4. Let be prmetriztion of the oriented curve C, nd let ds be the rclength form of C. If f : R n R nd F : R n R n re continuous mppings, then b f((t)) (t) dt f ds, nd b F((t)) (t) dt F T ds. 6
2 Green s Theorem Green s theorem is 2-dimensionl generliztion of the Fundmentl Theorem of Clculus. We wnt to construct notion of the differentil of differentil form defined in R 2. efinition 12. Given C 1 differentil form ω P dx + Q dy in two vribles, its differentil dω is defined by dω ( Q x P ) dxdy. y efinition 13. Given continuous differentil 2-form α dxdy nd contented set R 2, the integrl of α on is defined s α (x, y)dxdy. NOTE: Rel vlued functions re clled -forms, differentil forms of the type ω P dx + Q dy re clled 1-forms, nd differentil forms of the type α dxdy re clled 2-forms. 2.1 Informl Version of Green s Theorem Let be nice region in the plne R 2, whose boundry consists of finite number of closed curves, ech of which is positively oriented with respect to. If ω P dx + Q dy is C 1 differentil 1-form defined on, then dω () Wht does nice mens? (b) Wht does positively oriented mens? (c) How to compute the right-hnd side? ω. 2.2 Piecewise smooth curves efinition 14 (Piecewise smooth pths). The continuous pth : [, b] R n is clled piecewise smooth if there is prtition P { < 1 <... < k b} of [, b] such tht ech restriction i of to [ i 1, i ] is smooth. Note then tht 1 n is conctention of the i. For ω continuous differentil 1-form, we define ω ω i 7
efinition 15 (Piecewise smooth curves). A piecewise-smooth curve C in R n is the imge of one-to-one, piecewise-smooth pth : [, b] R n. The pth is prmetriztion of C, nd the pir (C, ) is clled n oriented piecewise-smooth curve. The curve C is closed if () (b). efinition 16 (Integrting 1-forms over piecewise smooth curves). Given n oriented piecewisesmooth curve C nd continuous differentil 1-form ω defined on C, the integrl of ω over C is defined s C ω where is ny prmetriztion of C. Moreover, for C (C, ) (C, β) with nd β the sme underlying pth but inducing opposite orienttions, we ω ω. 2.3 Nice regions C efinition 17. A nice region in the plne is connected compct set R 2 whose boundry is the union of finite number of mutully disjoint piecewise-smooth closed curves. Hence, ω ω, C r efinition 18. An oriented boundry curve C (C, ) of nice region is positively oriented with respect to if the region stys on the left of C s the curve is trversed in the direction given by its prmetriztion. NOTE: The conclusion of Green s Theorem cn be reformulted in terms of the divergence of vector field. Given C 1 vector field F : R n R n, its divergence div F : R n R is the rel-vlued function defined s C i ω. div F F 1 x 1 + + F n x n. For R 2 nice region with positively oriented, let N denote the unit outer norml vector to. Then for ny C 1 vector field F defined on. F N ds div F dxdy, (4) 8
2.4 Green s theorem on the nicest region Lemm 1 (Green s Theorem for the Unit Squre). If ω P dx + Q dy is C 1 differentil 1-form on the unit squre I 2, nd I 2 is oriented counterclockwise, then dω ω. I 2 I 2 Proof. Since dω ( Q x P ) dxdy, y we cn pply Fubini s theorem nd the FTC to get 1 1 ( ) Q 1 ( 1 ) dω I 2 x dx P dy y dx dy 1 ( 1 ) Q 1 ( 1 ) x dx P dy y dy dx 1 Q(1, y)dy 1 Q(, y)dy 1 P (x, 1)dx + 1 P (x, )dx. We now wnt to show tht the right hnd side of (4) reduces to the sme thing. Towrd this end, we define four curves 1, 2, 3, 4 : [, 1] R 2 by: 1 (t) (t, ), 2 (t) (1, t), 3 (t) (1 t, 1), 4 (t) (, 1 t). Then since ω P dx + Qdy, with P, Q : R 2 R C 1, we get ω I 2 ω + 1 ω + 2 ω + 3 ω 4 1 1 P (t, )dt + P (x, )dx + 1 1 Q(1, t)dt + Q(1, y)dy 1 1 P (1 t, 1)dt + P (x, 1)dx 1 1 Q(, 1 t)dt Q(, y)dy. where the second line follows from the first since 1 [ ω P ((t)) 1 (t) Q((t)) 2(t) ] dt nd the third line is obtined from the second by the substitutions x t, y t, x 1 t, y 1 t respectively. 9
2.5 Oriented 2-cells nd pullbcks efinition 19. The set R 2 is clled n oriented smooth 2-cell if there exists one-to-one C 1 mpping F : U R 2 defined on neighborhood U of I 2, such tht F (I 2 ) nd F (x) > for ech x U. NOTE: If the nice region is convex qudrilterl, then is n oriented 2-cell. More generlly, nice region is n oriented 2-cell if it hs single boundry curve which is piecewise smooth nd consists of exctly four smooth curves, nd the interior ngle t ech of the four vertices is < π. efinition 2. The pullbck of -form ϕ : R n R under the C 1 mpping F : R n R n is defined s (F ϕ)(x) (ϕ F )(x). efinition 21. Given 1-form ω n ϕ i dx i, its pullbck under the C 1 mpping F : R n R n is given by n (F ω) x (v) (F ϕ i )(x)dx i (F v) efinition 22. Given 2-form α g dxdy defined on R 2, its pullbck under the C 1 mpping F : R 2 uv R 2 xy is defined s (F α)(u, v) (g F )(det F )(u, v) du dv The following lemm compiles some useful properties of pullbcks: Lemm 2. Let F : U R 2 be s in the definition of the oriented 2-cell F (I 2 ). Let ω be C 1 differentil 1-form nd α C 1 differentil 2-form on. Then 1. ω I F ω 2 2. α I F α 2 3. d(f ω) F (dω). Proof. Homework. 2.6 Green s Theorem for nicer regions We now prove Green s theorem for oriented 2-cells. The point here is tht nice regions cn often be decomposed into oriented 2-cells, so we will be ble to reduce the most generl form of Green s theorem to the next lemm. 1
Lemm 3 (Green s Theorem for Oriented 2-cells). If is n oriented 2-cell nd ω is C 1 differentil 1-form on, then dω Proof. Let F (I 2 ) be s in the definition of the oriented 2-cell. Then dω F (dω) by (2) I 2 d(f ω) by (3) I 2 F ω by Lemm 1 I 2 ω by (1). ω. 2.7 Cellultions nd Green s theorem efinition 23. A smooth cellultion of the nice region is finite collection K { 1,..., k } of oriented 2-cells such tht k with ech pir of these oriented 2-cells re either disjoint, intersect in single common vertex, or intersect in single common edge on which they induce opposite orienttions. A cellulted nice region is nice region with together with cellultion K of. Exercise 1. A cellultion K of nice region induces n orienttion of. Moreover, ny other cellultion of induces the sme orienttion on. We re now redy to stte nd prove Green s theorem. Theorem 5 (Green s Theorem). If R 2 is cellulted nice region nd ω is C 1 differentil 1-form on, then dω i ω. 11
Proof. Let K { 1,..., k } be cellultion of by oriented 2-cells. We will pply Theorem 3 to ech 2-cell in K. This gives us dω dω i i ω The lst line follows from the first line becuse ll the interior edges of the cellultion cncel with ech other, since ech ppers s prt of 2 different 2-cells but with opposite (nd hence cnceling) orienttions. Proposition 2. Every nice region dmits cellultion. Proof. 1. Prove tht every nice region with one boundry component (which is thus exterior) with four pieces cn be cellulted. Hence tringle cn be cellulted. 2. Now prove tht every nice region with one boundry component cn be cellulted. o this by proving tht every such region cn be tringulted. 3. Finlly, prove tht every nice region cn be celluted by using induction on the number of boundry components. ω. Hence: Corollry 2. If R 2 is nice region nd ω is C 1 differentil 1-form on, then dω ω. Finlly, we give chrcteriztion of closed differentil 1-forms: Theorem 6. If ω P dx + Q dy is C 1 differentil 1-form defined on R 2, then the following three conditions re equivlent: () There exists function f : R 2 R such tht df ω. (b) Q/ x P/ y on R 2. (c) Given points nd b, the integrl ω is independent of the piecewise smooth pth from to b. 12
Proof. We lredy hve tht () implies (b) nd (c) by Corollry 1. It suffices to prove tht ech of (b) nd (c) implies (). Let (x, y) R 2 nd define x,y to be the stright line pth between nd (x, y). Now define f(x, y) ω. x,y Let α x be the stright line from to (x, ) nd β y the stright line from (x, ) to (x, y). Let T be the tringle in R 2 with sides x,y α x β y. Then either immeditely from (c) or by Green s Theorem 5 pplied to T if we re ssuming (b), we hve f(x, y) x,y ω ω + α x x β y ω P (t, )dt + y Q(x, t)dt. Hence f y y Q(x, t)dt Q(x, y) y by the FTC. Similrly, f x P, so df ω, s required by (). This completes the proof. 13