1 Line Integrals in Plane.

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1 MA213 thye Brief Notes on hpter Line Integrls in Plne. 1.1 Introduction urves. A piece of smooth curve is ssumed to be given by vector vlued position function P (t) (or r(t) ) s the prmeter t moves in n intervl I on the rel line. The smoothness mens the function is differentible nd the derivtive P (t) = v(t) = r (t) is non zero on I. If I is closed, then the curve hs well defined strting point nd n end point. A curve is sid to be simple if different vlues of t give different points P (t), except the beginning point is llowed to be the sme s the end point. For exmple, consider the curve: P (t) =< cos(t), sin(t) > for t [, b]. epending on the vlues of, b, this curve trces prts of the unit circle nd hs the strting nd end points s P () nd P (b). If [, b] = [0, π], then we get the upper semicircle. If [, b] = [0, 2π] then we get the complete circle where the strting nd end point re the sme. This is n exmple of simple closed curve, where ech vlue of t gives different point of the curve, except for the strt nd end. If we tke [, b] = [0, 4π], then this is not simple curve nd ech point is visited twice! The curve given by the formul P (t) =< cos(2π t), sin(2π t) > describes the sme set s, but trced in reverse. We denote such curve by or curve trced bckwrds. A curve in generl is composed by smooth pieces joined t one point. Given two curves 1, 2 such tht the end point of 1 is equl to the strt of 2, we define the curve s the curve obtined by trcing 1 followed by 2. We cn dd severl pieces s desired. By we men + ( ) i.e. the trvel on followed by return trip! Given piece of curve, we my describe it by different prmetriztion nd we will consider it s the sme piece of the curve if the points re in one to one correspondence. 1 For exmple, our circle is lso described by the prmetriztion P (2t) where t [0, π] Integrl on curves. Given piece of plne curve : r(t) =< u(t), v(t) > with t [, b] nd function f(x, y), we define integrl of f(x, y) on to be: f(x, y)ds = b f(u(t), v(t)) u (t) 2 + v (t) 2 dt. It is not hrd to show tht this integrl is independent of the choice of prmetriztion. If we hve vector field F (x, y) =< F 1 (x, y), F 2 (x, y) >, then integrl of F(x, y) on to be: F T ds = F (u(t), v(t)) T (t) u (t) 2 + v (t) 2 dt. 1 If our prmetriztion runs through our geometric curve multiple times, then the new prmetriztion hs to run the sme number of times. Usully, we void such techniclities nd use simple curves. 1

2 If we recll tht T (t) = <u (t),v (t)>, then it is esy to see tht u (t) 2 +v (t) 2 dt F T ds = (F 1 (u(t), v(t))u (t) + F 2 (u(t), v(t))v (t)) dt. This is clerly equl to F dr nd independent of prmetriztion. More generlly, we cn define the integrl of ny differentil P (x, y)dx + Q(x, y)dy (lso briefly written s P dx + Qdy ) on prmetric curve by nd < P, Q > < dx, dy >= P dx + Qdy = b For spce curve, there re obvious generliztions: P dx+qdy+rdz = F T ds = f(x, y, z)ds = b b 1.2 Fundmentl Theorem (P (u(t), v(t))u (t) + Q(u(t), v(t))v (t)) dt. f(u(t), v(t), w(t)) u (t) 2 + v (t) 2 + w (t) 2 dt (P (u(t), v(t), w(t))u (t) + Q(u(t), v(t), w(t))v (t) + R(u(t), v(t), w(t))w (t)) dt. (F 1 (u(t), v(t), w(t))u (t) + F 2 (u(t), v(t), w(t))v (t) + F 3 (u(t), v(t), w(t))w (t)) dt. If we hve vector field F = (f) for some function f, then we see tht F dr = b (f x (u(t), v(t))u (t) + f y (u(t), v(t))v (t)) dt = b d(f(u(t), v(t)) = f(u(b), v(b)) f(u(), v()). Thus, the integrl depends only on the vlues of f(x, y) t the two endpoints of. In prticulr, we get the sme integrl if we replce the curve by ny other curve which hs the sme strting nd ending point. We define vector fields of the form (f) to be conservtive. If F denotes the force field cting on prticle moving on curve from r() to r(b), then the work done is known to be given by F dr. Thus, if F = (f), then the work does not depend on the pth nd evlutes to f(r(b)) f(r()). This is the reson for the term conservtive. The function f is clled the potentil function of the field. Given field < P, Q >, when is it conservtive? If P, Q hve continuous derivtives in domin, then by lirut s theorem, we get P y = f xy = f yx = Q x s necessry condition. This my not be sufficient unless our domin of definition is nice. The technicl term is simply connected nd roughly mens it hs no holes. A stndrd exmple of non conservtive field which stisfies the necessry condition is given by F =< P, Q >=< y x 2 + y, x 2 x 2 + y >. 2 It cn be redily shown tht F dr = 2π if we let the unit circle r(t) =< cos(t), sin(t) > s t vries in [0, 2π]. ince the nswer is not zero even though the strting nd ending points re the sme, it is not conservtive. The problem is tht the vector field is not defined t the origin which is enclosed by the curve. 2

3 1.3 Green s Theorem One of the most importnt nd useful theorem bout line integrls is: Green s Theorem uppose tht is piecewise smooth, simple closed curved trced counterclockwise (positively oriented) nd is the enclosed region. If the functions P, Q hve continuous prtil derivtives on n open region contining, then P dx + Qdy = (Q x P y ) da. omments: One wy to formlly define positive orienttion is to require tht the unit norml N points towrds the region t ll points (where it exists). Note tht continuous differentibility is ssumed in slightly bigger region thn itself. Thus, the functions hve to be well behved in s well s in neighborhood of. ometimes the integrl round the positively oriented is denoted s. It is possible to define formlism whereby the oriented boundry cn be thought of s derivtive of the region. Moreover, it is possible to set up formlism of derivtives of differentils which give d(p dx + Qdy) = P y dx dy + Q x dx dy. Then the theorem cn be reworded s d(p dx + Qdy) = P dx + Qdy. The fundmentl theorem of lculus cn lso be rephrsed in the sme lnguge d(f) = f = f(b) f() I where we interpret the oriented boundry of n intervl I = [, b] s [b] []. Lter on we will see two more theorems with the sme philosophy Uses of Green s Theorem. Mny results cn be deduced using the theorem. I 1. If P dx + Qdy is closed, mening Q x P y = 0, then P dx + Qdy = 0. Thus, it proves the result bout the conservtive vector fields without finding the potentil function. 2. ydx = xdy = 1 (xdy ydx) = the re of the enclosed region 2 For exmple the re inside the ellipse r(t) =< cos(t), b sin(t) > is esily found by 1 ydx) = 2 (xdy 1 cos(t)(b cos(t) b sin(t)( sin(t)) dt = 1 2π bdt = πb Note how the third formul mde the work esier! 3

4 3. hnging the pth without chnging the integrl. uppose tht we hve P y = Q x in region round the origin. Then for ny simple closed positively oriented piecewise smooth curve going round the origin the integrl P dx + Qdy is the sme. The ide of the proof is to consider some smll circle totlly contined inside nd pply the Green s theorem to the region between the two curves, using mild generliztion of the theorem. 2 Line Integrls in pce. If we consider curve in spce lying over surfce, then it cn enclose region of the surfce. We now set up nottion for line integrls on such curves nd get generliztion of Green s Theorem known s tokes Theorem. 2.1 url nd iv. We wish to know whether three dimensionl vector field F =< P, Q, R > is conservtive, i.e. F = (f) for some f. We define vector opertion curl F =< R y Q z, P z R x, Q x P y >. We note tht this cn be formlly understood s cross product F, where is interpreted s vector opertor <,, >. x y z It is redily checked tht curl (f) = 0 by lirut s theorem in three dimensions. Thus, curl F = 0 is necessry condition for F to be conservtive field. In generl, it my not be sufficient, unless our field F is defined with continuous prtil derivtives in simply connected region. Another useful function is divergence, defined for vector field F =< P, Q, R > s div F = F = P x + Q y + R z. This is clled the divergence of the field F. It is esily checked (ssuming continuous prtil derivtives) tht F = div curl F = 0. Thus, div G = 0 is necessry condition for field G to be of the form curl F. 2 The mening of the url nd iv If F =< P, Q, R > describes the vector field of the velocity of fluid flow, then the field curl F denotes the xis of rottion of the fluid t given point nd the sclr quntity div F mesures the tendency to move wy from (diverge) the point. This ide cn be visulized by considering the simplest exmples of F. If F =< x, 0, 0 > then it is esy to see tht the flow from ny point (, b, c) is strictly long the x-xis, in the positive or negtive direction depending on the sign of. The fct tht div F = 1 expresses this nture. If we fix little sphere round point, then it cn be rgued by suitble integrtion tht there is net fluid movement in the positive x-direction. curl F = 0 nd there is no rottion. If F =< y, 0, 0 >, then curl F =< 0, 0, 1 > suggesting rottion in plne prllel to the xy plne in the clockwise direction. This rottion is cused by the fluid moving fster with the incresing vlue of y. There is simple wy to mesure this. hoose clockwise pth round point, sy (0, 1, 0) in this plne by r(t) =< cos(t), 1 sin(t), 0 > s t goes from 0 to 2π. Integrte our field long this pth, i.e. compute 2π 0 F dr = 2π 0 (1 sin(t))( sin(t))dt = π. 2 As before, we my sk for the converse, but it leds to more complicted notions of contrctible spces. This is beyond the scope of our course. 4

5 This integrl is clled the circultion of the fluid in our field. Along counterclockwise field, this would hve come out π nd this cn be shown to be the vlue of the curl F times the re enclosed by our pth. The divergence div F = 0, s the net fluid movement cross little sphere is zero cross ll lines prllel to the x-xis. A similr nlysis cn be done for F =< z, 0, 0 > to get curl F =< 0, 1, 0 > indicting rottion bout the y-xis. The divergence is zero for similr reson. Another useful function is the Lplcin ( or shortened to 2 ) which is defined s div ( (f)) = 2 (f) = < f x, f y, f z >= f xx + f yy + f zz. Even though we hve used three vrible definition, the definition nturlly mkes sense for ny number of vribles. A function f is sid to be hrmonic if 2 (f) = 0. 3 urfce Integrls. If surfce is given in prmetric form r =< x(u, v), y(u, v), z(u, v) > where (u, v), we define the integrl of function f(x, y, z) on by the formul: f(x, y, z) d = f(r(u, v)) r u r v da. However, surfces don t usully come with redy prmetriztion nd it is useful to hve more generl formul hndy. Let the surfce be described by n eqution g(x, y, z) = 0, so tht we hve the bsic reltion g x dx + g y dy + g z dz = 0. It is possible to think of x, y s prmeters t point where g z 0. In this cse, tking u = x, v = y, we see tht r u r v =< 1, 0, g x > < 0, 1, g y >=< g x, g y, 1 >= 1 (g). g z g z g z g z g z Thus, our integrl reduces to f(x, y, z) d = f (g) dx dy g z. It is esy to see tht similr formul holds when g y 0 nd we tke x, z s prmeters nd when g x 0 we tke y, z s prmeters. We define the Fundmentl differentil on to be: ω g = dx dy g z Now we hve single formul for ll cses: f(x, y, z) d = = dy dz g x = dz dx g y. f (g) ω g. Orientble urfce. We note tht (g) defines norml to our surfce t ll smooth points (points where (g) is defined nd non zero.) Also, it vries continuously if g is continuous. Then such surfce hs well defined unit norml vector field t ll smooth points, nmely n = (g) (g). 5

6 If our surfce encloses bounded solid (like sphere, then we cn even mke sense out of n outwrd or inwrd norml. We now define the surfce integrl of vector field F on the surfce by defining it s the integrl of the function F n = F (g). Our formul becomes (g) F n (g) ω g = F (g) ω g. For prmetric surfce, the formul becomes: F (r(u, v)) r u r v da. Note tht the bove formul presumes certin direction of the unit norml s determined by (g) or r u r v. We multiply by 1 if we wish to chnge the direction. 4 tokes Theorem. Green s theorem equtes the line integrl round plne curve with the integrl on the enclosed region, which is flt surfce. tokes theorem gives similr reltion for spce curve bounding surfce. Indeed, the Green s theorem is but specil cse of tokes theorem. The theorem sttes: F dr = curl F n d where is n oriented surfce enclosed by the curve nd we use the induced orienttion on. This is best imgined s follows. Assume tht you re wlking on the curve such tht the chosen norml points overhed. Then the curve is positively oriented, if the surfce lies on your left hnd side! Let us further note tht if g(x, u, z) = 0 is the eqution of the surfce, then curl F n d = curl F g ω g s explined bove. The proof of tokes theorem is usully deduced by cutting up the surfce into smll enough pieces so tht ech piece cn be rrnged to be prmetrized by (x, y) or (y, z) or (z, x). Then the proof reduces to plne integrl nd works s in the Green s theorem. The Green s theorem itself is lso estblished by cutting up the plne region into pieces of type I or II (i.e. where the double integrtion is done by verticl or horizontl sections between prllel lines.) One of the most useful consequences of the tokes Theorem is tht it sys tht if two surfces shre common oriented boundry curve, then the integrl of vector field of the form curl F is the sme on both of them. Then, integrls of such vector fields cn be conveniently evluted by chnging the surfce to something more convenient. Exmple. onsider the surfce : z = x 2 + y 2 1 below the (x, y)-plne. Let G =< 1, 1, 1 > nd consider G n d It is esily seen tht G = curl < z, x, y > the surfce integrl cn be esily evluted s line integrl long the boundry curve :< cos(t), sin(t), 0 > s t goes from 0 to 2π nd evlutes to π. Now the surfce cn be mde lot more complicted, sy z(1 + x 4 + x 2 y 2 + y 4 ) = x 2 + y 2 1 nd the corresponding surfce integrl is quite messy! But we know its vlue lredy. One of the trivil but useful consequences is tht the integrl of ny field of the form curl F on closed orientble surfce is zero. Thus, the integrl of constnt vector field on such surfce is seen to be zero! Edited April 13,