Week 10: Line Integrls Introduction In this finl week we return to prmetrised curves nd consider integrtion long such curves. We lredy sw this in Week 2 when we integrted long curve to find its length. Here we generlise this with prticulr emphsis on integrting over vector fields. 10.1 Line Integrls The bsic line integrl cn be motivted s follows. Given n intervl [, b] nd function f(x) which is positive over the intervl, b f(x)dx is the re under the grph y = f(x). Intuitively one understnds tht f(x) dx is the re of tll skinny rectngle of height f(x) nd width dx nd b dd these up for x s in the intervl [, b]. mens Why restrict ourselves to just integrting long stright lines? We known how to work with curves so let us generlise nd consider curve in the plne nd function f(x, y) tht is positive in some region contining the curve. A surfce is formed by f over the curve. Think of curtin hnging down from f to the curve. We wnt to compute the re of this curtin by integrtion. The formul for the integrl is esy once one reclls the formuls from Week 2. Recll the length of curve is given by l() = ds where ds is the infinitesiml rc length, or distnce, long the curve. Thus to find the re of the curtin formed from f over, we simply multiply the height f times the infinitesiml rc length ds nd integrte over the curve f ds In prctice, such n integrl is evluted by prmetrising the curve. Given prmetristion of the curve r(t), t [, b], the infinitesiml rc length
Week 10 108 ds cn be expressed in terms of the infinitesiml chnge dt vi ds = r (t) dt so tht f ds = b f(r(t)) r (t) dt It is not necessry to restrict to positive functions, nor does the method depend on dimension. The reltionship ds = r (t) dt holds in ny dimension. Thus we cn go directly to the generl formul. Given f : U R n R nd r : [, b] R n prmetristion of curve lying in U, the line integrl of f long curve is given by f ds = b f(r(t)) r (t) dt The comments from Week 2 pply here: the curve my be piecewise regulr nd one must prmetrise the curve in sensible wy. Note tht in our definition of the line integrl f is function defined on region of R n. It lso hppens tht one my hve f defined only on the curve. For exmple, f might represent the liner density (mss per unit length) of wire (the curve). Therefore f only hs mening on the curve. 10.2 Line Integrls for Vector Fields Given vector field F, it frequently occurs tht one wnts to compute line integrl where the function f is f = F T where T is the unit tngent vector to the curve. Exmples of this type of integrtion re work nd circultion discussed below. Hence we need to evlute F T ds To derive useful formul for such n integrl we recll (Week 3) tht T = r r
109 MA134 Geometry nd Motion Thus we cn write F T ds = F r r r (t) dt = F r dt The right-most expression is wht we will use in prctice to evlute this type of line integrl. However, to stress the independence of the line integrl of the prmetristion corresponding to chosen orienttion, it is common to write r dt s dr. Let F be vector field defined in some region of R n, nd let r : [, b] R n be prmetristion of curve in this region, the line integrl of F long is F T ds = F dr = b F(r(t)) r (t) dt One importnt feture of line integrls of vector fields is tht they re not independent of the orienttion of the curve. The reson is tht is tht if one reverses the orienttion of curve, then the tngent vector chnges sign. Denoting s the curve with the opposite orienttion, then F T ds = F T ds 10.3 Fundmentl Theorem of Line Integrls When we introduced vector fields it ws noted tht n importnt clss of vector fields ws tht obtined s the grdient of function of severl vribles: F = f. Such vector fields re clled conservtive vector fields. They re importnt becuse they rise in prctice nd becuse the following holds Fundmentl Theorem of Line Integrls (FTLI). Let be regulr curve prmetrised by r : [, b] R n nd let f be differentible function whose grdient vector is continuous on, then f dr = f(r(b)) f(r()) Note the nlogy with the Fundmentl Theorem of lculus (FT) b F (t)dt = F (b) F ()
Week 10 110 (See Week 2 notes). Proving the FTLI is not difficult s it primrily relies on the hin Rule (Week 4) nd the FT. The mnipultions re f dr = b = b f(r(t)) r (t)dt d f(r(t))dt = f(r(b)) f(r()) dt The FTLI tells use tht if we know our vector field F is conservtive vector field, nd hence given by grdient of some function f, then we cn evlute ny line integrl of F over simply by evluting f t the end points of. ll these points r nd r b. The importnce is not just tht it simplifies our clcultions, but the fct tht since the integrl depends only on the end points, it in fct must be the sme for ny curve tht strts r nd ends t r b. Tht is, if F = f, then F dr = F dr, 1 2 for ny two 1 nd 2 tht strt t r nd end t r b. The line integrl is sid to by pth independent. Note in prticulr tht if F = f then F dr = 0 for ny closed curve becuse r = r b for closed curve. With some mild conditions, it cn be shown tht if ll line integrls of vector field F re pth independent, or equivlently if the line integrl round ll closed curves is zero, then F is conservtive vector field nd there is function f such tht F = f. The converse is generlly esier, lthough perhps less importnt. If the line integrl of F round close pth is not zero, then F is definitely not conservtive vector field nd it cnnot be expressed s the grdient of function. 10.4 Work nd Potentil Energy Work is n importnt physicl concept tht you cn lern ll bout in mechnics module. It is clssic exmple of cse where one needs to do line integrls of vector field. If force F(r) cts on point prticle nd the prticle moves from position r to r b long curve
111 MA134 Geometry nd Motion, then the work W b done by the force on the prticle is W b = F dr This definition is independent of whether or not the force F(r) is conservtive vector field. In mny situtions (grvittionl fields for exmple), but not ll, the force F(r) is conservtive vector field. It cn be thus be written s grdient of function. One typiclly defines the function so tht F = V where V is potentil, nd more specificlly in this cse, V is potentil energy. The work done in moving from r to r b is given in terms of the potentil t the end points W b = ( V (r b )) ( V (r )) = V (r ) V (r b ) independently of how the prticle moved from r to r b. You should visulise the potentil V s the height of hill, or more generl lndscpe. Assume tht V (r ) > V (r b ). This mens the prticle strts out some high point nd moves to some lower point. The work done on the prticle is W b = V (r ) V (r b ) > 0, independently of the pth followed from r to r b. W b is the energy tht cn be extrcted from the prticle s it moves downhill from r to r b. ontrrily, if prticle strts t r b then one must expend energy to push it uphill to r. We must input energy equl to W b, All work (or energy) differences re encoded in the potentil V nd re independent of the pth tken by the prticle. Informlly, the force conserves mechnicl energy by converting work to potentil energy, nd bck. Grvittionl nd oulomb forces re two exmples of conservtive forces tht re frequently described in terms of potentils. 10.5 ircultion For mny vector fields, line integrls round closed curves hve physicl significnce. In fluid dynmics, for exmple, such integrls give wht is known s the circultion of the fluid round the curve. In electricity nd mgnetism, such integrls pper in the integrl sttement of Mxwell s equtions nd correspond to circultion of electric or mgnetic fields. We will focus of the fluids cse. Let v be vector field corresponding to the velocity of fluid in some
Week 10 112 region of spce (or could be confined to plne). Then the circultion Γ of v over closed curve is Γ = v dr Intuitively this integrl corresponds to the net mount the fluid is circulting round the curve. Knowing the circultion round body such s wing or spinning bll, one cn clculte the lift force on the body. In the cse of wing, the lift force is wht holds the eroplne up. In the cse of spinning bll, the lift force gives rise to deflection, or bending, of its pth through the ir. 10.6 Reltionship between Vrious Line nd Surfce Integrls First consider integrtion of sclr functions. Line integrtion of sclr functions nd surfce integrtion of sclr function re similr concepts. Surfce integrtion cn be thought of s the extension of line integrtion, just s double integrtion (Week 6) is n extension of single vrible integrtion. On the other hnd, line integrls of vector functions nd flux integrls through surfce correspond to very different concepts. For line integrls of vector functions, the integrnd is the dot product between the vector field F(r) n the tngent T to the curve. The integrl mesures totl component of F(r) in direction of the curve. Such integrls mke perfectly good sense in ny dimension lrger thn one. (Dimension n = 1 cn even be included where vectors just become sclrs.) A curve through R n mkes sense s does vector field in R n, for ny n. The physicl resons for computing the component of F long the curve do not depend on dimension. In ny dimension F dr gives the net mount by which F points in the direction of the tngent to the curve. For flux integrls, the integrnd is the dot product between the vector field n the unit norml n to the surfce. The integrl mesures totl component of F crossing the surfce. The relevnt physicl ide is sometime trversing or pssing through the surfce (whether nything ctully crosses the surfce of not). The surfce must be of dimension one less thn the dimension of the spce, i.e. we consider two-dimensionl surfce in R 3. In fct, it
113 MA134 Geometry nd Motion is perfectly sensible to define flux integrls for two dimensionl vectors field F : R 2 R 2. In this cse the flux is through curve ( one-dimensionl surfce in R 2 ) nd we hve ± (F n)ds where n is the principl norml to. The sign will be determined by context. Finlly, we end with look hed to Vector Anlysis. In tht module you will tke this further nd lern tht there re deep reltionships between certin line, surfce nd volume integrls. For exmple, F dr = curl F n ds nd S to be explined... S F n ds = Ω div F dv