hapte Vecto Integal alculus I. Lne ntegals. Defnton A lne ntegal of a vecto functon F ove a cuve s F In tems of components F F F F If,, an ae functon of t, we have F F F F t t t t E.. Fn the value of the lne ntegal shown fom A to B. oluton: We ma epesent b t + t t an F t + tt F t t F, when F + an s the ccula ac as t t t t t t outhen t Tawan t t t Unvest t t t t t B, A, Vecto hapte
E.. Fn the value of the lne ntegal fs, when f + an s the lne fom, to,. oluton: We ma epesent b + + s an f + + + fs 8 6 [Eecse] Evaluate the lne ntegal pont B,, along : the staght lne t + t + t. : the paabolc ac t + t + t. [oluton] F t + t + t, + + t F t t t t 7 F t + t + t, + + tt F t t t t 7 F, wth F + + fom pont A,, to outhen Tawan Unvest Vecto hapte
. onsevatve fels A vecto fel F s sa to be consevatve f thee can be foun some scala such that F,.e., F, F, F. Then s calle a potental functon o smpl potental fo F, an F The lne ntegal fom A to B along a cuve s F B A B A Ths shows that the value of lne ntegal s smpl the ffeence of the value of at the two enponts of an s nepenent of the path. Hence the lne ntegal aoun a close cuve of a consevatve fel s eo. nce F Thee follows a fact that f an onl f F, the vecto fel F s consevatve. E.. Fn the lne ntegal of F, wth F + + fom pont A,, to pont B,, along : the staght lne t + t + t. oluton: F the lne ntegal s nepenent of the path, an the potental s an F f g + + f, g f g g outhen Tawan Unvest B F,,,, 6 A Vecto hapte
II. uface ntegals Defnton: the flu of F though the suface to be the suface ntegal F s o Fns, whee n s the unt nomal vecto to. Fom the fgue shown, s, s the angle between the nomal of s an -as. n s n F ns F n n mlal, f an ae the angle between the nomal of s an an aes espectvel, Fns Fn Fn n n When the suface s paamete b u,v PQ u, P v u v an s PQ P uv u v F s F uv u v If F n, the suface ntegal s aea of the suface tself. u, v+ v P u, v u+ u, v+v R Q u+ u,v n s E.. Gven F + an s the poton of the plane + + n the fst octant, evaluate F s. oluton: Let f + + the unt nomal vecto n of s f n f outhen Tawan Unvest F s [ F n n ] Vecto hapte
Vecto hapte E.. alculate the suface ntegal of the vecto functon F + ove the poton of the suface of the unt sphee : + + above the -plane,. oluton: Let f + +, the unt nomal vecto of s f f n n F s F n Let,, we have,, s F Let,,, then + +, F +, an F outhen Tawan Unvest
Vecto hapte III. Volume ntegal Defnton: the volume ntegal of a functon f ove the volume V s V fv In atesan coonate sstem, v. If f, the volume ntegal s volume of V. E. 6. Fn the volume ntegal of f,, + ove the bo boune b the coonate planes,,, an +. 7 6 6 7 6 ] [ oluton : E. 7. Fn the volume of the egon of space above the plane an beneath the plane + +, boune b the planes,, an the suface. 7 ] [ oluton : = outhen Tawan Unvest
IV. Dvegence theoem Gauss theoem Dvegence theoem: Let V be a close boune egon n space whose bouna s a pecewse smooth oente suface. Let F be a vecto functon that s contnuous fst patal evatve n some oman contanng V. Then F s V Fv. E. 8. Evaluate [ ] s, wth : oluton: Let f + + n F ns [ ] [ [ [ V 6,. V v ] outhen Tawan Unvest F ns F n n Fv F ns F ] ] Vecto hapte 6
[Eecses]. Evaluate ] [ s, s the lateal suface of the poton of the clne + fo whch. [oluton] Let f +, then n + F n [ ] n V Fv [ F ] F [ ]. Evaluate 7 s, wth : + +. [oluton] n + + 7 n 8 8 8 7 8 outhen 8 6 Tawan 6 Unvest 8 8 8 [6 6 ] 8 7 8 8 7 7 8 7 6 8 7 7 6 7 s 6v 6[ ] 6 V Vecto hapte 7
V. toes theoem toes theoem: Let be a pecewse smooth oente suface n space an let the bouna of be a pecewse smooth smple close cuve. Let F be a contnuous vecto functon that has contnuous fst patal evatve n a oman n space contanng. Then F s F [Note] The postve ecton along s efne as the ecton along whch an obseve, tavelng on the postve se of, woul pocee n eepng the enclose aea to hs left. E. 9. Fn the suface ntegal. oluton : n F n F s F s, wth F + + an : the paabolo +,, F, F s n F, whee : Let, outhen F Tawan Unvest F Vecto hapte 8
E.. Evaluate F, whee s the ccle +,, oente counteclocwse as seen b a peson stanng at the ogn, an F +. oluton: Let, F + + 8 F 8 As suface boune b we can tae the plane ccula s + n the plane. Then n, an F 9 7 8, F n 8 F s 8s 8 VI. Geen s theoem If F F + F s a vecto functon that s contnuousl ffeentable n a oman n the plane contanng a smpl connecte boune close egon whose bouna s a pecewse smooth smple close cuve. Then the toes theoem can be euce to F F F n F an hence F s F F F F F F outhen Tawan Unvest whch s the Geen s theoem. F F F Vecto hapte 9
E.. Evaluate, along. oluton: [ ], s a close cuve fom, to, along an bac to [ ] [ ], E.. Evaluate the lne ntegal F counteclocwse aoun the bouna of the egon R, whee F, R s the ectangle:,. oluton: On :,, F F ; On :,, F F ; On :,, F F ; On :,, F F. F F F F F. F F F. outhen Tawan Unvest Vecto hapte