VIEN CAc TR110NG CAO DANG)

NHA xuiiT BAN GIAO Dl,JC

NGUY~N aiNH TRi (Chubi;;n)

L~ TRQNG VINH - DUONG THUYvY

Giao trinh

loAN: ate eM clPT~P2

(Sach dung cho sinh vien cfIe truong Cao ddng)

NHA XUAT BAN GIAO Dl,JC

1!ai noi tauSinh vien mdi vao nifm hoc tiuinhOt cdc tnrtJng D{Jihoc; Ow ding

thuifng gifp kh6 khlfn do phU'etngphep dilY, phtro'ngphap hoc J b#ch(JC nay co nhiiu di~'ukhJc biet so vuj J ~c Trung h9C. Toen hoc ceocAp Iei 18 mot mtJn hoc kh6 v{jj thai jltt;mg Ian cua nsm tinrnhAt dcdc Iruifng Dili hf'C, Ow ding kf thu(lt; nhJm ren Iuyen ttr duy khoah9C, cung cAp cdng c~ tosn hoc dlsinh vien hoc cae mdn khoa hockjthu(ltkhac va xOy d!bllJ Mm lire dl fiep tuc nrhoc sau nay.

&) giao trinh "Toen h!JC ceo cap" nay auoc bien soan can eli vaochsrong trinh khungdii duoc bsnhenh va th'!C f{lgiJngdilyJ M caoding ctia mot st{lrU'ongDeihockf thu(lt va din cti vao dnrong trinhman Teen hin nay cUa cdc wOng Trunghoc Phdthong, nhJm girlpcho sinh vien h~ Ow ding h9C tOtman hoc nay.

Do y~u diu daD tso hin nay cue h CaDdi~ mat s6'phin aiaToen hoc cao cOp nhtr csu trtic dili so; di'flg todn phuong; tich phOnphi: thuoc tham so; tich phsn ba Jdp, tich phOn mift chu6i Fourier;...khong dur;tc due vao giao trinh nay. Nhiing khai niem Tosn hoc coMn, nhii'ng phLfdng phdp co b8n" nhilng k~t qua co bin cua cdcchuang deiJ dU''!c trinh biiy do'y d&. M(Jt'so dinh If khong diracchring 111izJ4 nhungf nghia cua nhiiN; dinh If quan trpng dU''!c giJithich ro ranEY nhfu vi du minh hOil auoc dus ra. Nhfu ring d!bllJctIa If thuyet vao tinh go'n dring dur;tc trinh biiy J dOy. Rieng voinhilng la~n thiic vlgiai tich ma sinh vien duuc hoc J Trong h9Cph{l

\ thong, giao trinh nay chinhic Jili mot cach h thong cdc dilm chinhva trinh bay cae kj~n tluic nAng cso. Phin ceu hoi 6ntfJp J cuoi m61chtro'ng nhJm girlp sinh vien hoc t4p va f'!kilm Ira ket qua hf'C t4Pcua minh. Lam nhiing biii t4p dlra J cuol m6i chtro'ngBegirlp ngu'ifihf'Chilu sOusichdn cac khai nim Tmin hf'C, ren Juy~nkf niing tinhtoan va kha nang v~ dlj11gcae khiini(}m Jr. Cae hai tfJp d6 se dltficgiJi Irong b(J biii t4p kem thea b(J giao trinh nay"

3

B giao trinh nay dl1'f,fc viet thanh hai t{lp va l.i c6ng trinh t{lp thlcus ba nhe giao : Nguy~n Dinh Tri (chu bi~nJ Le TT9ng Vinh vaDuong lluly Vf. 6ng L~ Trang Vinh vi~t cdcchuang L IL n-; l-; (JngDU'dng lluly Vf vi~t cdc chU'O'ng IlL VL VIIL IX; (JngNguy~n DinhTri via cdc chuang Vli: X Xl

Khi xAy d!hW d~'

MljCLljC

Trang, 6 LOIN IDAU 3

Chrnm~ VIIHAM 86 NHIEU BIEN 86

I. Khai niem rna dau 72. Dao ham rieng. Vi phan toan phan 183. Dao ham cue ham s6 hop, Dao ham cua ham s6 an 284. Dao ham thea huang. vectc gradien 325. O,rc tri 386. val eng dung cua phep tfnh vi phan trong hlnh hQC 44Cau hoi Oil t~p 50Bai t~p 52Dap s6 )9

Chuang VlllTiCH PHAN KEP

I. Biti loan dan dOn khai niem tfch phan kep :1M tfch vat tM hinh tru eong , 66

2. Dinh nghia tfch phan kep : 673. Ole tlnh chat cua tieh phan kep 684. Cach tfnh tich phan kep trong he tea d De-cac 705. Cach tinh tlch phan kep trong h~ tea d cue 78

6. Dng dung hinh hoc cua tlch phan kep 86Cau hoi On tap 93Bai l~p 94Dap s6 97

ChucmgIXrica PHAN DOONG

.I. Bai roan dan den khai niem tfch phan ducng :GOng cue mot luc bien d6i 99

2. Dinh nghia tfch phan duong 1013. Cach tlnh tlch phan duong 102

5

4. Cong thuc Green 1075. Dieu kien de tich phan dtrcng khong phu rhuoc dtIang cong

lliy tich phan 109

6. Ullg dung cua tich phan dirong 1147. rich pharr dtIang trong khcng gian ; 116CAu hoi on t~p 118Bititap 119Dap s6 123

Chuun!!XCHU6I

1. Dai ctrcng ve chuei s6 1242. Chuoi s6 duong 1283. Chuoi co s6 hang vOi dliu blit ky 1354. Chuai1uythua 140CAu hoi on ~p 154Bai tap 156Dap sc 160

Chuang XlPHWNG rRl:NH VI PHAN

1. Dai cirong ve phirong trlnh vi phan 1632. Phuong trlnh vi phan clip mot 1643. Phuong trlnh vi phan clip hai tuyen tfnh 183CAu hoi on tap 201Bai~p 203Dap s6 207

rAI LI~U TRAM KHAo 211

6

Chlldng VII

HAM s6 NHIEU BIEN s6

a'Iil\U! ctlaf. !JEu cd...- Trinh bo~y nhung khai ni~m co ban va k8t qua co ban vB' phep tinh vi

ph~n cua ham 68 nhia'u bign 65: dfnh nghl:a ham 68 nhiSu biBM s8.mi2'n xac djnh, each biiu dian hlrth hQC, gibi han va tfnh li8n tuc cuaham s8 nhilu biBn 68, cl

Hlnh 7.1

1.2. Mii!n xiic d!nh

Neu ngirci ta eho ham s6 hai bien s6 bili biEu thuc z = f'(x, y) rna khongn6i gl ve mien xac dinh cua no thi mien xac dinh cua ham s6 do ducc hieu littap hop nhilng cap (x, y) sao eho biEu tlnrc f(x, y) e6 nghia. Neu xem (x, y)lit tea d cua diem M trong mar phang thi mien xac dinh cua ham s6 f(x, y)

Ia tap hop nhilng diem M sao eho bieu thee f(M) e6 nghia. D6 tlunmg Ia met~p hop D lien thong trong lR?, nrc III mot tap hop rna hai diem M I , M2 ba'tky ella n6 luon co the n61 vai nhau boi mot ducng cong lien rue nam hoantoan trong D (hinh 7.1). Trir tnrcng hop mien xac dinh D = 1Il.', D thucngduoc giai han bCri met ducng cong L kin hay khong. Mien D ducc goi lit mdneu nhilng diem cue bien L deu khong thuoc D, Ia dong neu moi diem cuebien L dell thuoc D.

Mien lien thong D duoc gci Ia danlien neu no hi gioi han hOi mot ducngcong kin, da lien neu n6 bj gi61 hanbci nhieu duong cong kin roi nhau ttrngdoi mot.

Vi du 1. Ham s6 z = Zx - 3y + 5xac dinh vex, y) E 1Il.2, mien xac dinhella no Ia toan Ix! mat phang.

Vi du 2. Ham s6 z = Jrl -_- X-;2;-_- y"'2 xac dinh khi 1 - x2 - l ~ 0 hayx2 + y' ,; 1. Mien xac dinh ella noIa hlnh tron d6ng, tarn 0, ban kfnh 1 (hinh 7.2).

Vi du 3. Ham s6 z = In(x + Y-I) xac djnh khi x + Y-I > 0 hay x + Y> 1.Mien xac dinh ella no IA DlIa m~t phjng rna nam aphfa tren ducng thangx + y = 1 (hlnh 7.3).

.~,

8

y .

Hinh 7.2

1

y

o1

Hinh 7.3

x

Y

Y

B(O,2,0)

M(x, ,0)

C(O,O,3)

D

s , P(x, .f(x,Y)),,,

z

Hl'nh7A

,

i,i,'0J. _

o

z

A(1,0,0)x

x

Sau nay cec khai niem se ducc trlnh bay eho tnrcng hop n = 2 hay n = 3.Cac khai niem ay cling duqc rna rong cho trucng hop n nguyen duang bAt ky.

1.3. Biiiu dilln hlnh h9C cua ham so hal bien so

Gi. sit z = ftx, y) lA ham s6 xac dinhtrong mien 0 ella m~t phang xOy. Quadiem M(x, y. 0) trong mien D, dungdutmg thang song song va cung hu6ng vOi

true Oz va lay diem P tren do sao choMP = f(x, y) = z. Khi diem M bien thientrong mien D thl ditm P bien thien trong IR3

va sinh ra met mat S nao do. goi la d6 thicu. ham 56z =f(x, y). T. n6i rang z =f(x, y)la plurong trlnh cue mat S. Moi duong thang

song song vOi true Oz eiit m~t S akh6ng quamot diem (hlnh 7.4).

Vi du 4. Ham s6 z = 3 - 3x - ~Y xac

dinh tren toan mat phang xOy. D6 thi ella nola m~t phang cat ba true tea dQ theo thu tl;l aba diem A(l, 0, 0), B(O, 2, 0), oo, 0, 3).Phdn cua d6 thi ella ham so do nam trongg6c phan tamthll nhai (hlnh 7.5).

Hinh 7.5

1.4, M~t b~c hai

M~t b~c hai Ia nhiing mat rna phuong trinh cua chung la bac hai del vOix, y, z.

1.4.1. Mijt elipx6it

M~t elipxoit Ia m~t eo phuong trlnh

x2 y2 z2-+-+-~l

a 2 b2 e2(7.1)

trong do a. b, c la nhiing s6 duong. VI x, y, z c6 mat trong phuong trinh (7.1)voi so mil chan nen mat elipxoit nhan cac mat phang tea d lam mat phangd6i xitng, nhan g6c 0 lam tam d6i xirng.

9

alt mat elipxcit hOi cac m~t phang tea d xOy, yOz, zOx, cac giaotuyen theo thli'tg la cac ducng elip :

.x2 yZ-+-=1 z=O;aZ bZ '

y2 z22 + 2'" = 1, x == 0 ;b e

z2 XZ-+-=1 y e O.CZ aZ '

Cat mat elipxcit boi mat phang z == k, k 18 hang 00, glao tuyen cophuong trlnh

y

(7.2)

z

Hinh 7.6

x

D6 la phuong trlnh cue duang elip cotam tai di~m (0.0, k). co cac ban true Iii

r-kl r-klaVI-~, bVI-~.

XZ y2 k2-+-=I--,z=k.a2 bZ c2

Neu k < -;; hoac k > c, phucng trinh (7.2) vo nghiem, m~t phang z = kkhong cat mar elipxoit.

Neu k = e, giao tuyen thu vI! diem (0, 0, c).

Neu -c < k < c, phuong trinh (7.2) co the viet

x2 2

a2(1-::r b2(-::r l,Z=k.

Khi k tang til den e, cdc ban true nho dan tm 0. Khl k tang til -; den c,giac tuyen di chuydn va sinh ra mi;it elipxcit (hinh 7.6). a. b, c goi Ia cac bantrue cua elipxoit.

Neu hal trong ba ban true bang nhau, chang han a == c, ta co mat elipxeit2 2

tron xoay, sinh bel duong elip ;-+.;. =1, z =0 quay quanh true Oz. Neua b

a = b = c, mat elipxnit tra thanh mi;it cau tam 0 ban kfnh a.

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(7.4)

(7.3)

z

,,, .-,--,..,..-+_9~-:'_' (O,b,O)

, y,,,,~.---i-----_.,,,

1.4.2. Mijt hypebiil6it m9t tdng

Do III m~l bac hai co phuong trlnhxl y2 z2-+----1a2 b2 c2 -

trong d6 a, b, e 1. nbilng h~g s6 duong, M~t d6 nhan cac m~t pMng tea dlam m~t phang d6i xrmg. nhan g6c toa d lam tarn d6i xrrng. N6 cat matphang toa d xOy theo ducng clip

x2 y22+2=I,z=0,a b

cat cac mat phang toa dO yOz, zOx Ilin hrct thee cac dutmg hypebon :

y2 z22-""2 = I, x=O,b e

x2 Z22-"2 = 1, y e D.a e

Giao tuyen cua mat hypecbolcit vcri mat phang z = k ia dueng elip

x2 y2 k22+2=1+ 2,z=k.abc

Khi Ikl tang tir 0 den +00, ceo ban true ellaelip do theo tha nr tang tll: a den +00 va tir bden +00. Khi k bien thien tu -co den +00 giaotuyen do dich chuyen va sinh fa m~tbypebolcit mot tang (hlnh 7.7).

Nu a ~ b, ta e6 mat hypeboloit mtx2 Z2

tang trcn xoay, do hypebOn 2 - """2 = 1a e

y = 0 quay quanh true Oz sinh ra.Htnh 7.7

1.4.3. M~t hypeb6l6it hai tang

D6 I. m~t bac hal e6 phtrcng trinhxl y2 z2-+-----1 (7.5)a2 b2 c2 - ,

trong do a, b, C III nhUng hang s6 dirong. No nhan cac mat phang to"," dQ Hunmat phang d6i xung, nhan gee tea d lam tarn d6i ximg.