Midterm 3 Notes PDF

Ch16 1. Vector Fields a vector field F assigns to each (X y) =IP , a 2-D rector...

Ch16 1. Vector Fields a vector field F assigns to each (X y) =IP , a 2-D rector : F(x , y) EIR2 F(x , y) = P(X , y)i + Q(x , u)] (x y z) EIR3 , , - F(x y z)EIR3, , F(x , y , z) = P(x , y , z)u + Q(x , y , z)j + R(x , y , z)k F(x , y) is the direction of arrow NOT the terminal example : F(x , y) = ( y - , x) X T ⑨ (x , y) F(x , y) = ( - y , x) ⑧ ⑨ (1 , 0 (0 , 1) A A & G (0 , 1) ( - 1 , 0)1 ( - 1 , 0) (0 , - 1)b 10 -1). (1 , 0) - (P Q) example : X =2 , F(x , y) = A i C > - - - y = - π = 1(y x) · G , (P , 2(X - 2 , x + 3) = = 40 = 0 3 siny) (1 , 4(y Yx) y = - i , > - 8 - - - & g XCO y PTOQO >0 - A * A x70 , yo -P P & Q1 example : F(x , y) = (p Q) , = (x , - y) > 0 xcoy x70y > 0 P30Q0 P > OQLO A L N ↑ Xcoyco X > Oy0 P10Q0 P > 0Q > 0  y) & example : F(x , = (x , - 2) C - x , - 1) X0 X > 0 XL0 X >0 P10Q0 POQ0 PT0QLO PLO QLO 2 IL Gradient Field Vector f(x , y) of = (fx fy) , = fx(x y(x , + fy(x y)] ,  Ch 16 2 Line. Integrals wrt : w/respect to... f(x , y) c - C : r(t) = (x(t) y(t) , at + = D ⑨ Sf(x(t) , y(t)(dt = S , fat S f(x y)d, , as : wrt arc length parameter Sof(x(t) , y(t)) (x'(t))2 (y'(t)) 2dt + Scf(x y(dx , + g(x y(dy , = SYf(x y(x'(t) , + g(x y(y'(t)] , example : C right : half x2 + y = 4 Counterclockwise SoX as -X = 2 cost 2 Sint y = v = 2 7 4 = 2 cost (sint 12cst)2 + = Syst 4 sin = = 4 - ( -4) = example Syax : 02X4 IEX *2 (1 1) So xax S, ( X , E = y + x 2)dx = Syax C -(2X y 2 + S - + = yax + , 10 ,0 (2 0 , f(x , y , z) C r(t) (x (t) : =. y(t) , z (t) a = + b Scf(x , y , z)ds = Saf(x(t) , y(t) , z(t))(x)) + (y)2 (z)2at + SP(X , y , z)dX + Q(X , y , z)dy R(x + , y , z)dz S[P(X , y , z)x'(t) + Q(x y , , z)y'(t) + 2(x y , , z)z'(t)] example S xyzds : TO , C : line segment (0 , 0 , 0) - (2. 3. 4) X = It P + tQox + = 3t ↑ y = Q z = 4t ⑧ P S0(2t)(3t)(4t) 2 = 3 2 + 42dt = 24299. t at = * (24)(29)(i) = 629 0) (2 3) example : C : (1 0 , , , 2 , , 20 , 30 S & zdX (1 0, 8) t() + 2 3)x = t + 1 + Xay + yaz , , , 2t y = , z = 3t So[3t + (t 1)(z) + + (2t)(3)]at So'(llt + 2]dt = + 2 = example : line integral of vector field ScFpar = SaCEsrCt)or'(t)) at dot product F(r(t)) = (t3, t5, t") y dot product F = (Xy , yz , xz) ri(t) = (1 2t , 3t2) , integral C % +3 27 C r(t) : = + 2t 3tdt + = 28  example : SF. ar c : arc of circle X + y = 1 from 11 0) , to 10 , 1) , 1) 10 , (3 5) (5 3) T = + + = 0 - (1 , 0 - example : F(x , y) (ye" siny)i (e* = + + + xcosy + 2y)] findf such that of = F PYysin + 2y f(x y) , = Sye + Sinyax = ye + xsiny + g(y) fy(X y) (ye* siny g(y))y , = + + ex xcoSy g'(y) = + + = ex + xcosy + 2y g'(y) 24 = g(y) = yz + ktakek = 0 : f(x , y) = ye" siny + + y ** ** example : F(x , y , z) yze i e j + (xye = + ** + 1)k find f(x y z) Such that If , , = F fx(X y) , = yzexz f(x , y , z) = Syze ** dX ** fy(X y) , = exz *= = yz(z)e * + constant fz(x y) , = xye + 1 = ye + C +z ** = ye + g(y z) , fz(x , y z) , = (ye + n(z(z) (f(x z)) (ye z))y *= ** = xye + n'(z) + akek , y, = + g(y , = xyexz + 1 yaso = exz + gy(y z) , n'(z) = 1 : n(z) = z + k = eXz , L gy(y , z) = 0 g(y z) h(z) , = f(x , y , z) = yexz + z - f(x , y , z) ye h(z) = = + Ch16. 4 Green's Theorem C oriented closed positively : curve (Counterclockwise example : S ye zedy ax + , * = SoS. (ae)x- (yet) y dyax S. So'ze = * -e * dyax = SoSo'e dydy * = Coe dx = e * 1 = e2-1 example S Sindy : (3y e(dx + + (4x + , ↓ SoSxi(Q) (P)y x - SoSaz" I dyax (4x + Siny)x = 4 >y - 3 = (3y )y + + e = 3 So x-x = ax - - 5 - = example : F(x , y) (y3 = , - x3) SoF-drewe want C: x+ y2 = 4 SSDQx PydA - = ( x 3)x - - (43)y L - = 3x2 - 3yz r = 2 = SSx - 3(x2 + y =)dA 7 -2 - rarde = (2π)( - -)r") = - 24π exampa b , (20 = (2x y)x (xyz)y - = 4xy 2xy - = 2xy - - fSpzxydA = -SoS zxyayax - SoSexxy2ax -9. 4x3 = = - x*ax Ch 16 5. Cur1 Divergence F(x , y , z) = P) vector field (3D) · vector vector cur1 : curl (f) = (Ry Qz)i (Pz Rx)j - + - + (Qx - Py) divergence : div(F) = Px + Qy + Rz vector oscular gradient : of = fxc + fyj fzk + oscular ovector y = wa + yay + 42 curl (f) = - xF across product example : F(x , y , z) = Sxyn + (4x2 + byz)y + 3y2 IS F conservative ? it is conservative iji Curl (F) = 222 2x 87 by 8xy4x byz3y2 + 22 - by zk - 2xaz 28t 2x8Yz 4x + 6yz3y2 Sxy 3y2 Sxy 4x + 64z (by - by) = 0 0 -0 = 08x - 8x = 0 2 find f such that F = Of of = (fx(x , y , z) , fy fz). F = (P , Q , R) fx(x , y , z) = 8xy fy(x , y , z) 4x 0yz = + 5z(x , y , z) 3y2 = f(x y z) , , = 4x2y 3y z + f(x , y , z) = S8xydz = 4x2y + gu fy(x , y , z) = (4x2y g(y + , z)(y = 4x2 + gy(y z) , = 4x2 + 6yz gy(y z) byz , = = g(y , z) = Soyzdy = 3y 2 z h(z) + f(x y z) , , = 4x2y 3y2z h(z) + + fz(x , y , z) (4x y 3y2z h(z))z = + + = 3yz + n'(z) 3y = 2 n'(z) = 0 - h(z) = k constant... take k = 0  Cn 16 6 Parametric Surfaces. (curves r(t) = (cost , Sint) circle radius r(n v) , = x(n v(z , + y(u v(j , + z(u , v)E is defined over D : E(u v) I conditions , = example : upper %2 of Sphere : x+ y + z =1 x* z = 1 yh - - approach 1 : r(x , y) = (X y , , 1 - xz - yz) = D : x2 + - y approach 2 rcos y =Using : = x (X , y , z) - (rcost , Using , 1 - rz) = u(r a) , D : Orel 02022 , example : S : the part of the cylinder y + z = 4 lies , between x = 0 & X = 1 y = 2018 , z = Ising , x = X D : [ v(X 0) = (X ICSE IsinG) , , , example : S : the part of the plane z = 2x + 1 lies inside Ch16 6 Parametric. Surfaces & Area S : r(u v) , = (X (u v) y(u v) z(n v) , , , , , (u v) , ED Theorem : A(S) = SSplruXrvldA  example : S : r(u v) , = (42 , ur , 202) D : 0443 0 = V43 A(S) ru = (2u , v , 0) ru = (0 , u , v) t Ji O Y 2y 2 242) zu v = - + = (v2 - zuv , Ou V ruXVv = 14 + 44222 444 + = v2 + 24 ? 133(v + Zudu)dv = vin + =3 13 " 32 + 18dv = v3 + 18v) ! = 27 + 54 = 81 example the of function graph : a z = f(x y)r(x y) (x , , = , y , f(x , y) fx) To rx = (1 , 0 , ↑ I ry = (0 , 1 , fy) 0 I fy (rxx y) (( fx = - , - fy 1)) , = 1 + fx fy + S r(x y) (x : , = , y , f(x , y)) (x , y)ED Als) = SSD 1 + fx2 + fy dA example : S : part of 3x + 4y-z = 1 lies inside x + y == 4 , z = 3x + 4y - 1 = f(x y) , (x y , , 3x + 4y 1) - = r(x , y) = D : x2 + y 14 A(S) = SSx 1 + 3 2 + 42dA = 426iT = 26SSD IdA = 26 A(D) = 26it2 X = rosf y = rsing garde =S.= (2)(2) = 4 example : S : part of z =* (x* + y) lits inside x = + y = = 4 D : x + y 2 24 r(X , y) = (x Y ,.2) of (X 4) , Als) = SSD 1 + xi + y 2dA = So S. 1 + rerardo = So "1 + u · du = * u = r2 23/2) ? 2rdr (2)(5)(1 * 5)1 au = + u) = + = 2π)5 * -  Ch 16 7 Surface. Integrals r(u , v) , (u v)ED , f(x , y. z) SSsf(x , y , z)aS = SSpf(rcu v(/ruxrvldA. f(x y) , C : r(t) = a = + 1b SofdsSaf(r(t) Ir '(t)(dt - arc length S : r(u v). , (n. r)ED f(x , y , z) function SS. f(x y z)ds ,. = SSpf(r(u v)) (ruxroldA , example : S : part of 6x + 3y + z = 6 , lies in the 1st octant X , U , z 20 compare SSs XdS , z = - 6x - 3y + 6r(X , y) = (X y , D : 6X + 3y = 6 , x20y20 , 2x + y = 2 | rx + y) = 1 + 62 + 32 = 46 4690x(2 =x)) ! * 46SS : x. (rxxrylaydx = - 2x)ax = 46(x2 - D example : r (0 4) , = (psinYcos O psing sing , , pros 4) 0012 , 024 IT sphere with radius p : Iroxryl = pising Alsphere radiusp) : SS1 as = So Sop since dy do (p4)(2π) SosinGd4 Php2)-cosco) 1 -( 1) = = 2 - =  example : SS g 1 + 2z3ds , S : x = y + zozy , z1/ - (rxX (y) = fx fy + + 1 r(y z) (y , = + z2 , y , z) D : 01 y , z -1 S r(x y) (x : , = , y , f(x , y) r + y = ( fx - , - fy , 1) the graph of x = y + z (ryx +z) = 1 + (xy(z + (xz) = 1 + (1)2 + (4z2) = 2 + 47 Sof ! 1 + 2z22 4zdzay + = SoS. 2 (1 + 2z4dzay = So 2(1 224dz+ = 1 + 222) 2 1 + 2z2) 2(1 5) + = =(2) 2(1 2z2) + example : SSSF-d) , F(x y z), , = (y , - X , z) X Y , z , S : r(u v). = Curosv , usinv , v) D : OUEl , OVE IT F(r(u v)) , = Jusinv , -ucosv , v) = j * u(cosv)2 u(sinv)2 u(1) + = = 4 ru = cosv , sinv o ? , = (sinv , -cosu , u = rux rv = usinv ucosv , - , SSa(ucosvsinv-usinucosv nu)dA So S.uvavau = () += + =  P Q R SSsF , , example : - dS F(x y z) ,. = (xy 4x uz) , , S : z = xe3if(x Y)02X11024Ez ,. D S : r(u v). (UUSED f(x y z) , , function F = (4 , Q , R) r(x , y) = (x , y f(x, , y) SoSo -xy(xe3(x " - 4x * (xe)y + yz = SoS. - xye" - 4x * 23 + xyedyax = SoS-4x edydx 3 = (e2-1) 1-e = Ch16 8. Stoke's Theorem example : SS , curi(F) - as F(x y z) , , = (2ycosz , e"sinz xe3) , S x+ y = + : + z = 9 z20 , x = C : 9 = 0 y z + = , r(f) (3c0s8 = , 3 sing , 0) 0012 SSScurICF). as = So F. ar = SaF(r(t)) - r'(t)dt F(r(0)) = (GSinOcosO e3ctSinO , , 3 CoStessing r'(o) (-3Sing = , scost , 0 F(r(f)). r' (0) = - 18(sing) - 181sindo = -1891-cos(2dgsin example : ScF - ar F(x Y z) (X , , = + y2 , y + z2 , z +