b = and their properties: b 1 b 2 b 3 a b is perpendicular to both a and 1 b = x = x 0 + at y = y 0 + bt z = z 0 + ct ; y = y 0 )

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1 ***************** Disclimer ***************** This represents very brief outline of most of the topics covered MA261 *************************************************** I. Vectors, Lines nd Plnes 1. Vector rithmetic; directed vector P 0 P 1 from P 0 to P 1 ; dot product of vectors ( 1 i+2 j+3 k) (b1 i+b2 j+b3 k)=1 b b b 3 ; ngle between two vectors, b cos θ = b ; cross product i j k b = nd their properties: b 1 b 2 b 3 b is perpendiculr to both nd 1 b, 2 b = re of tringle spnned by nd b; projections pr b = b ; v = v (cos θ i +sinθ j) Eqution of line contining (x 0,y 0,z 0 ), direction vector L = i + b j + c k: () Vector Form: r = r 0 + t L, (b) Prmetric Form: x = x 0 + t y = y 0 + bt z = z 0 + ct x x 0 (c) Symmetric Form: = y y 0 = z z 0 b c (if sy b = 0, then x x 0 = z z 0 c ; y = y 0 ) where r 0 = x 0 i + y0 j + z0 k 3. Eqution of plne contining (x 0,y 0,z 0 ), norml vector N = i + b j + c k: N P 0 P =0 or (x x 0 )+b(y y 0 )+c(z z 0 )=0. 4. Sketching plnes (look t intercepts : x + y b + z c = 1). II. Vector-Vlued Functions 1. Differentiting nd integrting vector-vlued functions nd sketching the corresponding curves. 2. Prmeterizing curves of the form sy y = f(x), x b ( : r(t) =t i + f(t) j, t b). 3. Unit tngent vector T(t) = r (t) r (t) ; length of curve b r (t) dt.

2 III. Prtil Derivtives 1. Domins of functions of severl vribles; level curves f(x, y) =, level surfces f(x, y, z) =; sketching surfces using level curves. 2. Qudric surfces. 3. omputing limits, determining when limits exist. 4. Prtil derivtives; HAIN RULE (consider tree digrms). 5. Implicit Differentition, for exmple : () If y = y(x) is defined implicitly by F (x, y) = 0, then (b) If z = z(x, y) is defined implicitly by F (x, y, z) = 0, then dy dx = x x = x nd =. 6. Grdients: f(x, y, z) =f x i + fy j + fz k; the grdient f(x, y) is perpendiculr to level curve f(x, y) = nd f(x, y, z) is perpendiculr to level surfce f(x, y, z) =. 7. Directionl derivtive : D u f(x, y, z) = f(x, y, z) u, where u is UNIT vector; f D u f f ; f(x, y, z) increses fstest in the direction f. 8. Norml vector n to surfces : () is level surfce, F (x, y, z) =, then norml is n = F (x, y, z). (b) is the grph of z = f(x, y), then norml is n = f x i fy j + k 9. Tngent plnes to surfces; Tngent Plne Approximtion Formul: f(x + h, y + k) f(x, y)+f x (x, y) h + f y (x, y) k. 10. riticl points of f(x, y, z) : points where f(x, y, z) = 0 or f(x, y, z) does not exist.

3 11. Finding reltive extrem of f(x, y) t those prticulr criticl points (x 0,y 0 ) where f(x 0,y 0 )= 0 using 2 nd Prtils Test: let D(x, y) = f xx f xy () If D(x 0,y 0 ) > 0ndf xx (x 0,y 0 ) > 0 f hs rel minimum vlue t (x 0,y 0 ) (b) If D(x 0,y 0 ) > 0ndf xx (x 0,y 0 ) < 0 f hs rel mximum vlue t (x 0,y 0 ) (c) If D(x 0,y 0 ) < 0 f hs sddle point t (x 0,y 0 ). 12. Finding bsolute extrem over closed, bounded regions: find interior criticl points, find points on the boundry where extrem my occur, mke tble of vlues of f t ll these points. 13. onstrined extreml problems: Mximize nd/orminimize f(x, y) subject f = λ g to the condition g(x, y) =; Lgrnge Multipliers: g(x, y) = IV. Multiple Integrls 1. Double integrls; verticlly nd horizontlly simple regions, iterted integrls; double integrls in polr coordintes (da = rdrdθ) 2. Applictions of double integrls: res between curves, volumes, surfce re S = fx 2 + f y 2 +1dA. R 3. hnging the order of integrtion in double integrls. 4. Triple integrls; iterted triple integrls; pplictions of triple integrls: volumes, mss m = δ(x, y, z) dv. D 5. Triple integrls in Rectngulr, ylindricl, nd Sphericl oordintes: () Rectngulr oordintes: dv = dz dy dx or dv = dz dx dy, etc (b) (c) ylindricl oordintes: Sphericl oordintes: x = r cos θ y = r sin θ z = z x = ρ sin φ cos θ y = ρ sin φ sin θ z = ρ cos φ dv = rdzdrdθ f xy f yy dv = ρ 2 sin φdρdφdθ

4 V. Vector Fields 1. Vector fields F = M i + N j + P k ; divergence nd curl of vector field F: div F = F = M x + N y + P z curl F = F = i j k x M N P ; Lplcin of f =div f = 2 f = f xx + f yy + f zz. 2. onservtive vector fields F = f; how to determine if F is conservtive : check tht curl F = 0 (if region hs no holes ); given tht F = f, know how to determine the potentil function f(x, y, z). 3. Line integrls of functions f(x, y, z) ds = b f(x(t),y(t),z(t)) r (t) dt; line integrls of vector fields F = M i + N j + P k : F dr = b F( r(t)) r (t) dt or equivlently Mdx+ Ndy+ Pdz= b Mx dt + Ny dt + Pz dt, where : r(t) =x(t) i + y(t) j + z(t) k, t b. 4. Fundmentl Theorem of Line Integrls: f dr = f(p 1 ) f(p 0 ); independence of pth (check if F = f or curl F = 0) ; pplictions to work W = F dr. 5. Green s Theorem :If is closed curve trversed counterclockwise, then M(x, y) dx + N(x, y) dy = R ( N x M ) 6. Surfce integrls : if is the grph of z = f(x, y) with (x, y) R, then g(x, y, z) ds = R g(x, y, f(x, y)) fx 2 + f y 2 +1 da. da.

5 7. Flux integrl of F = M i+n j+p k over the surfce, the grph of z = f(x, y) with (x, y) R, nd n = upper unit norml vector to : F n ds = R { M f x Nf y + P } da. 8. Divergence Theorem (Guss Theorem) : If D is solid region nd is its closed boundry surfce, n = outer unit norml to, then F n ds = D div F dv.