eview for 263.02 finl 14.1 Functions of severl vribles. Find domin nd rnge. Evlute. ketch grph. rw nd interpret level curves. (Functions of three vribles hve level surfces.) Mtch surfces with level curves. 14.2 Limits nd continuity. The limit is undefined if two pths to point suggest different vlues. ompute limits (vrious techniques: multiply nd divide by conjugte, squeeze theorem, convert to polr, etc... ). A function f is continuous t point if the limit nd the function vlue both exist nd re equl. Find points of continuity. 14.3 Prtil derivtives. Find prtil derivtives by differentiting with respect to one vrible while treting the others s constnts. Estimte derivtives from grph or contours. Mixed prtils. lirut s theorem: If f xy nd f yx re both continuous in, then f xy f yx in. Implicit differentition. heck solutions for prtil differentil equtions by substitution. 14.4 Tngent plnes nd liner pproximtions. Find tngent plne t point. The totl differentil: dz Bz Bx dx Bz dy, nd similrly in higher dimensions. Use the totl differentil to estimte errors, pproximte functions. 14.5 The chin rule. For z f px, yq, x gptq, y hptq, we hve dz Bz dx dt Bx dt For z f px, yq, x gps, tq, y hps, tq, we hve Bz Bz Bt Bx Use chin rule for higher order derivtives s well. 14.6 irectionl derivtives nd the grdient vector. Bx Bt Bz dy dt. Bz. We find Bz{Bs similrly. Bt The grdient of f: f px, yq xf x, f y y. A similr sttement pplies in 3 dimensions. Grdient vector points in pth of fstest increse, so moving in the direction of the grdient gives the pth of steepest scent. irectionl derivtive: u f u, where u is unit vector. Find tngent plnes to level surfces F px, y, zq k. Grdient of F is in norml direction to the surfce.
14.7 Mximum nd minimum vlues. f x, f y re 0 t locl mx or min (provided they exist). riticl points re the plces where ll first order derivtives re 0. econd derivtives test. onsider f xx f yy fxy. 2 If 0, f xx 0 t criticl point, then locl min. If 0, f xx 0 t criticl point, then locl mx. If 0, neither locl mx nor min. To find bsolute mx or min, check criticl points nd the boundry. 14.8 Lgrnge multipliers. Know method. gpx, y, zq k. Used for mximizing nd minimizing subject to one or more constrints, e.g. 15.1 ouble integrls over rectngles. ouble iemnn sum definition. Approximte vi midpoint method. 1 Averge vlue: f verge Arepq f px, yq da. Bsic properties: linerity, order preserving. 15.2 Iterted integrls. Fubini s theorem: If r, bs rc, ds, then i.e. da becomes dx dy. f px, yq da b d c f px, yq dy dx d b Evlute inside integrl by treting other vribles s constnts. evlute s the product of two integrls. 15.3 ouble integrls over generl regions. c f px, yq dx dy, If f px, yq gpxq hpyq, cn Extend functions over generl region to be over rectngle by tking them to be 0 outside of their domin. Brek region into pieces to mke esier to integrte. Bounds for inner integrls my depend on outer vribles, but not the other wy. Exchnge the order of integrtion when necessry. Properties of double integrls. Integrting 1 gives re. 15.4 ouble integrls in polr coordintes. For f defined in the polr region 0 r b, α θ β, becomes r dr dθ. f da Use x r cospθq nd y r sinpθq to convert crtesin problems to polr. β b α f r dr dθ, i.e. da 15.5 Applictions of double integrls. Mss is the integrl of density. For density ρpx, yq, moment bout the x-xis is M x y ρpx, yq da. imilrly, M y x ρpx, yq da.
enter of mss is t pm y {m, M x {mq. (Yes, M y goes with the x-coordinte, nd M x with the y.) For moment of inerti (second moment), I x bout the origin is I 0 I x I y. Probbility within region is the integrl of the joint density function. ompute expected vlues given joint density function. 15.6 Triple integrls. y 2 ρpx, yq da. imilrly for I y. Moment of inerti Fubini s theorem extends to higher dimensions. (When integrting over box, integrte the x, the y, nd the z.) efine integrl in generl region by working in box, tking the function to be zero inside the box but outside the old domin. dv becomes dx dy dz. Integrting 1 gives volume. Iterted integrls s with two vribles. ompute probbilities. 15.7 Triple integrls in cylindricl coordintes. Especilly useful for solids of revolution. dv becomes r dz dr dθ. 15.8 Triple integrls in sphericl coordintes. Especilly useful for cones nd spheres centered t the origin. dv becomes ρ 2 sinpφq dρ dθ dφ. r ρ sinpφq, so x ρ sinpφq cospθq, y ρ sinpφq sinpθq, nd z ρ cospφq. Hence x 2 y 2 z 2 ρ 2. Note tht 0 φ π nd 0 θ 2 π. 15.9 hnge of vribles in multiple integrls. The Jcobin of the trnsformtion x gpu, vq nd y hpu, vq is Bpu, vq g u g v h u h v, 16.1 Vector fields. with similr definition holding for trnsformtions of three or more vribles. (Note: The Jcobin is sclr vlued function.) f px, yq da f pxpu, vq, ypu, vqq du dv. A similr stte- Bpu, vq ment holds for triple integrls. Bpu, vq du dv, i.e. da ketch vector fields: A vector field F is function tht ssigns vector to every point in its domin. The output is the sme dimension s the input. 16.2 Line integrls. If is the prmetriclly defined curve x xptq, y yptq, t b, then b 2 f px, yq ds f pxptq, yptqqd dx 2 dy dt. dt dt
Insted of integrting with respect to rc length s, we cn integrte with respect to x : f px, yq dx b f pxptq, yptqq x 1 ptq dt, nd similrly for integrting with respect to y. (Follows from the chin rule.) Integrting 1 with respect to rc length gives the totl rc length. Line integrl of vector field F P i Q j k long is b Fprptqq r 1 ptq dt F T ds Work to move prticle long the curve defined by rptq is W P dx Q dy dz b. 16.3 The fundmentl theorem for line integrls. A conservtive vector field is field F f for some function f. f dr f prpbqq f prpqq. Tht is, the line integrl of conservtive vector field is independent of pth. onversely, if line integrl of continuous vector field F is independent of pth, then F is conservtive vector field. is independent of pth if nd only if 0 for every closed pth L in the domin. L If F P i Q j is conservtive vector field nd P nd Q hve continuous derivtives, then P y Q x. (ttement is true on ny domin; converse only holds for open simply-connected sets.) If force is described by conservtive vector field, then energy is preserved. (onservtion of energy.) 16.4 Green s theorem. onverts line integrls over the boundry to integrls over the re. P dx Q dy pq x P y q da curl pp i Q jq k da, for positively oriented. If is negtively oriented, then sign is flipped from the bove. Positive orienttion: ounterclockwise rottion. (egion to left of direction of motion.) Negtive orienttion: lockwise rottion. (egion to right of direction of motion.) ometimes useful to clculte res enclosed by prmetric curves. Just pick ny Q nd P such tht Q x P y 1. Exmples include: Q x nd P 0, Q 0 nd P y, or Q x{2 nd P y{2. 16.5 url nd divergence. lculte curl nd divergence: curl pfq F. div pfq F. Positive divergence t P mens net flow ner P is outwrd. pcurl pvqqpp q points in the direction of the xis of rottion of v t P. curl p f q 0, provided f hs continuous second derivtives. Tht is, the curl of conservtive vector field is 0. The converse is true s well: If curl pfq 0, then F is conservtive vector field. div pcurl pfqq 0.
Vector forms of Green s theorem: ¾ pcurl pfq kq da, ¾ pf nq ds div pfq da. 16.6 Prmetric surfces nd their res. Find prmetric representtion for surfces. Find tngent plne to surfce. r u r v is the norml vector. Are of surfce defined by rpu, vq where pu, vq P is A pecil cse: Are of surfce z f px, yq where px, yq P is A 16.7 urfce integrls. ompute: f px, y, zq d Note: The surfce re of is pecil cse: Integrting over z f px, yq: f prpu, vqq r u r v da. d r u r v da. r u r v da, s bove. f px, y, zq d onvention: Positive orienttion is for outwrd norml vectors. urfce integrls of vector field F, i.e. the flux of F cross : F d pf nq d b1 pf x q 2 pf y q 2 da. f px, y, gpx, yqqbpf x q 2 pf y q 2 1 da. F pr u r v q da. pecil cse: If is the grph of z f px, yq, nd F xp, Q, y, then F d p P f x Q f y q da. 16.8 tokes theorem. Use to convert surfce integrl to line integrl round the boundry, or vice-vers. curl pfq d, for the positively oriented boundry of. orollry: If 1 nd 2 shre the sme boundry with the sme orienttion, then 1 curl pfq d 2 curl pfq d. 16.9 The divergence theorem. Use the divergence theorem to convert surfce integrls to volume integrls, or vice-vers. ½ F d div pf q dv, for the region bounding E with outwrd orienttion. E