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# Calculus III MATH 2243

GSU

GPA 3.94

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This 3 page Class Notes was uploaded by Miss Emerald Langosh on Monday October 12, 2015. The Class Notes belongs to MATH 2243 at Georgia Southern University taught by Staff in Fall. Since its upload, it has received 6 views. For similar materials see /class/222049/math-2243-georgia-southern-university in Mathematics (M) at Georgia Southern University.

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Date Created: 10/12/15

s 1vt Arc length parametrized curves have unit speed7 ie7 1 for all s E 07L Unit Tangent T 15 r tlr tl Note T T 1 i T T T 0 Unit Normal Vector N T lT l B gtlt T a7 IT TaN In R2 take N r gt0 Unit Binormal Vector B T gtlt N r gtlt r lr gtlt r l Acceleration a aTT aNN with IT 1 r r lr l and 1N K02 lr gtlt r llr Curvature H lT lv lr gtlt r llr l3 Torsion 739 r gtlt r r lr gtlt r l27 lTl Signed curvature of planar curves rt ltxt7ytgt is H b x y 7 z y lr l3 quot Signed curvature of functions y f is H y 1 y 232 Osculating circle Radius p 1H7 Center c r pN Conic Sections 1 b d z ayaz22bzycy22dx26yfXTQXr y1 b c e y 0 d e f 1 Nondegenerate if D detQ 77 0 Let A ac 7 b2 Ellipse D 77 0 and A gt 0 Parabola D 77 0 and A 0 Hyperbola D 77 0 and A lt 0 Quadric Surfaces Plane ax by 02 d Sphere x2 y2 22 p2 Cylinder ay 0 in IRS Ellipsoid 7 717 357 7 1 Elliptic Paraboloid z 7 717 Hyperbolic Paraboloid z 7 7 2 Elliptic Cone 7 Hyperboloid of one sheet 7777 1 Two sheets 7j7g7j7i72 1 Gradient Directional Derivative Normal Vector to the surface Fyz 0 is the gradient VF ltfm7 fy7fzgt The vector u is a direction77 if lul 1 Directional Derivative of Fx at x0 in the direction u DuFx0 VFX0 u DuFx is maximized when u VFXlVFxl7 which is the direction of greatest increase steep est ascent and perp to the contours Fx c ie7 normal to the curve or surface Fx 0 Chain Rule for w frt with rt ltztytztgt diu 3w dz 3 d 3w dz VF t if iwiy A77 dt 31 dt ay dt 32 dt t r Chain Rule for w fst7ystzst ia a g ia al 341 a as azas3yasazas and t aat ayatazat Tangent Planes TP to z ay at 11 2 fab fwa7bx 7 a fyaby 7 b TP to Fyz 0 at X0 VFX0 x 7 X0 0 TP to ruv ltxuv7yuvzuvgt at 11 TPuv rab u 7 a ruab 1 7 b rvab Linear and Quadratic Approximations to z fzy at 071 Linear LW 24 fa7 5 Mm 5W 7 a fya7 My 7 b Nab V m 5 90 7 a y 7 5 Quadratic cm a 7 Mm mm W 7 a 2 Ma we 7 agtlty 7 b am we 7 12gt 7 fwmlta7b f1 a7b mia Local MaxMinSaddle for z fzy Critical points 071 such that fmab fyab O l fma7b JenGob A Let Aab 7det h M MW7Mawyyw7ltfmyltabgtgt2 2 If A gt 0 and fmm7 b gt 0 then 11 is a local minimizer and c2z7 y is an elliptic paraboloid opening up If A gt 0 and fmm7 b lt 0 then 071 is a local maximizer and c2z7 y is an elliptic paraboloid down If A lt 0 then 11 is a saddle point and c2z7 y is a hyperbolic paraboloid Constrained Max Min Extremize fx subject to the constraint gx 07 Lagrange Multipliers Solve Vf A V9 and g 0 simultaneously Vector Fields Vector Fields FX ltP7y7Q9671gt 0r FX 030671172XQ967y727R967y72gt Del Operator V 7 or V avail731 Gradient of fz7y7z gradf 3777 Divergence divF V F i j k Curl curlFVgtltF 331 77777 P Q R VVgtltF0 Identity divcurlF Identity curlVf 0 ie7 gradient vector elds are conservative Conservative Vector Fields De nition The vector eld F is conservative if F Vf for some potential function f Corollary F is conservative iff V gtlt F 0 IrrotationaI Vector Field V gtlt F 0 Conservation of Mass V F 0 Source V F gt 0 Sink V F lt 0 Path Independence The value of C F dr is independent of the path 0 Fundarnental Theorem of line integrals f0 Vf dr Vfrt r t dt frb 7 fra Corollary F is conservative gt C F dr f0 Vf dr frb 7 fra and c F dr 0 Theorem F conservative iff path independent iff circulation 0 iff curl F0 irrotationaI Line Integrals of curves rt xzt7 yzt7 zt Unit Tangent T r r Unit Norrnal N Scalar line integral f0f7y d5 ffrtr t dt Arc length when f E 1 Differentials dr ds7 dy or dr dz7 dy7 dz Vector line integrals W CF Tds Frt r t dt ch dr Green7s Theorern circulation 0FTds faRszQdy ffRQm7Py dA ffRVXFdA Green7s Theorern ux c F Nds 313 de 7 Qd ffRPm Q dA ffR V FdA Stoke7s Theorern gs F Tds 5 curlF N dS xu7v7yu7v7zu7v Unit Norrnal Vector outward N ru gtlt rvIru gtlt rEI Scalar Surface integrals 5 fru71 dS fforu7vru gtlt rEI du d1 Surface area when f E 1 Rectangular f S f7y41 2 25 d dy Cylindrical f S fr7 9 I M r223 292 dr d0 Vector Surface integrals Flux ffSF N dS ffD Fru71 ru gtlt rv dude ffSdedz deds Rdzdy Divergence Gauss7 Theorem for closed surfaces ffav F N dS f f V V FdV Surface Integrals of ru71

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