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# 484 Outline for MA 26100 at Purdue

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This 5 page Class Notes was uploaded by an elite notetaker on Friday February 6, 2015. The Class Notes belongs to a course at Purdue University taught by a professor in Fall. Since its upload, it has received 53 views.

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Date Created: 02/06/15

Disclaimer This represents a very brief outline of most of the topics cov ered in this course To be fully prepared you must read your class notes7 the book and correctly work as many problems as possible CHAPTER 11 1 Vector arithmetic directed vector P0131 from P0 to P1 dot product of vectors a1ia2j a3k b1isz bgk 1161 azbzagbg angle between two vectors a B u a 1 J k i cross product a X b a1 a2 a3 and their properties Hall b b b 1 2 3 a X b is perpendicular to both a and b gtlt b area of triangle spanned a a it a a by 1 and b projections p39rgb aw a 7 7cos9i sian 1 2 Equation of line containing 60 yo 20 direction vector i af 53 CE a Vector Form F f0 t i where f0 Of yoj 2012 SC 0 at b Parametric Form 3 yo bf 2 20 of 55 550i3J 30iZ 20 c Symmetric Form a b C ifsayb0 then m39 yy0 C 7 3 Equation of plane containing 60 yo 20 normal vector N af 55 CE NPOP0 or a 0by y00z ZO0 4 Sketching planes look at intercepts E E 1 a C CHAPTER 12 1 Differentiating and integrating vector valued functions and sketching the cor responding curves 2 Parameterizing curves of the form say 3 f a 3 SC 3 b C t tiftj a gig b ft llft l a b 3 Unit tangent vector Tt length of a curve dt CHAPTER 13 1 10 Domains of functions of several variables level curves fv y 0 level sur faces fv y 2 C sketching surfaces using level curves Quadric surfaces Computing limits determining When limits exist Partial derivatives CHAIN RULE consider tree diagrams Implicit Differentiation for example 3F 82 371 82 ray 3 and 3 W a 12 y 32 Gradients Vfvy 2 fxf fyj fz 12 the gradient Vfcy is perpen dicular to level curve fcy C and Vfvy 2 is perpendicular to level surface fv y 2 C Directional derivative D fcyz Vfcyz 11 where u is a UNIT vector Vfll S D f S fv y 2 increases fastest in the direction Vf Normal vector F1 to surfaces 2 a Z is a level surface FU y 2 C then a normal is n VFUy b 2 is the graph of 2 fcy then a normal is IT fxf fyj 12 Tangent planes to surfaces Tangent Plane Approximation Formula Critical points of fv y 2 points Where Vfvy 2 6 or Vfv y 2 does not exist 11 12 13 Finding relative extrema of fcy at those particular critical points 60 yo fame fxy fxy fyy a lfDU0 30 gt 0 and fmwo 30 gt 0 i f has rel minimum value at 60 yo Where Vfv0 yo 6 using 2 d Partials Test let DU y b If DU0 30 gt 0 and fmwo yo lt 0 i f has rel maximum value at 60 30 c If DU0y0 lt 0 i f has a saddle point at 030 Finding absolute extrema over closed bounded regions nd interior critical points nd points on the boundary Where extrema may occur make a table of values of f at all these points Maximize andor minimize fv 3 subject Vf A Vg w0 Constrained extremal problems to the condition g y C Lagrange Multipliers CHAPTER 14 DRAW PICTURES FOR THIS CHAPTER 1 Double integrals vertically and horizontally simple regions iterated integrals double integrals in polar coordinates dA rd d6 Applications of double integrals areas between curves volumes surface area SampM 1WL Changing the order of integration in double integrals Triple integrals iterated triple integrals applications of triple integrals vol D 6yz dV Triple integrals in Rectangular Cylindrical and Spherical Coordinates a dV dz dy dry or dV d2 div dy etc 1111168 mass m Rectangular Coordinates SC 7 cos 6 b Cylindrical Coordinates y 7 sin 6 dV 7 dz d7 d6 2 2 SC psin cos6 y psin sin6 2 pcos c dV p2 sinodp do d6 Spherical Coordinates CHAPTER 15 DRAW PICTURES FOR THIS CHAPTER 1 Vector elds P Mf Nj PE divergence and curl of a vector eld P m vitwnma ff E curlPVgtlt 36 83 82 M N P Laplacian of f diva sz fm fyy fzz 2 Conservative vector elds P Vf how to determine if P is conservative check that curlF 0 if region has no holes given that F Vf know how to determine the potential function fv y b 3 Line integrals of functions 0 fyz d5 ftytzt ftdt line integrals of vector elds P Mf Nj PE L amp m rww b or equivalently O Mdrv Ndy sz MSCdt Nydt Pzdt where C t t f ytj 4012 a g t g b a 4 Fundamental Theorem of Line lntegrals O Vf dr fP1 fP0 inde pendence of path check if P Vf or curl 6 applications to work a WLFdL 5 GREEle THEOREM If C is a closed curve traversed counterclockwise then C MltcygtdwNltwaygtdyR M 6 Surface integrals if 2 is the graph of 2 fy with ray 6 R then 2 gm 2 d3 R gm my ifi f 1 cm Flux integral of MlNjP12 over the surface 2 the graph of 2 fv y With in y E R and upper unit normal vector to Z 39 dSR fo nyPdA DIVERGENCE THEOREM GAUss THEOREM If D is a solid region and Z is its Closed boundary surface outer unit normal to 2 then Z dSDdiv dv

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