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# 37 Study Guide for MA 26100 at Purdue

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COURSE
PROF.
No professor available
TYPE
Study Guide
PAGES
5
WORDS
KARMA
50 ?

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This 5 page Study Guide was uploaded by an elite notetaker on Friday February 6, 2015. The Study Guide belongs to a course at Purdue University taught by a professor in Fall. Since its upload, it has received 26 views.

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Date Created: 02/06/15
Disclaimer This represents a very brief outline of most of the topics covered MA261 WWWwwwwwwwwwwwwwwwwwwww LVECTORS LINES AND PLANES 1 Vector arithmetic directed vector P0131 from P0 to P1 dot product of vectors a1ia2j a3k b1isz bgk 1161 azbzagbg angle between two vectors 113 a 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 3 and b projections p39rgb llz a 7 7Cos9i sian 1 2 Equation of line containing 60 yo 20 direction vector i 611 53 CE a Vector Form f f0 t E Where f0 601 yoj 2012 SC 0 at b Parametric Form 3 yo bt 220Ct C Symmetric Form SC 0 y 90 i Z 20 a b C ifsayb0 then m yy0 C 7 3 Equation of plane containing 07M 20 normal vector N 611 53 CE NPOP0 or a 0by y00z ZO0 4 Sketching planes look at intercepts E 3 E 1 a C II VECTOR VALUED FUNCTIONS 1 Differentiating and integrating vector valued functions and sketching the cor responding curves 2 Parameterizing curves of the form say 3 fv a 3 SC 3 b C t tiftj a gig b a r t b 3 Unit tangent vector Tt length of a curve dt r 74 III 10 PARTIAL DERIVATIVES Domains of functions of several variables level curves fcy 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 Vfv y 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 fv7 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 fm0y0 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 IV MULTIPLE INTEGRALS 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 S 1w1 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 sin6 dV 7 dz d7 d6 2 2 SC psin cos6 y psin sin6 2 pcos c dV p2 sinodp do d6 Spherical Coordinates V VECTOR FIELDS 1 6 Vector elds P M i N 5 P 12 divergence and curl of a vector eld P div VFMxNyPZ f I E curl Vgtlt 36 83 82 M N P Laplacian of f diva sz fm fyy fzz Conservative vector elds P Vf how to determine if P is conservative check that curlF 6 if region has no holes given that F Vf know how to determine the potential function fv y 5 Line integrals of functions 0 fcy 2 d5 ftyt ftdt line integrals of vector elds P Mf Nj PE C 1 d ab ftftdt b or equivalently O Mdrv Ndy sz MSCdt Nydt Pzdt where C t t f ytj 4012 a g t g b a Fundamental Theorem of Line lntegrals C Vf dr fP1 fP0 inde pendence of path check if P Vf or curl WC 7 applications to work F dr GREEle THEOREM If C is a closed curve traversed counterclockwise then 0 MwydrcNltrv7ygtdyR lt gt dA Surface integrals if 2 is the graph of 2 fy with ray 6 R then Z gxyzgtds R Warm m f 1 cm Flux integral of MlNjP12 over the surface 2 the graph of z 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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