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# Multivariate Calculus MA 26100

Purdue

GPA 3.97

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This 5 page Class Notes was uploaded by Dorothea Bode on Saturday September 19, 2015. The Class Notes belongs to MA 26100 at Purdue University taught by Staff in Fall. Since its upload, it has received 39 views. For similar materials see /class/208129/ma-26100-purdue-university in Mathematics (M) at Purdue University.

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Date Created: 09/19/15

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 a 1 J k 1 b a cos6 i cross product a X b a1 a2 a3 and their properties llallllbll 51 52 53 a X b is perpendicular to both a and B gtlt area of triangle spanned a ll ll2 a 7 x7cosefsmej by 1 and B projections p736 lt D 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 55 550 ZJ ZJO i 2 20 c Symmetric Form C a b if say 5 0 then 9 Z Z L y yo 1 c 7 3 Equation of plane containing 60 yo 20 normal vector N af 55 CE a NP0P0 or a 0by yoCZ Zo0 4 Sketching planes look at intercepts E E 1 a C CHAPTER 12 H Differentiating and integrating vector valued functions and sketching the cor responding curves 3 Parameterizing curves of the form say 3 f a 3 SC 3 b C t tiftj a gig 5 1W00 llf tll 0 a b Unit tangent vector Tt length of a curve dt CHAPTER 13 H Domains of functions of several variables level curves fv y C level sur faces fv y 2 C sketching surfaces using level curves 2 Quadric surfaces 3 Computing limits determining When limits exist 4 Partial derivatives CHAIN RULE consider tree diagrams 5 Implicit Differentiation for example dy l a If y is defined implicitly by FU y 0 then X gTZi L39 7 3y b If 2 2Uy is defined implicitly by FU y 2 0 then 3F 32 d 32 Ty 7 7 an 7 7 3F 3F 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 m 1 Directional derivative D fcyz Vfcyz 11 where u is a UNIT vector Vf S Daf S IIVfll fv y 2 increases fastest in the direction Vf 00 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 frv 7173 k ay MSW 71 May h 1 O Critical points of fv y 2 points Where Vfvy 2 6 or Vfv y 2 does not exist 11 Finding relative extrema of fcy at those particular critical points 60 yo 1 D 1 OJ Constrained extremal problems fame fxy fgcy 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 to the condition in C La ran e Multi liers 99 g g p mwc CHAPTER 14 DRAW PICTURES FOR THIS CHAPTER H 3 OJ 4 OT 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 SE4 1wr Changing the order of integration in double integrals Triple integrals iterated triple integrals applications of triple integrals vol umes mass m D 6y 2 dV Triple integrals in Rectangular Cylindrical and Spherical Coordinates a Rectangular Coordinates dV dz dy dry or dV d2 div dy etc 36 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 p er1ca oor 1nates p s1n p Sh39lC d39 dV Z39dddQ CHAPTER 15 DRAW PICTURES FOR THIS CHAPTER 1 Vector elds P MT Nj PE divergence and curl of a vector eld P 3 OJ 4 9quot m Surface integrals div V MxNyPZ T f E curlPVgtlt 7 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 curl 6 if region has no holes given that P Vf know how to determine the potential function fv y b Line integrals of functions 0 fcyz d5 fctytzt ftdt line integrals of vector elds P MT Nj PE u a b a a a O F dr Frt rt dt 5 or equivalently O Mdrv Ndy sz MSCdt Nydt Pzdt where C t t T ytj 4012 a g t g b a dr fP1 fP0 inde 5 applications to work Fundamental Theorem of Line lntegrals O Vf pendence of path check if P Vf or curl WT Fdr C 7 GREEle THEOREM If C is a closed curve traversed counterclockwise then C MltcygtdwNltwaygtdyR M if 2 is the graph of 2 fcy with ray 6 R then 2 gm 2 d3 R gm my ifi f 1 cm 1 00 DIVERGENCE THEOREM GAUss THEOREM Flux integral of MiNjP12 over the surface 2 the graph of 2 fv y With in y E R and upper unit normal vector to Z ZFndSA Mf nyPdA If D is a solid region and Z is its Closed boundary surface outer unit normal to 2 then Z dSDdiv13dv

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