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# Multivariate Calculus MA 26100

Purdue

GPA 3.97

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This 5 page Study Guide was uploaded by Dorothea Bode on Saturday September 19, 2015. The Study Guide belongs to MA 26100 at Purdue University taught by Staff in Fall. Since its upload, it has received 30 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 covered MA261 LVECTORS LINES AND PLANES 1 Vector arithmetic directed vector P0131 from P0 to P1 dot product of vectors a1ia2j a3k b1iszbgk 1161 azbzagbg angle between two vectors a B a 1 J 9 cross product a X b a1 a2 a3 and their properties llallllbll bl 52 53 a X B is perpendicular to both a and B gtlt area of triangle spanned a a r b a a by a and b projections p39rgb llz a 7 7cos6i sian 1 2 Equation of line containing 60 yo 20 direction vector i ai 53 CE a Vector Form f f0 t E Where f0 Oi yoj 2012 SC 0 at b Parametric Form 3 yo hi 2 20 ct z z c Symmetric Form m w 70 a b if say 5 0 then 7 Z ZO y 30 a c 7 3 Equation of plane containing 07M 20 normal vector N ai 53 CE 1 NPOP0 or a 0by y00z ZO0 4 Sketching planes look at intercepts E g E 1 a C 5 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 gtg b b length of a curve dt a t 3 Unit tangent vector Ht W r III H 3 OJ 4 9quot m 1 00 Directional derivative PARTIAL DERIVATIVES Domains of functions of several variables level curves fcy 0 level sur faces fv y z 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 a If y is defined implicitly by FU y 0 then b If 2 2Uy is defined implicitly by FU y 2 0 then g E Gradients Vfy 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 I D fwjyjz Vfcyz 11 where u is a UNIT vector Vfll S D f S quot367 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 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 IV H 3 OJ 4 01 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 DU0y0 gt 0 and fm0y0 lt 0 i f has rel maximum value at 60 30 c If DU0 yo lt 0 i f has a saddle point at 60 yo 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 Constrained extremal problems Maximize and or minimize f 3 subject Vf A V9 to the condition in C La ran e Multi liers m m g g p mwc MULTIPLE INTEGRALS 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 636 3 z dV 1111168 mass m 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 sin6 dV 7 dz d7 d6 2 2 SC psin cos6 y psin sin6 2 pcos c p er1ca oor inates p sin p Sh39lC d39 dV Z39dddQ V VECTOR FIELDS 1 Vector elds P Mi Nj PE divergence and curl of a vector eld P 3 OJ 4 9quot m Surface integrals divFVFMxNyPZ I E curl Vgtlt 36 83 82 M 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 b Line integrals of functions 0 fcyz d5 fctytzt ftdt line integrals of vector elds P Mi Nj PE u u b u C F dr Frt rt dt 5 or equivalently O Mdrv Ndy sz MSCdt Nydt Pzdt where C t t i ytj 4012 a S t g b a Fundamental Theorem of Line lntegrals C Vf dr fP1 fP0 inde pendence of path check if P Vf or curl W Fdr C applications to work GREEN S THEOREM If C is a closed curve traversed counterclockwise then 8N 8M if 2 is the graph of 2 ay with ray 6 R then 2 gm 2 d3 R gm fm m 13 1 cm 1 00 DIVERGENCE THEOREM GAUss THEOREM Flux integral of MiNjP12 over the surface 2 the graph of z fv y With 6 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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