ENGINEERING MATH III
ENGINEERING MATH III MATH 251
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This 3 page Study Guide was uploaded by Vivien Bradtke V on Wednesday October 21, 2015. The Study Guide belongs to MATH 251 at Texas A&M University taught by Staff in Fall. Since its upload, it has received 34 views. For similar materials see /class/226053/math-251-texas-a-m-university in Mathematics (M) at Texas A&M University.
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Date Created: 10/21/15
Math 251JZh0u Keys to Exam III Fall 2008 739 1 7r1 1 Form A Z4 COSZ tN4COS2 t 4sin2 tdt 7139 Form B Z4 sinz tN4COS2 t 4sin2 tdt 7T 0 0 2 FormAD473dAD7T FormBD773dA4D47T 3 OCCOStyS1Ht200 tS27139 27r 27r Form A sin t0cos xy 7sin tcos t 0 dt 7sin2tdt 77T 0 0 27r 27r Form B 0003 t e 7 sin t cos t 0 dt COSZ tdt 7T 0 0 4 FormA3722dv3E 47r FormB321dv6E87T E E 5 Forms M6133 a N b V C S d N e N f V g S h N i V j N k N 1 S 6 Form A 01 rt t00 Cg rt 12t7t Cg rt 1217t 3001 010dt0 OZA107t212t71dt72 Og010012dt2 Answer07220 Form B 01 rt t00 Cg rt 13t 7t OgiTlttgt 1317t 3001A10dt0 OZA107t373tdt73 03A10013dt3 Answer07330 Or nd FVf with fyz So Fdr f1230 7f000 0 7 a Check V gtlt F 0 Form A b Find F Vf with f y22ysinx c F dr g 12 7 f000 3 0 Form B b Find F Vf with f 22xsiny c F dr f1 g 1 7 f000 2 C 8 S is not Closed no Divergence theorem Downward 9192171 22y 71 Form A S z gxy 2 y2 716 DP 91 Q gy 7 R dA Dx2y3xy22zyz 2z2y 71 dA D2x3y 6xy3 7 2xyx2 y2 716dA 02 014xy3 32zy dydx 34 Form B S z gy 2 y2 7 9 DP gm Q gy 7 R dA D2y3zy22zyz 2 2y 71 dA 2 1 2z3y 6xy3 7 2xyx2 y2 7 9 dA 4zy3 18ydyd 20 D 0 0 Bonus See review sheet Math 251JZhou 1x 9 03 F 0 7 OD ForrnsAampBAgC go 1 m Form A m z2y2dydx 0 ix Keys to Exam II Fall 2008 Forrns AampB Straightforward evaluation 32 Switch order 4 x5 Form A 4 ysin 2 dy dx 17 cos 16 04 o Form B 4 y cos x2 dy dx sin 16 0 0 05 39 rdrdt 1 12 Straightforward evaluation Form A 27 Form B 9 Forrns AampB 07 since E is symmetric about xzplane and the integrand is odd in y Skipped a7 since most students did not do well For b and C7 1 27r 39rcos 92 Form A 7 rzrdzdr d0 7W 4 0 1 0 4 27r 2 39rcos 92 Form B rzrdz dr d0 157T 0 1 0 g l m Form A 3 pzpz sin bdp dab d0 0 0 0 g t m Form B pzpz sin dpd d 0 0 0 1 m 8 My zx2y2dydzE7Mm0 So 0 ix WHO 9279 370 y 2 2 2 1 y 2 2 8 Form Bm x ydzdy7Mw yz yddyi7My0 So 4 0 7y 3 0 7y 15 i71707g 1 uxiy7 z 2uv7 6x7yil ltagtv2y gtTyyiul b6u7173 12 l l Cux7y17vz2y 2 ForrnA v idvdu ForrnB Lidvdu 0 0 0 0 cosu3 0 0 cosy MATH 251 section 505 Fall 2006 Final TEST Calculators are OK but not necessary There will be 10 problems of weight 5 and 10 pts for a maximal score 75 pts For full credit you need to show the whole work The problems are aimed to test your knowledge de nitions and main results under standing of the material main mathematical ideas and techniques and some basic appli cations The test will be based on your homework assignments In particular the problems will cover the following material H F 9 4 U 03 5 00 p Vectors in 2 and 3 dimensions Dot and cross products properties applications Equations of lines and planes in 3 D Tangent planes and normal lines Quadric sur faces identi cation characterization intersections Vector functions and space curves derivatives integrals Arc length and curvature various parameterizations normal vector Functions of many variables limits and continuity a must Partial derivatives chain rule Directional derivatives and gradient vectors Local extrema of functions of 2 variable Critical point saddle point Absolute mini mum and maximum values Method of Lagrange multipliers for minimizing or maxi mizing a function subject to constraints this whole part is an absolute must Double integrals over rectangle iterated integrals Fubini7s theorem Triple volume integrals Application of multiple integrals in mechanics mass center of mass etc Vector elds in 3 D conservative vector elds line integrals of functions of two and three variables and application and line integrals of vector elds this is an absolute must Surface area and surface integrals of functions Surface integrals of vector elds and application an absolute must Differential operators over vector elds curl grad and div and their properties Green7s Theorem and applications this is an absolute must Stokes7 Theorem and applications a must Good luck
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