Analysis 1 MATH 3001
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This 4 page Study Guide was uploaded by Cydney Conroy on Thursday October 29, 2015. The Study Guide belongs to MATH 3001 at University of Colorado at Boulder taught by Staff in Fall. Since its upload, it has received 49 views. For similar materials see /class/231830/math-3001-university-of-colorado-at-boulder in Mathematics (M) at University of Colorado at Boulder.
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Date Created: 10/29/15
Gale 3 Exam 3 review Doug Lipinski Fall 2007 Exam 3 is on Wednesday7 11287 so now is a good time to begin studying if you havent already done so To help in your studying l7ve created the following review sheet with a list of important topics and formulas to get you started As always7 I believe one of the best ways to study is to do some of the old exams which are on the website The exam will cover Sections 131 142 excluding 135 from the textbook Capter 13 Chapter 13 covers the topic of multiple integrals in various coordinate systems including cartesian7 cylindrical and spherical coordinates as well as general coordinate transformations and the use of the Jacobian operator to account for the stretching which takes place in coordinate transforrnations 131 Double integrals p 1001 Key topics 1 Double integrals over rectangles 2 Double integrals as volumes 3 Double integrals over non rectangular regions Deterrnining best order of intergration Deterrnining lirnits 132 Areas Moments and Centers of Mass p 1012 ARdA Area of a closed7 bounded region R Average value of f over R 1 A dA V8 areaofR Rf Mass rst moments used for centers of mass center of mass second moments moments of inertia and radii of gyration are included in table 131 p 1014 and you should be familiar with them all Centroids found the same way as center of mass but setting 6xy 0 133 Double Integrals in Polar Form p 1020 Key topics 1 Polar coordinates x rcost9 y rsint9 r x2 yz t9 arctanyx 2 dA 7 dr d0 3 Determining limits of integration in polar coordinates 7 Changing cartesian integrals to polar coordinates Draw a pictureto help with the limits 134 Triple integrals in Rectangular Coordinates p 1026 This is mostly the same as double integrals except nding the limits of integration can be more dif cult It helps to think of an integration in a certain variable as smashing the region of integration in that direction For example if we integrate in 2 over the cube 0 S x S 1 0 S y S 1 0 S 2 S 1 we have effective smashed77 the cube in the z direction leaving the square 0 S x S 1 0 S y S 1 for our last 2 integrations Also note that a triple integral can be used to nd the volume of a region and that our average value equation translates to triple integrals just as you would expect 136 Triple Integrals in Cylindrical and Spherical Coordinates p 1039 Be familiar with the coordinate transformations between cartesian cylindrical and spherical coordinates table on p 1044 It would be a good idea to get some practice nding limits for spherical integrations since these are sometime dif cult to determine Exercises 33 38 on page 1046 offer some practice with this Finally be sure not to forget to change your incremental volumes dV appropriately dV d dy dz 7 dr d0 dz p2 sin dp dab d0 137 Substitutions in multiple integrals p 1048 Be able to perform a general coordinate transformation using the Jacobian and transform the region of integration appropriately Here is the general procedure it is laid out in more detail in the book 1 Transform the boundary a Draw the region in the original coordinate system usually cartesian b Write down the equations for the boundary of the region this could be a circle ellipse three lines to form a triangle 4 lines to form a quadrilateral etc c Plug in for z and y in terms of u and 1 in these boundary equations and solve the resulting equations for something you recognize such as a line parabola ellipse circle etc and plot these new equation in the uv plane to nd your new region of integration 2 Transform the integral a g ay d c dy ffa f96uw7yuw NONM du dv b The region G and your limits of integration come from the picture you drew in the uv plane above c You must compute the Jacobian Juv p 1048 d Plug into you function f for z and y in terms of u and 1 You should also be able to do this for 3d regions and coordinate transformations see p 1051 Chapter 14 The exam will cover only sections 1 and 2 from Chapter 14 including line integrals work circulation and ux 141 Line Integrals p 1061 Key ideas 1 Line integrals over smooth curves 2 Additivity of integrals and integrating over piecewise de ned curves 3 Mass and moment formulas see table 141 p 1064 Given 0 I39t gti htj ktk a S t S b C agam fltglttgthlttgtklttgtgtvlttgtdt You may also want to review parameterizing curves and be sure to note that we can write d5 Vt dt 142 Vector Fields Work Circulation and Flux Key ideas 1 Vector elds7 know what they are 2 Gradient elds7 a special type of vector eld de ned as Vf i j k 3 Work W gums ijdr 5F fTEdt ijdz Ndy sz 4 FlowCirculation a Circulation is just ow around a closed loop b Flow F Tds is evaluated the same way as work integrals 5 Flux a Flux of F across 0 C F nds b for a closed loop that moves counterclockwise once7 Flux c Mdy 7 Ndx Although this material was kind of squeezed in before the break7 don7t neglect it as it is fair game for the test Finally7 I cant stress enough that it is a VERY good idea to do some old exams once you7ve done some preliminary studying This will give you some practive in applying your knowledge to harder problems which is the most common problem students have on the exams Study hard7 good luck and have a great break
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