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# VECTOR CALCULUS MATH 241

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This 3 page Class Notes was uploaded by Cassidy Grimes on Monday October 26, 2015. The Class Notes belongs to MATH 241 at University of South Carolina - Columbia taught by M. Boylan in Fall. Since its upload, it has received 8 views. For similar materials see /class/229518/math-241-university-of-south-carolina-columbia in Mathematics (M) at University of South Carolina - Columbia.

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Date Created: 10/26/15

Math 241 Final Exam Information Saturday May 2 2 5 pm LC 115 Review April 30 6 pm LC 112 The Final Exam will be based on 0 Sections 121 126 131 133 141 143 145 149151 153 155157 158 o The corresponding assigned homework problems see httpwwwmathsceduboylanSCCourses241Sp09241html At minimum you need to understand how to do the homework problems Useful materials 0 Exams 1 2 3 and their solutions 0 Quizzes 1 7 9 and their solutions New Topic List not necessarily comprehensive Consult review handouts for Exams 1 II III for a list of old topics You will need to know how to de ne vocabulary wordsphrases de ned in class 155 Triple integrals Be able to set up and evaluate triple integrals in rectangular coor dinates de my fem2 d2 dA G R 9mm Note that the limits on the inner integral are functions of at most the outer variables m and y We typically try to View G as a simple zy solid In particular we try to identify 0 The top of G 2 92zy the bottom of G 2 91zy We must have 919679 S 92zy for all my 0 The projection R or shadow of G on the zy plane We try to nd equations for the boundary of B To set up the limits you may need to do two separate sketches a sketch of the solid G in zyz space and a sketch of the projection R in the zy plane Once we evaluate the inner integral a double integral remains which we must integrate 4 Fzy dA This can be done by Viewing R as a type I region integrate with respect to y rst or as a type 11 region integrate with respect to z rst as in section 152 As a triple integral the volume of a solid G is VGdV 157 Triple integrals in cylindrical and spherical coordinates Cylindrical coordinates The cylindrical coordinates are 7362 These should be viewed as extending polar coordinates r 6 by throwing in the rectangular height coordinate 2 The relevant formulas are these 2 2 z roos67 y rsin6 r V22 yz 6 tan 1 Further the volume element dV is given as dV 7 d2 dr d6 Triple integrals in cylindrical coordinates take the form gg39r6 62 726 gg39r6 de fr62d2 dA G R 9106 61 716 9106 Note that the order of integration is 2 r 6 As for integrals in rectangular coordinates you will need to identify The top of G 2 920 6 the bottom of G 2 910 6 The difference is that now the equations for the top and bottom are in the polar variables r and 6 f0 7 67 2 7 d2 dr d6 The projection R or shadow of G on the zy plane We try to nd equations for the boundary of R viewed as a polar region As in the case of rectangular coordinates it is helpful to do a sketch of G and a sketch of R Spherical coordinates The spherical coordinates are p 6 b The relevant formulas are these 2 psin cos 67 y psin gtsin6 p V22 y222 Further the volume element dV is given as 2 pcosab7 dV p2 sin as dp d d6 Triple integrals in spherical coordinates take the form 92 152 WWW de G 9i 1 91 9 me 45 pzsiwdpda d Note that the order of integration is p b 6 As in the case of other coordinate systems it is helpful to do a sketch of the solid G to determine the limits In this case however we do not need to sketch R the projection since the triple integral does not involve R 158 Change of variable in multiple integrals Jacobians The Jacobian is a necessary ingredient in the change variables formula for multiple integrals Suppose that a change of variables is given by 96 Mum y WM so we want to change from Ly to uv The Jacobian of this change of variables is de ned by m aim Wu 6966 al l 6mm 5215 011611 u 62 62 au39 Now the change of variables formula for double integrals is f7ydydlfu717yu71 In particular note that d1 du o The new integrand is fzuvyuv o dy dz becomes d1 du o The region over which the integration is to be done changes from the region R in the zy plane to the region S in the uv plane The third bullet tells us how the z and y limits change to u and v limits Speci cally the change of variables will take the boundary of R to the boundary of S Noting this it is usually not hard to determine S and therefore the u and v limits For this purpose you will probably need to sketch both R and S One can appropriately specialize the change of variables formula for double integrals to single integrals giving the usual u substitution from rst semester l variable calculus Similarly one can appropriately generalize the formula to triple quadruple etc integrals An important change of variables for double integrals is given by the polar coordinate trans formation z rcos y 7 sint97 which yields ay dy dx frcost9rsin0 rdrd R rectangular S polar We studied this in 153 Further a computation shows that the Jacobians for cylindrical and spherical coordinates are 59672172 7 59672172 7 p2 sin b 60 a z am 6 lt1 We also know this from our work in 157

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