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# MULTIVAR CALCULUS MATH 053

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This 6 page Class Notes was uploaded by Kavon Feest on Thursday October 22, 2015. The Class Notes belongs to MATH 053 at University of California - Berkeley taught by Staff in Fall. Since its upload, it has received 41 views. For similar materials see /class/226613/math-053-university-of-california-berkeley in Mathematics (M) at University of California - Berkeley.

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Date Created: 10/22/15

Math 53 Spring 2000 sections 107 amp 109 Review This le and others are available at httpwwwmathberkeleyeduquoththallSS for download in pdf format Here are the problems which were submitted for extra credit some of them have been slightly edited 1 Tommy Sy Find the area of the portion of the plane 31 2y 62 6 which is bounded by the coordinate planes 2 Tommy Sy Evaluate the integral I y dA Rzy where R is the square with vertices 02 11 22 and 13 Hint Use the transformation u z 7 y 1 z y 3 Eric Ng The plane I y 22 2 intersects the paraboloid 2 12 y2 in an ellipse Find the points on this ellipse that are closest to and furthest from the origin 4 Eric Ng 1f 2 fu v where u my 1 g and f has continuous second derivatives show that 822 822 822 82 277 2774 7 2 i I 812 y 8y2 uvau 81 v81 5 Tuji Chang Find 5 curlF dS where Fzy 2 2zyi zzj yzk and S consists of the following three pieces with outward orientation z2y210z0z2 2 00 m2y2 1 220 S 112y2 S 1 6 Tuji Chang Evaluate the line integral f0 y dzz dyz d2 where C consists of two line segments from 000 to 112 and from 112 to 31 4 7 Allison Ryan Prove that the maximum volume of a box with xed surface area is achieved when all side lengths are equal 8 Allison Ryan Find the mass of a sphere of radius a whose density in the appropriate units is equal to 2T2 where 7 is the distance from the origin 9 Allison Ryan Find the ux of the vector eld Fzyz zy22z 5 through the cylinder 12 y2 1 71 S 2 S 1 oriented outward 10 Bobby Young Evaluate f5 curlF dS where Fzy 2 zyzi zyj IQka and S is the top and four sides but not the bottom of the cube with vertices i1 i1 i1 oriented outward 11 Bobby Young Find the area of the surface de ned by the parametric equations 1 uvy u 112 u 7 1 over the region u2 v S 1 12 Bobby Young Show that every plane that is tangent to the cone 12y2 22 passes through the origin 13 Albert Lee amp Steve Hong Find the volume of the solid that lies above the cone 2 412 y2 and below the sphere 12 y2 22 2 14 Albert Lee amp Steve Hong Evaluate f0 Fdr where Fz y 2 z iyj2k and C is the intersection of the surfaces 2 z y2 and z 2y 2 6 oriented counterclockwise as viewed from above 15 Albert Lee amp Steve Hong Find the shortest distance between the following two skew lines 1 z2t y57t 23t Z2 1473 y68 27sli 16 Matthew long If a b and C are constant vectors r is the position vector 11 yj 2k and E is the region de ned by the inequalities 0 S a r S a 0SbrS and0SCrSyshowthat 7 WT EarbrCrdV7 S alb X 17 Matthew long Evaluate C y sin 1 dz 22 cos y dy 13 d2 where C is the curve rt lt sin tcos tsin 2t gt 0 S t S 27L Hint C lies on the surface 2 21y 18 Matthew long a Evaluate dA where n is an integer and D is the region between D WNW circles centered at the origin of radius T and R 0 S T S Rf b For what values of n does the integral in part a have a limit as T A 0 1 c Find m dV where n 1s an integer and E 1s the reglon between spheres centered at the origin of radius T and R 0 S T S Rf d For what values of n does the integral in part c have a limit as T A 0 19 Yasemin Salavatcioglu Find the volume of the region bounded by these equations 31 4y 7 22 0 z 2y I 2 2 0 20 Yasemin Salavatcioglu Find the area of the surface 412 4y2 2 l which lies above the z y planei 21 Easan Drury Evaluate the integral 5 curlF dS where Fz y 2 12y2i y22j 236111 k and S is that part of the sphere 12 y2 22 5 that lies above the plane 2 l and S is oriented upwardi 3 22 Easan Drury Evaluate ffR12 7zyy2 dA Where R is the region bounded by the ellipse 12 7 my y2 2i Hint Find a change of variables Which transforms the boundary to an equation of the form u2 v2 ll 23 Greg Beck Find the volume of the region bounded by the surfaces 12 7 61y274y727l3 and2 24 Amy Platt Evaluate 5 V X F dS Where S is the portion of the surface 12 y2 7 22 1 Which lies inside the cylinder 12 y2 4 oriented outward and Fzy2 2izjykl 25 Amy Platt Find the minimum and maximum values of the function fzy 3x2 7 2y2 over the region I4 y4 S 2 26 Amy Platt Find the volume of the tetrahedron bounded by the equations gzl20z0andzyl 27 Amy Platt Evaluate 5 my d5 Where S is the surface 2 12y 7 fr over theregion0 z 20 y 3l 28 James Junio Find 5 de if fzy2 xzy2 122 and S is the cone 222 y ngg2 29 James Junio Find 0 ydz 2 dy 1 d2 if C is the line segment Which runs from the point P 223 to the point on the plane I 2y 32 1 Which is closest to point Pt 30 Damian Sowinski Find the mass of the solid inside the sphere 12 y222 a2 but outside the cylinder 12 y2 2 whose density is given by pzy2 a2ix27y2722 31 Damian Sowinski Describe the set of level surfaces of the function fz y 2 12 7 y2 2 32 Hank Fung Find the maximum and minimum values of the function fzy2 3x 7 y 7 32 subject to the constraints 1 y 7 2 0 and 12 222 ll 33 Hank Fung Evaluate the surface integral 5 122y22 d5 Where S is that part of the plane 2 4 z y Which lies inside the cylinder 12 y2 4 34 Hank Fung Evaluate the triple integral 2dV Where E is the region in the rst octant bounded by the cylinder y2 22 9 and the planes 1 0 2 0 and y 31 35 Chris Wahl Find the ux of F through the surface S Where Fzy2 11 Zyj 32 k and S is the surface of the cube With vertices i1 i1 i1 36 Chris Wahl Determine Whether or not the vector eld Fz y 1y2 i 21y y2j is conservative If it is nd a function such that F Vfl 37 Scott Heise Use Stokels Theorem to evaluate c F r Where Fzy2 z2i 21yj 3zyk and C is composed of three line segments traversing the points 1 0 0 0 3 0 0 0 3 and l 0 0 again in order 38 Scott Heise Use Stoke s Theorem to evaluate f5 curl F dS where Fz y 2 IQyZi y22j 236111 k and S is the portion of the sphere 12 y2 22 1 above the plane 2 2 39 Scott Heise Evaluate the surface integral 5 zyz d5 where S is the portion of the sphere 12 y2 22 1 which lies above the cone 2 412 y2 40 Laura Gambs A silo in the shape of a cylinder with radius 7 sits in a eld of grass At one point on the outside of the silo a cow is tethered by a rope just long enough to wrap halfway around the silo allowing the cow to reach the opposite side What is the total amount of area in which the cow can graze 41 Laura Gambs In a few sentences describe all of the major theorems of Chapter 14 and how they relate to each other Include a physical interpretation of each equation 42 Mujtaba Saifuddin Find the equations of a the tangent plane and b the normal line to the surface 12 7 2y 2 3 at the point 711 72 43 Mujtaba Saifuddin Find a parametric representation for the portion of the elliptic paraboloid z y2 222 4 with z 2 0 44 Nicole Foletta Calculate the work done by the force eld Fzyz 11 22 i yy 12j 22 y2 k when a particle moves under its in uence around the boundary of that part of the surface 12 y2 22 4 which lies in the rst octant in a counterclockwise direction as viewed from above 45 Nicole Foletta Verify by direct calculation that Stokels Theorem is true when applied to the vector eld Fz y 2 y izjz k and the surface zyz 1 in the rst octant oriented upward Here are a few additional problems that 1 came up with sometimes based on the problems above especially dif cult problems are marked with an asterisk A Find the center of mass of the surface described in Problem 1 if the density is proportional to the distance from the origin B Find the ux of the vector eld Fwy yy 2 M 12 y2 22 2 through the surface in Problem 1 with upward orientation 1f you7re curious the ux of this particular vector eld through a surface measures the amount of light hitting it from a point source at the origin Hint In order to get an integral you have some hope of evaluating calculate the ux through an octant of a sphere together with pieces of the three coordinate planes and then use the divergence theorem to prove that the two integrals are equa C Calculate the ux of the vector eld from Problem B through the unit sphere oriented outward Why doesn7t this contradict the divergence theorem Herels a related question Why isnlt there really such a thing as a point source77 of light 12wyy2 wry or demonstrate that the limit does Di Evaluate the limit lim y400 not exist Hint The limit doesnlt exist Saying the limit is or the function is unde ned along the line y 71 1s NOT enough to show that the limit doesnlt exist 977 0 El What are the best angles of boat and sail two different angles for tacking straight into the wind The force produced by the wind on the sail is perpendicular to the sail7 with a magnitude proportional to the ux of the wind velocity eld through the sail The motion of the boat is determined by projecting the force vector onto a vector perpedicular to the keel of the boat7 and the amount of forward progress of the boat against the wind is obtained by projecting the boats velocity onto a vector in the opposite direction of the wind How many sailors realize that sailing against the wind is only possible because of the mathematical curiousity that the direction of a vector may be reversed by applying a series of three projections F If my 2gt ruv is a parametrized surface7 then the tangent plane at the point A ltabcgt can be parametrized as lt1 y 2gt A sru trvi Explain why Expand this vector equation into three scalar equations and eliminate s and t to get a single equation in z y and 2 Read the normal vector to the plane off of the coef cients of your equation7 and verify that it is a multiple of the vector ru gtlt rvi G Draw a picture of Stoke s Theorem in action Explain it to someone who is not in Math 53 Hi Draw a picture of the Divergence Theorem in action Explain it to someone who is not in Math 53 ll Prove that Stoke s Theorem applied to a surface in the z yplane is equivalent to Greenls Theorem J What is the twodimensional analogue of the Divergence Theorem Show that this is equivalent to Greenls Theorem Ki What is the onedimensional analogue of the Divergence Theorem This is also equivalent to some wellknown theorem i i i L Use Stokels Theorem to prove that ffs curlFdS is zero for any closed surface Mi Use the Divergence Theorem to prove that 5 curlF dS is zero for any closed surface S Math 53 Midterm 1 Review GSl Santiago Canez 1 Eliminate the parameter to nd a cartesian equation of the curve given by the parametric equations z 2 sinht and y 3 cosht 2 Find parametric equations for the curve of intersection of the surfaces y 22 7 2 and z 3 7 m2 Are there any points on the curve with zero velocity 3 Sketch the curve given by the polar equation r 1 2 cos0 and nd the area of the region inside the outer loop but outside the inner loop 4 Suppose that lal 4 lbl 2 and a b 5 Compute the area of the parallelogram spanned by a and b 5 Find the equation ofthe plane containing the points 1 O 72 111 and 723 1 6 State the relations between rectangular and cylindrical coordinates and between rectangular and spherical coordinates 7 Identify each of the following surfaces a 9 7r4 b j 7r4 c z2 2x 7 y2 4y 22 0 8 Show that the curve z x cost 7 1 y 5 z x sint lies on the surface z22m7y24y220 9 Let m y 1 7 z2 yz Show that 1 satis es the partial differential equation yum 7 mug 0 Draw a contour map of f and describe its graph 10 Find the tangent plane to the graph of m y e m2y 7 sinm 7 at 00 Use this to approximate the value of f01 701 11 Compute the following limit or show it doesn t exist my lim wya0gt0 V m2 y2 Date February 20 2006

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