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by: Mrs. Dedric Little


Mrs. Dedric Little
GPA 3.79


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Class Notes
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This 8 page Class Notes was uploaded by Mrs. Dedric Little on Monday October 19, 2015. The Class Notes belongs to MTH 255 at Oregon State University taught by Staff in Fall. Since its upload, it has received 55 views. For similar materials see /class/224451/mth-255-oregon-state-university in Mathematics (M) at Oregon State University.




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Date Created: 10/19/15
20012002 Catalog Data MTH 255 VECTOR CALCULUS II MTH 255 4 credits Double integrals in polar coordinates triple integrals in rectangular cylindrical and spherical coordinates Introduction to vector analysis divergence curl line integrals and work surface integrals conservative elds and the theorems of Gauss and Stokes PREREQ MTH 254 Note The material on double integrals and coordinate systems is now covered in MTH 254 The catalog will be updated next year Prereguisites by Topic Textbook 1 Ability to handle standard algebraic trigonometric and vector calculations at the college level 2 Pro ciency with differentiation and integration of functions of one two and three variables James Stewart Calculus Early Transcendentals BrooksCole fourth edition 1999 Course Learning Objectives By the completion of this course students are expected to 1 Topics Prepared by J W Lee Use scalar and vector fields to model a variety of physical situations and be able to draw field strength diagrams and infer qualitative properties of divergence and curl from the diagrams Evaluate divergence and curl for particular vector fields and interpret the physical meaning of these quantities for uid ows Set up and evaluate line and surface integrals and use them to find physical quantities such as mass charge work ux and center of mass Know the integral theorems theorems of Green Gauss and Stokes and be able to motivate why these results hold with physical plausibility arguments Explain and use the relationships among independence of path integrals around closed loops curl conservative fields and existence of potential functions 1 Scalar fields Vector fields divergence and curl 4 classes 2 Properties of line integrals the fundamental theorem of calculus for line integrals potential functions and applications such as mass charge and work 8 classes 3 Green s theorem and applications 3 classes 4 Parameterizes surfaces tangent planes surface area surface integrals and applications such as finding mass charge and ux 4 classes 4 Gauss divergence theorem and applications 3 hours Stokes theorem and applications 3 hours 6 Catch up and review 2 classes V39 Date February 2002 MTH 255 VECTOR CALCULUS 11 Schedule Lecture 1 hour three times per week Recitation 1 hour 20 minutes each week Prepared by J W Lee Date February 2002 Prepared by J W Lee Date February 2002 1 Potential for correspondence instructor dependent s Substantial correspondence ISUMMARY s Ob ective l Ob ective 2 Ob ective 3 Ob ective 4 Ob ective 5 Re quirem ents Course L earning Obj ectiv e s ABET S a Ability to apply math science and engineering b Ability to design and conduct experiments as Well as to analyze and interpret data 0 Ability to design a system component or process to meet desired needs d Ability to function on multidisciplinary teams 8 Ability to identify formulate and solve engineering problems Understanding ofprofessional and ethical responsibility f g Ability to communicate effectively 11 Broad education necessary to understand the impact of engineering solutions in aglobal and societal context 1 Recognition of the need for and an ability to engage in lifelong learning 139 Knowledge of contemporary issues k Ability to use the techniques skills and modem engineering tools necessary for engineering practice 1 Abil39 to apply advanced mathematics through mult ariate calculus and differential equations m Familiarity with statistics and linear algebra n Knowledge of chemistry and calculusbased physics with depth in at least one 0 Ability to Work professionally in the thermal systems area inclu ing the design and reali ation of such systems p Ability to Work professionally in the mechanical systems area including the design andrealization ofsuch systems Student Self Assessment of Capability Course Learning Objectives Mapped to ABET Goals MTH 255 VECTOR CALCULUS II HO 9 g a a MTH SAMPLE PROBLEMS 255 VECTOR CALCULUS 11 FOR MIDTERM II Review your class notes and work through the examples in the text and the HW problems Use Green s theorem to nd the area of the region in the xyplane bounded by the simple7 closed curve rt costi sin3 tj7 0 S t S 27L Recall sin2t 2sintcos t7 cos 2t cos2 t 7 sin2 t 2 Compute f0 Pdm Qdy7 where P my siny cos m37 Q I m cos y e yz7 and c is the trefoil curve m 4 cos li5t cos t7 y 4 cos li5t sin t7 0 S t S 47L Find the Jacobian of the transformation m uv7 y 1 Use the transformation in part a to compute the integral ffR my dA7 where R is the region in the rst quadrant bounded by the lines y m7 y 3m and by the hyperbolas my 17 my 3 Example 3 in the section on change in variables in multiple integrals Let fmy2 x1 m2 y27 and let S be the helicoid parametrized by r ucos vi usinvj vk7 where 0 S u S 17 0 S v S 7r Convert the integral f5 de into a double integral over the parameter domain Remember to specify the limits of integration You do not need to evaluate the resulting integral Let Fmy7 2 lt 7m 7y2722 gt and let S be the ellipsoid m2 4y2 229 l a Use the divergence theorem to express the outward ux of F across 5 as a triple integr Apply a change of variables to convert the integral in part a into a triple integral over the unit sphere Hint m au y by 2 cw where a 127 c are some constants Finally express the integral in part b in spherical coordinates with appropriate limits of integration You do not need to evaluate the resulting integral b e Let F17 F2 be two vector elds whose components possess continuous partial deriva tives and let S be a smooth7 oriented surface that is bounded by a simple closed curve 0 Suppose that F1P F2P for all P on 0 Then VgtltF1ndSVgtltF2nd5 S 5 Explain Let F17 S c be as in part a7 and suppose that Squot is another smooth7 oriented surface bounded by 0 Then VgtltF1ndSVgtltF1nd5 S 5 Explain Let Fmy7 2 lt y2997m2 y2ln2 gt and let S be the part of the upper hemisphere 12 y2 22 2 2 gt 07 that is contained in the cylinder m2 y2 l oriented 0quot V n V H MTH 255 VECTOR CALCULUS II SAMPLE PROBLEMS FOR MIDTERM II With the upwards pointing normali Use the identities in parts a and b to compute ffs V X F ndS Without explicitly evaluating any integrals d Problem 16 in the section on Stokes7 Theoremi Find the surface area of the part of the hyperbolic paraboloid z 22 7 y2 that lies between the cylinders y2 22 1 and y2 22 4i i Let Fzy iyi y2i a Let c be a cirle of radius 7 centered at the origin Compute L F dri b Let E be any simple closed curve enclosing the origin Show that f5 F dr L F dri c Let cl be the trefoil curve I 4 cos 1 5t cos t y 4 cos115t sin t 0 S t S 47L Find 01 Fdri Hint No integration is needed simply read off the value of the integral from the graph of the curve d Let Cg be the gure 8 curve consisting of two circles of radius 1 centered at 0 732 and 0712 Compute fczF dr provided that V lt 10 gt is tangent to Cg at 0 752 1 a Use the curloperator to show that the vector eld Fyzy1izzzjzy22k is conservative b Find a potential fz y 2 for F c For Which values of the parameters a and b is the vector eld G z ayzi 12 y3j my bzk conservative d Let F coszi 3y zezzj my ln12 y2 k Is there a vector eld G such that F V X G Explain The solid torus T2 generated by rotating a disk of radius a With center 1200 b gt a around the z axis can be parametrized y z 12 acosa cost9 y 12 acosa sint9 2 asina WhereOSt9lt27r0Salt27r0Saltai a Find the Jacobian of the above parametrizationi b Compute ffng adVi a Find a parametric representation for the surface obtained by rotating the curve rt lt 0 cos tt gt about the z axis b Let S be the surface given in parametric form by ruv lt u2 1 v3 1 u 1 gt1 Find the equation for the tangent plane to S at 121 c Sketch the surface given in parametric form by ru v lt cos u cos 2v sin u cos 2v sin 1 gt by graphing u and vgrid curves 11 a The Mobius strip M is given in parametric form by r lt 47 using cost947 using sint9ucos gt 0 S 9 lt 2mi1lt u lt11 2 0quot 12 a 0quot n V MTH 255 VECTOR CALCULUS II SAMPLE PROBLEMS FOR MIDTERM 11 Suppose that the density p of M is proportional to 2 Where 2 is the usual 2 coordinatei Find the total mass 0 i Find the ux of the vector elds E lt zy2 gt and F lt zy22 gt across the cone 2212y20lt2lt2i Compute the ux of the vector eld F lt yz y sinz2 ln12 y2 gt across the torus in problem 9 Let r be the position vector eld r lt zy2 gti Show that the volume of a simple solid region E is one third of the outward ux of r across the boundary 0 Let F rlrlgi Compute the ux of F across the unit sphere centered at the origin directly from the de nition Hint you don7t need to compute any integralsi Next show that V F 0 The results seem to contradict the divergence theoremi Explain HO 9 00w MTH 255 VECTOR CALCULUS II SAMPLE PROBLEMS FOR MIDTERM I Review your class notes and work through the HW problems Let c be the curve with vector equation rt 6 sin ti 6 cos tji a Find the arc length function 8 3t for 5 measured from t 0 and reparametrize c with respect to 8 b Find the curvature of c at the point z 0 y 1 Let F1z y 2 zzi zyzj 7 ka and F2z y 2 zy21 IQyj Cos 2k a Show that there is no function f fzy 2 so that F1zyz Vfzyzi b Find a function g gzyz so that F2zyz Vgzyzi c Evaluate the integral f0 F2 dr where the curve c is given by rt e it4 sin tj tk 0 S t S 27L a Find the maximum rate of change of the function fzyz 12 4y2 922 at the point P2 71 0 and the direction in which it occurs b Next consider the ellipsoid 12 4y2 922 1 Find the equation of the tangent plane to this surface at the point Pl2 13 imlS Suppose that the problem is to nd three positive numbers I y z the sum of which is 100 so that the product P zy22 is maximumi a Use the method of Lagrange multipliers to set up equations that the critical points of the maximum problem satisfyi b Solve the equations you obtained in part a What is the answer to the maximum problem i a A wire takes the shape of a circle 12 y2 l 2 0 Find the total mass of the wire provided that the linear density is pz yz kl 7 y where k is a constant b Find the work done by the force eld Fz y 2 z2e zzii ij Zcoszyk in moving a particle along the quarter circle r cos ti sin tj 0 S t S 7r2i A particle moves along the curve r lt sin t cos t 4t gt where t stands for time a Find the unit tangent vector T and unit normal vector N for the path of the particle at t 7r2i b Find the tangential and normal components of the acceleration of the particle at t 7r2i Section 158 Lagrange multipliers Example 5 Problems 15 16 i a Find the directional derivative of the function fzy 2 11 y2 sinzz in the direction of the vector V lt 11 71 gt at the point P 202 b Let fz yz be as in part a and let 5 be the trefoil knot parametrized by rt lt 2 cosli5t cos t 2 cosli5t sin t sinli5t gt i 9 H H H H H la MTH 255 VECTOR CALCULUS II SAMPLE PROBLEMS FOR MIDTERM I Find the rate of change of the restriction of fz y 2 to c at the point r7r2i Let 312y y3 7 312 7 3y2 2 a Find the critical points of b Use the second derivatives test to determine the local maximum and minimum values and saddle points of fz A rectangular box Without a lid is to be made from 12 m2 of cardboard Use the method of Lagrange multipliers to nd the maximum volume of such a boxi a Find the arc length of the curve rt lt 2sint5t2 cost gt 710 S t S 10 2 3 t5 7 gt at a general point and at 00 0 Find the absolute maximum and minimum values of the function fz y 5 7 31 4y on the closed triangular region With vertices 0 0 40 and 45 b Find the curvature of the curve rt lt t Suppose that you are looking for the extremum values of the function fz y 12 y2 subject to the constraint 14 y4 1 using Lagrange multipliers a Write out the Lagrange conditions for the problem that is equations for the points at Which the extrema must occur b Solve your equations in part a to nd the extremum valuesi Let F 7731 l Ji Determine F d39r for the circle at 9 0 a 12 y2 2 b I742y2 4 c Does F have a potential function in the region R2 00 How about in the right halfplane z gt 0 Suppose that you are standing at the point 1225 on a hill Whose shape is given by 2 36 312 2y In What direction on your topographic map do you need to go in order to descend as steeply as possible Use the curloperator to show that the vector eld F yzyli 12zjzy22 is conservative Find a potential fz y 2 for F For Which values of the parameters a and b is the vector eld b c G z ay2i 12 y3j my bzk conservative


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