É' B > Page 1 Calculus III : Project 1 /40 Due 9/09/2016 "Þ Ð3 points) Compute the following derivatives.

Transcription

1 Page 1 Calculus III : Project 1 /40 _ Due 9/09/2016 "Þ Ð3 points) Compute the following derivatives. The Mathematica website shows how to calculate derivatives "step by step". It can be used to easily verify your results. a) f ÐBÑ œ %B # " w È f ÐBÑ œ $ B 2 & sinðbñ w b) f ÐBÑ œ B for B Á 0 f ÐBÑ c) f ÐBÑ œ B & ln Ð/ #B Ñ f w ÐBÑ # B # w d) f ÐBÑ œ B e f ÐBÑ œ e) f ÐBÑ œ BB w f ÐBÑ É' B > f) f ÐBÑ œ / # w.> f ÐBÑ!

2 Page (3 points) Evaluate the following integrals. Both Wolfram Research On Line Integrator at and Wofram Alpha evaluates indefinite integrals symbolically, but do not supply the arbitrary constant, hence their results may differ by an algebraic rearrangement from any other correct answer. The WinPlot program or a graphing calculator can evaluate definite integrals numerically. To use Winplot to evaluate the definite integral, ', 0ÐBÑ db use the 2-dim Window. Enter the integrand, i.e., 0ÐBÑ, + under the 1. Explicit y = f(x) Equa format. From the View menu choose Vew/Set corners to set left less than + and right larger than,. The down and up values depend on the extremes of 0ÐBÑ on Ò+ß,Ó. From the View menu, Grid may be chosen to set convenient tick marks and tick labels. To numerically evaluate the definite integral choose Measurement/Integration from the One menu. Set the lower limit to + and the upper limit to,. Choose the desired number of subintervals (i.e., the 8 value for the numerical quadrature), select the approximation method to be used (left endpoint, midpoint, right endpoint, trapezoidal, parabolic (Simpson's rule) and random) and then press the definite button to see the numerical result. Answers to indefinite integrals can also be checked this way by seeing if the calculated value of the associated definite integral with chosen upper and lower limits agrees with the numerical definite integral (see ). a) ' cos % ÐBÑ sin ÐBÑ d B b) ' sinðèbñ ÈB d B c) ' _ 0 B/#B db d) ' " "! B(B"# #.B e) ' " È.B #BB# f) ' B" $ B" #.B

3 Page Ð4 pointsñ Evaluate the following limits.wofram Alpha may prove helpful. sin a) lim # Ð! BÑ # BÄ! B b) lim BÄ_sin # Ð! BÑ B# c) l im BÄ_ % #B lnðb / Ñ ÈB* # d) lim BÄ_ Ð"! B Ñ B tanð# 1BÑ e) lim BÄ" B" # f) lim BÄ! sinð#bñ sin" ÐBÑ g) lim tanh( #B) BÄ_ h) lim BÄ! ' 1B sinð>ñ! >.> B

4 Page (6 points) Give the equations for the following geometrical objects : a) A sphere of radius % centered at the point ( $, #, & ). b) A parametric equation for the line passing through Ð ", #, $ ) and (", &, #Ñ. c) The symmetric equations for the line of part b). d) The plane containing ( #, &, "%Ñ, Ð$, ", *Ñ, and Ð#, $, "!Ñ. e) A parametric equation for the line of intersection of the two planes : B# CD œ# and BC Dœ". f) The angle between these same two planes. 5. (3 points) Consider the plane, defined by the equation and the point with coordinates. ß U #B#CDœ"& T Ð""ß (ß'Ñ a) Find a parametric equation for the line through T normal to UÞ b) Where does this normal line through T intersect U? c) What is the distance from T to U? 6. (4 points) For each equation below in either cylindrical ( <ß ) ß DÑ or spherical Ð3 ß ) ß 9Ñ coordinates identify and/or describe the surface. a) <œ+! b) 3 œ+! 1 1 c) 9 œ ' d) ) œ $ # # # e) Dœ< f) D œ< # # # # # # # # g) D < œ + h) 3 Ðsin 9 cos ) cos 9Ñ œ +

5 Page Þ (4 points) Match each figure with an equation from the list below which could generate the surface shown. Write the answer in the blank beneath the figure. # # # # # # # # # # # a) C D œ " b) B C D œ! c) B C D œ" d) B C D œ " # # # # # # # # # # e) B C Dœ! f) B C Dœ! g) B C D œ" h) B C D œ " # # # # # # # # # i) B C D œ! j) B C Dœ! k) B C œ " l) B D œ " Figure A Figure B Figure A Figure C Figure B Figure D Figure C Figure D

6 Page 6 6 Figure E Figure F Figure E Figure F Figure E Figure F Figure G Figure H

7 Page (2 points) Sketch the surface generated by each equation. # # # # # a) B D œ* b) B C D œ* # # # # # # c) B C D œ* d) B C D œ *

8 Page (5 points) In cylindrical coordinates <ß ) ß and Done can form the following vectors : Y< œ cos ) s3 sin ) s4 ; Y) œ sin ) s3 cos ) s4 ; a) Evaluate the following : Y Y œ Y Y œ Y 5s œ Y Y œ < < < ) < ) ) Y 5 s œ Y Y œ Y 5s œ 5 s Y œ ) < ) ) < p p b) The position vector VœBs3 C s4 D s5 can also be written as Vœ< Y s < D5. What are the transformation equations for the Cartesian coordinates Bß Cß and D in terms of < ß ) ß and D? In spherical coordinates 3ß9 ß and ) one can form the following vectors : Y3 œ sin 9 cos ) s3 sin 9 sin ) s4 cos 9s5 ; Y9 œ cos 9 cos ) s3 cos 9 sin ) s4 sin 9s5 ; Y) œ sin ) s3 cos ) s4 ; c) Evaluate the following : Y 3 Y3 œ Y 3 Y9 œ Y 3 Y) œ Y 9 Y9 œ Y 9 Y) œ Y ) Y) œ Y 3 Y9 œ Y ) Y3 œ Y 9 Y) œ p p d) The position vector VœBs3 C s4 Ds5 can also be written as Vœ3Y 3. What are the transformation equations for the Cartesian coordinates Bß Cß and D in terms of 3ß 9, and )?

9 Page (6 points) For each equation below i) Sketch the surface generated. ii) Give a valid parametric representation of the surface with a single set of three equations of the form: B œ 0Ð?ß >Ñ C œ 1Ð?ß >Ñ D œ 2Ð?ß >Ñ Þ Include in your representation the domain of the parameters? and >. iii) Generate and attach with your project a Winplot 3D graph of your parameterization. # # # # # a) The cylinder: C D œ% with #ŸBŸ# b) The elliptic hyperboloid B C D œ% # # # c) The hemisphere: B C D œ % with B Ÿ!

10 Page Self-Assessment: (2 bonus points) a) Describe three strengths in your perfomance on this project. Include why each is a strength. b) What are three things that could be improved about your performance on this project. Explain specifically how you will make these improvements. c) Identify two things about this project which are still unclear to you. d) Identify two insights that you have acquired in doing this project.

11 Page 11 Calculus III : Project 2 /40 _ Due 9/16/ (2 points) Consider the curve CœEsin Ð,BÑ, where E and, are positive constants. a) Determine the curvature,,, as a function of B, E and,. b) Determine the radius of curvature, 3, as a function of B, E and,., œ 3 œ c) For what values of B is the curvature a maximum? Explain the relationship between this answer and the graph of the curve. d) For what values of B is the curvature a minimum? Explain the relationship between this answer and the graph of the curve. 2. (8 points) Consider the parabola Cœ,B #, where, is a positive constant. a) Determine the curvature,,, as a function of B and,., œ b) What is the maximum value of the curvature and where does this occur on the parabola? c) Determine the unit normal vector to the parabola at the point Ð>ß,> # Ñ. # d) Determine the radius of curvature, 3, as a function of > and, at the point Ð>ß,> ÑÞ 3 œ

12 Page 12 2 # e) Determine the coordinates of the center of curvature to the parabola at the point Ð>ß,> Ñ Þ f) Using Winplot generate and attach the following two graphs. i) For,œ" the plot of the parabola and a parametric plot showing all the centers of curvature (called the "evolute") to the parabola. ii) For,œ# the plot of the parabola and a parametric plot showing all the centers of curvature to the parabola. 3. (3 points) Consider the polar curve <œ0ð) Ñ, where 0is any twice differentiable function. a) Determine an explicit formula for the curvature, in tems of 0 and its derivatives.,) ÐÑ b) What does your answer to part a) give for 0Ð) Ñ œ +, where + is a positive constant? Is this the expected answer? c) What does your answer to part a) give for 0Ð) Ñ œ #+ cosð) Ñ, where + is a positive constant? Is this the expected answer?

13 Page 13 3 It is possible to view and generate computer plots of three dimensional trajectories. In WinPlot use the 3-dim Window and the Curve space curve option of the Equa menu to enter equations for the Bß C, and D components of the curve in terms of the parameter >. 4. (4 points) For the curve below find the parametric equation for the tangent line to the curve at the given point and at this point calculate the curvature Ä,. V Ð > Ñ œ #/ > #>/ > 4 Ð> # 3s s >$Ñ s 5 at Ð #ß!ß $Ñ Tangent line equation, œ In the remaining problems the following notations will be used: Ä Ä V Ð>Ñ œ BÐ>Ñs3 CÐ>Ñs4 D Ð>Ñ s 5, where V is the position vector of a given object at time >. The Cartesian Ä components of V are also considered to be functions of time. Ä Ä Ä ± V l œ É V V œ ÈB2 C2 D 2 is the length of the position vector at time >. Ä œ V V B C 4 D d> ŠÄ œ Ä œ Ð>Ñs3 Ð>Ñs Ð>Ñ s 5 is the instantaneous velocity vector of the object. ± ± œ Ä œ È B. 2 C. 2 D. 2 is the speed of the object. Ä d + œ d> is the instantaneous acceleration vector of the object. 5. (10 points) A positively charged particle moving in a uniform magnetic field directed along the positive D axis will travel in a helical trajectory of the form : Ä V ÐÑ > œ<! cos Ð = > Ñ 3s <! sin Ð = > Ñ > s! 5, where and = are positive constants. a) What is the particle's velocity in Cartesian coordinates? b) What is the particle's acceleration in Cartesian coordinates? c) What is the particle's speed?

14 Page 14 4 d) What is the unit tangent vector of the particle's motion at time >? 21 e) How far does the particle travel along its trajectory in one turn of the helix (i.e., over any = time interval )? f) What is the curvature,,, of the trajectory? g) For fixed = and, what is the maximum value of, as v varies? r!! h) What is the radius of curvature, 3, of the trajectory? i) What is the principal unit normal vector of the trajectory? j) What is the binormal vector of the trajectory? k) What is the torsion, 7, of the trajectory? l) For fixed = and r!, what is the maximum value of l7l as v! varies? m) What is the tangential component of acceleration? n) What is the normal component of acceleration?

15 Page (4 points) Given that E, F, Gß! ß " ß # are all constants, and 5is a positive constant, solve the following vector differential equation for Ä V Ð>Ñ : Ä + œ 5 Ä with V Ð!Ñ œ E s3 F s4 Gs5 and Ð!Ñ œ! s3 " s4 # 5sÞ 7. (3 Bonus points) The following table gives astronomical data for the four inner planets of our solar system in units where the earth's distance to the sun is taken as "Þ!! and time is measured in earth years. As a simplification, all the planets are assumed to orbit the sun in the ecliptic, the plane of the earth's orbit (i.e., the angle of inclination is taken as zero). The angle?= is the sum of the longitude of ascending node and the longitude of perihelion of each planet minus the longitude of perihelion of the earth. In the approximation that the angle of inclination of a planet is zero,?= is the angular distance between the perihelion of the earth and the perihelion of the planet. The radial distance from the sun at perihelion is designated by r!. Planet Semimajor Axis r! Eccentricity Orbital Period?= Mercury!Þ$)(!Þ#!&' ##Þ#( Venus!Þ(#$!Þ!!') "!%Þ*) Earth "Þ!!!!Þ!"'( "Þ!!!!Þ!! Mars "Þ&#%!Þ!*$% #)#Þ") a) Determine the relationship between the semimajor axis, <!, and the eccentricity. From this equation calculate and fill in the values of r! missing from the above table. b) From Kepler's third law calculate and fill in the values of the orbital period missing from the above table. c) Using a computer plotting program generate and attach with your project a plot of the orbits of the four inner planets using the data in the above table. For convenience, take the perihelion of earth to occurr at a polar angle of ) œ!. Remember to convert all angles to radian measure before inputing to a trig function.

16 Page (4 points) Simplify the following expressions : d a) d> Š Ä œ b) d d> œ c) d ± l œ d) d d> ± l œ 9. (5 points) Ä Given V œ< s< D s 5, with s< œ cos ) s 3 sin ) s 4 ; s ) œ sin ) s 3 cos ) s 4 ; ) œ = >, = a constant ; ) Dœ," Ò cosð # ÑÓ ; <œ +" Ò cosð) Ñ Óà + and, are positive constants. a) Using a computer plotting program generate a three dimensional graph of this parametric space curve when +œ,œ". b) What is the period of this trajectory? c) Express in terms of s< ß s ) and s 5. d) Express Ä + in terms of s< ß s ) and 5 s. e) Is this trajectory the result of a central force? Explain.

17 Page 17 7 Self-Assessment: (2 bonus points) a) Describe three strengths in your perfomance on this project. Include why each is a strength. b) What are three things that could be improved about your performance on this project. Explain specifically how you will make these improvements. c) Identify two things about this project which are still unclear to you. d) Identify two insights that you have acquired in doing this project.

18 Page 18 Instructions for Group Work The following guidelines should be adhered to in forming your group, doing the work and writing up the project. Group Requirements: Each group must consist of at least two individuals but no more than four individuals. You are free to form your own groups, but if you can't find a partner see me and I'll assign you to a group. Some class time will be devoted to group work, but much of it will have to be done outside of class. It is up to the group to decide any internal division of labor, eg., who is responsible for what parts of the problem, who will be the 'algebra expert', who will check the work, who will write up what parts of the report. It is possible one group has one individual write the entire problem, while in another group everyone writes up different parts. It is in your own best interest to insist that you understand the solution of the whole problem. It might be a good idea for each person to have a copy of the finished report, so that when the assignment is due there will be at least one person present who can hand it in! You are free to use any written resources or computing technology in solving the problem. Report Requirements: Each group must hand in one solution for a given project which should include the following : 1. The names of all group participants. If the report writers feel an individual did not perform his/her assigned task, you are free to delete that person's name from the report. I will arbitrate all appeals on such disagreements and reserve the right to give either a written or oral exam to decide the issue. 2. The conclusions of any questions stated neatly in complete sentences which are both concise and complete in expressing the answers. 3. The mathematical work to each problem attached in a way which is both neat and clear. Solutions should be presented in the report in the same order that the associated problems appear in the project. Grading: 1. Each person in the group will receive the same point total out of the point value of the group portion of the project. Thus, it is the responsibility of everyone in the group to review the answers to all of the questions. 2. Grades will be based on the mathematical correctness both of the results and the methods used to arrive at them. Thus, a right answer arrived at by accident using faulty mathematics will not count for much. Points will be deducted for incomplete, illegible, sloppy or incomprehensible answers.

19 Page 19 Calculus III : Group Project 1 ( 9 points ) Due 9/26/2016 Choose any two of Problems 1 through Problem 5. You may do all five problems for up to 14 bonus points. 1. A carnival ride called the "Screeches" consists of a central arm of length %Þ& m which rotates at an angular velocity of & rpm (revolutions per minute). Attached to the outer end of the central arm is a second arm of radius $Þ! m which rotates in the same direction with an angular velocity of #) rpm. A passenger carrying gondola is located at the outer end of this second arm. a) What is the maximum magnitude of the velocity in meters per second experienced by a person riding in the gondola? b) What is the minimum magnitude of the velocity in meters per second experienced by a person riding in the gondola? c) What is the maximum magnitude of the acceleration experienced by a person riding in the gondola? Express the answer in units of 1 œ *Þ)! m per sec 2 Þ d) What is the minimum magnitude of the acceleration experienced by a person riding in the gondola? Express the answer in units of 1 œ *Þ)! m per sec 2 Þ e) What is the time interval between the experience of maximum magnitude of acceleration and minimum magnitude of acceleration?

20 Page 20 Calculus III : Group Project 1 ( 9 points ) Due 9/26/ The Twisted Cubic. Consider the space curve defined parametrically as Ä # #> V Ð>Ñ œ > ß $ ß > a) Determine an expression for the unit tangent vector, X s Ð:Ñ, at >œ: Þ $ b) Determine an expression the curvature,,ð:ñ, at > œ :. c) Determine an expression for the unit normal normal vector, R s Ð:Ñ, at >œ: Þ d) Determine an expression for the unit normal binormal vector, F s Ð:Ñ, at >œ: Þ e) Determine an expression for coordinates of the center of curvature at > œ:þ f) In Winplot generate a graph of of this space curve for #Ÿ>Ÿ#. The Winplot graph needs to include the TNB vectors Ä displayed as attached to the trajectory at the point V Ð:Ñ for # Ÿ : Ÿ # Þ This Winplot file should be attached to an sent to me at alehnen@matcmadison.edu. I need to receive this file the day the Group Project is due. Bonus (5 points): Ä g) Include in your Winplot graph the osculating circle attached to the trajectory at the point V Ð:Ñ for # Ÿ : Ÿ # Þ

21 Page 21 Calculus III : Group Project 1 ( 9 points ) Due 9/26/ In this problem the following notations are used: Ä The relative position vector from mass Q to mass 7 at time > is V Ð>Ñ œ B Ð>Ñ s 3 C Ð>Ñ s4 D Ð>Ñ s 5, where the Cartesian Ä components of V are functions of time. The instantaneous separation velocity vector is œ d V V B C 4 D d> ŠÄ œ Ä œ Ð>Ñs3 Ð>Ñs Ð>Ñ s 5. The instantaneous separation acceleration vector is Ä d + œ d> Ä After removing the motion of the center of mass from the problem, Newton's Second Law of Motion states F œ. Ä Ä +, where F is Q7 the net force acting on mass 7due to mass Q and. is the "reduced" mass given by. œ Q7. If the force on 7 due to Q is gravitational, it is given by Ä KQ7 Ä F œ Ä V, where K is the universal gravitational constant. l V l$ An object of constant mass, 7, is acted upon by a constant mass Q by some force Ä F. Ä Ä The angular momentum of the system is defined as P œ V. Ä dp Ä Ä a) Show that d> œ V F Ä Ä b) Suppose that F is a central force. This means that F 0б Ä Ä V œ V ±Ñ Ä, where 0 is a function only of the distance from Q lvl to 7 and not the direction. Show that Ä P must now be a constant vector. Ä c) Explain why this proves that in a central force the vector V is confined to a plane. d) Show that d d> Ä Š ± V œ Ä V ± Ä V±. e) The kinetic energy, K, is given by the expression : O œ Ä Ä do. Show that œ Ä F.

22 Page 22 2 f) Consider an object of mass, 7, attracted by gravity to an object of mass, Q. The potential energy is given by GMm dy GMm Ä V Ä Yœ Ä ; Show that d œ Ä. ± V± > œ F ± Vl $ g) Show that the total energy, IœOY, is a constant. In establishing Kepler's first law, the eccentricity of the orbit took the form, & œ 2! GM Ð 7Ñ ". h) What is the total energy, I, of the orbit at perihelion? i) What is the total energy at any other point along the orbit? j) Show that & " œ 2 r! I KQ7. k) If I!, what do you conclude about the orbit? l) If Iœ!, what do you conclude about the orbit? m) If I!, what do you conclude about the orbit? o) The probability of a comet having a parabolic orbit is zero. Explain why.

23 Page 23 Calculus III : Group Project 1 ( 9 points ) Due 9/26/ A planet follows a Kepler orbit (i.e., it satisfies Kepler's three laws) on an ellipse of semi-major axis + and eccentricity & about a star located at a focus of the ellipse. a) Calculate the mean distance from the star to the planet when averaged over the arc length of the ellipse. b) Let the planet be at position Tß the star is at the focus J, and let the position of closest approach (called perihelion if the star is the sun) be G. Let ) be the angle TJGÞCalculate the mean distance from the star to the planet when averaged over ) for a full revolution. [Hint: Look up the Weierstrass substitution for evaluating integrals of rational functions of sine and cosine.] Bonus (4 points): c) Calculate the mean distance from the star to the planet when averaged over time for a full revolution.

24 Page 24 Calculus III : Group Project 1 ( 9 points ) Due 9/26/ An ant at the bottom of an almost empty sugar bowl eats the last few remaining grains. The poor creature is now too bloated to climb vertically as it normally can. The steepest incline it can manage is to climb at a 45 angle with respect to the horizontal. The sugar bowl # # BC is a paraboloid satisfying the equation Dœ +, with!ÿdÿ%+þ a) Assume the ant starts out of the bowl by traveling along the positive Baxis ( ) œ!) and proceeds at ) œ! until it turns to its left when the angle of ascent equals 45. The ant then maintains a constant angle of ascent of 45. Find the path the ant takes. Hint : Use cylindrical coordinates Ð<ß ) ß DÑ and think of the ant's path as parameterized by <. Find the relation between the differentials.) and.< ß then integrate this relation to obtain ) as a function of < Þ The element of arc length in cylindrical coordinates is given by.= œ È.<# < #.)#.D# Þ b) At what point on the rim does the ant emerge from the bowl? c) How many multiples of #1 did the ant wind around in order to escape the bowl? d) Set up and evaluate the integral which gives the total distance travelled by the ant from the bottom to the rim.

25 Page 25 Calculus III : Project 3 /40 _ Due 10/07/ (4 points) Below are eight contour diagrams of surfaces each made with a constant difference between contours. Match each contour with the correct 3-D surface graph on pages 3 and 4 as well as an equation from the list below which could generate the surface shown. Write the two answers for each contour in the blank provided. # # B C # # B C # # # a) Dœ % * b) Dœ * % c) B C D œ" d) DœcosÐBCÑ # # # # e) Dœcos ÐBCÑ f) DœcosÐBÑcos ÐCÑ g) DœexpÐB ÐC"Ñ Ñexp ÐB ÐC"Ñ Ñ # # # # # # h) DœBC i) DœB C j) DœC B k) DœBC Contour A Contour B Contour C Contour D

26 Page 26 2 Contour E Contour F Contour G Contour H

27 Page 27 3 Surface Graph I Surface Graph II Surface Graph III Surface Graph IV

28 Page 28 4 Surface Graph V Surface Graph VI Surface Graph VII Surface Graph VIII

29 Page (2 points) For the given functions (1) State the functions's domain (2) State the functions's range. (3) Sketch the level curves of the function. (4) Sketch the surface Dœ0ÐBßCÑ a) 0ÐBßCÑœÈ B# C# " 10 y x b) 0ÐBßCÑœ È "B# C# y x

30 Page (4 points) For the given functions evaluate the following : # a) 0ÐBß CÑ œ B sinðbcñ lim ÐBß CÑ Ä Ð"ß 1Ñ 0ÐBßCÑ `0 `B `0 `C # `0 `C# # `0 `C`B ` `B Ð Ñ `0 `C # # b) 0ÐBßCÑ œ / #ÐB C Ñ, where 5is a positive constant. lim ÐBßCÑÄÐ!ß!Ñ "0ÐBßCÑ BC # # `0 `B # `0 `C# ` `C Ð Ñ `0 `B # # # # c) 0ÐBßCÑ œ CB ÈBC lim 0ÐBßCÑ ÐBß CÑ Ä Ð!ß! Ñ `0 `B `0 `C d) 0ÐBßCßDÑœ " È BCD # # # à T œðbßcßdñ lim 0ÐBßCßDÑ T ÄÐ!ß!ß!Ñ `0 `B # `0 `D# # # # `0 `0 `0 `B# `C# `D#

31 Page (3 points) Consider a right circular cylinder of radius < and height 2. a) If the radius increases by!þ&!% while the height decreases by!þ(&!%, what is the approximate percent change in the cylinder's volume? b) If there is no change in volume for a!þ&!% increase in radius, what is the approximate % decrease of the height? c) When < œ!þ&!! m and 2 œ "Þ&!! m, the radius is increasing at a rate of "Þ!! mmper minute, while the height is increasing at a rate of!þ(& mm per minute. At this instant how fast is the volume changing? 5. (3 points) The perpendicular component of the alternating electric field inside a metal varies according the equation : -B IÐBß>Ñ œ I! cos Ð = >Ñ/ ß where > is time, B is the distance from the surface of the metal, - and = and I! are positive constants. a) What is the physical meaning of = (omega)? b) What is the physical meaning of I!? c) What is the physical meaning of -? d) `I `B e) `I `> f) What is the physical meaning of `I `B? g) What is the physical meaning of `I `>?

32 Page (6 points) # # # a) Given?œBC D BCß Ð `CÑ `B?ßD # # # # b) Given 0ÐBßCßDÑœB CÐBDÑand AÐBßCßDÑœB C D, evaluate the following: Ð `0 Ñ `B CßD Ð `0 Ñ `D BßC Ð `0 Ñ `B CßA Ð `0 Ñ `D BßA c) The equation of state for an ideal gas is TZ œ 8VX ß where T is the pressure, Z is the volume, X is the absolute temperature, 8is the number of moles of gas (proportional to the mass of gas present) and V is a universal constant. Suppose that a fixed mass of gas occupies a spherical balloon. When the radius is $!Þ! cm, the temperature is $!! K, and the pressure remains constant what is the rate of increase of the radius with respect to temperature. 7. (2 points) Given 0ÐBßCßDÑ œ " Ä È and V œ s 4 5 BCD # # # 3 # s s a) f0 Ä b) the derivative of 0 at Ð#ß "ß #Ñ in the direction of V

33 Page (3 points) # # Let C be the intersection of the surface DœB #C with a sphere of radius three centered at the origin. a) Verify that Ð #ß "ß #Ñ lies in C Þ b) Give a parametric equation for the line tangent to C at Ð #ß "ß #Ñ Þ # # c) What is the angle between the tangent plane to D œ B #C and the tangent plane to the sphere at Ð #ß "ß #Ñ? d) What is the maximum value of D on C?

34 Page (4 points) For each function below find all critical points and determine if each critical point is a maxima, minima or saddle point. a) 0ÐBßCÑ œ / C # # ÐB %Ñ # # b) 0ÐBßCÑœB C %B#C"# B# C# # # # # Ð Ñ # # BC c) 0ÐBßCÑœÐB #C Ñ/ # œðb #C ÑexpŒ # % $ d) 0ÐBß CÑ œ B )BC 10. (1 point) Find the absolute extrema of the frunction, 0ÐBßCÑœC %BC%B, in the closed region bounded by CœBBœ%,, and Cœ! # #.

35 Page (4 points) Consider the "lumpy" surface D œ cosðcñ cosðbñ. a) Find all maxima, minima and saddle points of this surface. b) What are the level curves for D œ!þ c) Consider going from the origin to Ð# 1 ß!ß!ÑÞ If the path chosen stays on level curves for Dwhat is the length of the shortest path? d) Consider going from the origin to Ð# 1ß!ß!Ñalong the curve D œ"cosðbñþ Set up and numerically evaluate the integral which gives the length of this path.

36 Page (2 points) # A mass 7 is attached to a spring of spring constant 5 œ 7=!. The total energy I is a constant and satisfies the equation #I 7 œ@ # # B # =! is the speed of the mass and B is the displacement of the mass from equilibrium. The instantanious power # supplied to the mass by the spring is given by Tœ7 terms of Iand =, what is the maximum power?!! 13. (2 points) A building has a pentagonal cross section as shown below. Find the values of + ß, and 2 which minimize the perimeter of the cross section while keeping the cross sectional area at $'Þ! m 2.

37 Page Self-Assessment: (2 bonus points) a) Describe three strengths in your perfomance on this project. Include why each is a strength. b) What are three things that could be improved about your performance on this project. Explain specifically how you will make these improvements. c) Identify two things about this project which are still unclear to you. d) Identify two insights that you have acquired in doing this project.

38 Page 38 Calculus III : Group Project 2 ( 9 points ) Due 10/17/2016 Choose two of problems 1, 2, 3 or 4. You may do extra problems for up to 9 bonus points. 1. In thermodynamics one defines 'state functions' whose values are path independent. This means that their differentials must be 'exact'. For example the internal energy, Y has the differential form.y œ X.W T.Z, where W is the entropy, X is the absolute `Y `Y temperature, T is the pressure and Z is the volume. Since Y is 'exact', Ð `W ÑZ œ X and Ð `Z ÑW œ T Þ 2 2 ` Y ` Y `T `X Since `W`Z œ `Z `W, we are lead to the result Ð `W ÑZ œ Ð `Z ÑW Þ This equation is called a 'Maxwell Relation'. There are three other state energy functions related to Y Þ They are as follows : Enthalpy Helmholtz Free Energy Gibbs Free Energy LœYTZ JœYXW KœLXW a) Expess LJ ß and K in their differential forms.lœ.j œ b) Determine from each differential the corresponding 'Maxwell Relation'..K œ Maxwell Relation from.l Maxwell Relation from.j Maxwell Relation from.k c) Assuming the ideal gas equation of state TZ œ 8VX, Vis a universal constant and 8 is kept constant, determine the following : Ð `Y `Z Ñ Xß8 œ Ð `Y `T Ñ Xß8 œ d) The molar heat capacity at constant volume is G Z œ " Ð `Y Ñ 8 `X Zß8, while the molar heat capacity at constant pressure is " $V GT œ 8 Ð `LÑ `X Tß8. For a monatomic gas such as helium GZ œ #, assuming it behaves as an ideal gas, determine GT for helium. GT œ

39 Page 39 Calculus III : Group Project 2 ( 9 points ) Due 10/17/ A trash bin with no lid has five sides and is in the shape of a trapezoidal prisim as shown in the figure below. If the volume of the bin is to be "'Þ! m $, determine the values of + ß, and - which minimize the surface area.

40 Page 40 Calculus III : Group Project 2 ( 9 points ) Due 10/17/ A radio controlled four wheeled droid with an attached GPS system is being used on the face of a glacier which is nearly a perfect plane. The GPS system outputs the B (East), C (North) and D(elevation) coodinates of the droid to the controllers. When the droid makes a horizontal displacement of #Þ!! m (i.e., È Ð? BÑ# Ð? CÑ# œ #Þ!! m) in the direction N%& W (North West), the vertical displacement is a positive!þ&! m (i.e.,? D œ!þ&! m). When the droid makes a horizontal displacement of $Þ!! m in the direction E%& N (North East), the vertical displacement is a negative!þ%! m. a) What direction would make the droid move up the glacier as quickly as possible, i.e., what is the direction of steepest ascent? b) If the droid moves at a horizontal speed of!þ%! m per second in the direction of part a) what would the rate of ascent be? c) If the droid moves off at an angle ) measured counter clockwise with respect to East and travels with a horizontal what is the rate of ascent? d) In what directions can the droid move and neither ascend or descend the glacier?

41 Page 41 Calculus III : Group Project 2 ( 9 points ) Due 10/17/ # # # a) Find the equation of the tangent plane to the cone D œ+ðb C Ñ at ÐB ßC ßD Ñ assuming ÐB ßC ßD Ñ is not the origin.!!!!!! b) Does this plane pass through the origin? Explain your answer. C c) Consider the surface DœC0ÐBÑ for any differentiable function 0Þ Show that all tangent planes to this surface pass through the origin.

42 Page 42 Calculus III : Project 4 /40 _ Due 10/28/ (6 points) Evaluate the following multiple integrals over the regions indicated. a) ' " ' $!! # Ð#B C Ñ.C.B _ b) ' '!! _ # BC / ÐB #CÑ.C.B œ c) ' '!! B B / #ÐBCÑ.C.B d) H is the disk of radius + centered at the origin and 0ÐBßCÑ œ '' H 0ÐBßCÑ.E " %+# B# C# Þ e) H is the region in the first octant bounded by the coordinate planes and the plane #BC$D œ 'Þ ''' H BC.Z f) H is the region in the first octant bounded by the coordinate planes, the cylinder < œ # and the plane BD œ $Þ ''' H BC.Z

43 Page (2 points) To analytically evaluate the following integrals, first sketch the region of integration then change the order of integration in some appropriate way. Èln a) ' ' # Ð&Ñ ÈlnÐ&Ñ B! # / C #.C.B œ # È1 1B B b) ' ' '!!! sinðdñ 1D.C.D.B 3. (2 points) Consider the following integral : ' 1 ' 1 sinðbñ, the integral with respect to as written can not be done analytically.! C B.B.C B However, an analytic answer can still be obtained. a) Sketch the region of integration of this integral. sinðbñ b) What is the infinite Maclaurin series expansion for B? What is the radius of convergence of this expansion? c) Using the answer for part b) integrate term by term to obtain an infinite series answer for ' 1 ' 1! C sinðbñ B.B.C Þ d) Evaluate the integral directly by changing the order of integration. e) Show that your answer in part d) is equivalent to your infinite series of part c).

44 Page (2 points) Transform the following integrals from Cartesian coordinates to either polar (for 2D) or cylindrical (for 3D) coordinates and then evaluate them. a) ' ' + È + # B# È+ # B# C #.C.B +! 2 + È+ B b) ' ' ' 2 + # # È+B # # ln Ð # # Ñ BC + # ".C.B.D 5. (2 points) Transform the following integrals from Cartesian coordinates to spherical coordinates and then evaluate them. È a) ' ' ' + + # B # È+ # B# C# + È+ B È+ B C # # # # # # C.D.C.B È b) ' ' ' + + B È+ B C + È+B # #! # # # # # # D.D.C.B 6. (3 points) B _ One can calculate the area under the Gaussian curve 0ÐBÑ œ / by noting that 0ÐBÑ.B œ É 0ÐBÑ.B 0ÐCÑ.C!!! then transforming the expression under the radical to polar coordinates. # ' ' ' a) _ ' B # /.B!

45 Page 45 4 b) ' B # /.B c) For a constant mean (average),., and constant standard deviation, 5, the Normal Distribution Probability density function is given " B. # by :ÐBÑ œ EexpÐ # Ð 5 Ñ Ñ, where E is a constant chosen so that the area under the curve C œ :ÐBÑ from _ to _ is one. Determine the value of E. (Þ (7 points) Calculate the areas (for 2 D regions) or volumes (for 3 D regions) of the following : 1 a) The region in the first quadrant bounded by the lines Cœ!, ) œ ' and the cardioid <œ"cosð) ÑÞ b) The "crystal" with six planar faces and seven of its eight vertices at Ð!ß!ß!Ñ ß Ð"ß #ß!Ñ ß Ð!ß!ß "!Ñ ß Ð!ß #ß!Ñ ß Ð!ß #ß %Ñ ß Ð"ß!ß!Ñ and Ð"ß!ß)ÑÞ c) The region in the first octant bounded by the planes BCDœ" and #B#CDœ#Þ d) The spherical cap which is the intersection of D, and the interior of a sphere of radius +, +,, centered at the origin Þ e) The intersection of the region bounded by a sphere of radius + centered at the origin and a right circular cylinder of radius,,, + ß whose axis is the D axis.

46 Page 46 5 f) The region in the first octant bounded by the plane Dœ!ß the cylinder <œ#, and the plane which intersects the coordinate axes at Ð!ß!ß%Ñß Ð$ß!ß!Ñ and Ð!ß$ß!ÑÞ 1 1 g) The interior of the sphere, 3 Ÿ+, that lies between the two cones 9 œ ' and 9 œ $ Þ 8. (2 points) Consider the 4 points in the BC plane whose polar coordinates are Ð< " ß )" Ñ ß Ð< " ß )# Ñ ß Ð< # ß )" Ñ and Ð< # ß )# Ñ Þ Take?) œ )# ) "! and? < œ < # < "!. a) Using trigonometry determine,?e, the area of the quadrilateral with these four vertices. b) Determine the area,? F, of the region defined by ) Ÿ) Ÿ ) and < Ÿ<Ÿ< Þ 1 # 1 # c) Explain which of? F or? E is larger.? E d) Evaluate lim Ð?) ß? <Ñ Ä Ð!ß!Ñ? F 9. (3 points) Let H be the region in the first quadrant bounded by the B axis, C œ B and C œ * B Þ a) Evaluate '' C.E H b) Given? œ BC œ B, determine the Jacobean for the transformation from ÐBß CÑ to Þ

47 Page 47 6 c) Using the substitution of part b) sketch the region of the?@ plane which is the image of H Þ '' d) Calculate the value of C.E in the transformed H 10. (3 points) Let be the region and. H "ŸBCŸ# "ŸBCŸ$ a) Evaluate '' H # # ÐB C Ñ.E b) Given?œBC determine the Jacobean for the transformation from ÐBßCÑto Ð?ß@ÑÞ c) Using the substitution of part b) what region of the?@ plane is the image of H? '' d) Calculate the value of ÐB C Ñ.E in the transformed coordinates H # Þ 11. (1 point) B C D Using the transformed Aœ, evaluate ''' +, - lbcdl.z over the solid ellipsoid # # B C # D + #,# -# œ "Þ

48 Page (7 points) a) Calculate the centroid of the region in the first quadrant bounded by the coordinate axes and Cœ/ #B Þ b) Calculate the centroid of a uniform density right circular cone of base radius + and height 2, if the cone axis is the D axis and the base of the cone rests in the BC plane. c) Calculate the radius of gyration of a uniform density right circular cone of base radius + and height 2, if the axis of rotation is the cone axis. d) Calculate the radius of gyration of a uniform density solid sphere of radius + rotated about an axis through its center. e) Calculate the radius of gyration of a uniform density solid sphere of radius + rotated about an axis tangent to the sphere.

49 Page 49 8 f) Calculate the radius of gyration of a uniform density hollow sphere of inner radius + and outer radius, rotated about an axis through its center. g) Calculate the radius of gyration of a uniform density hollow sphere of of inner radius and outer radius rotated about an +, axis tangent to the sphereþ

50 Page 50 9 Self-Assessment: (2 bonus points) a) Describe three strengths in your perfomance on this project. Include why each is a strength. b) What are three things that could be improved about your performance on this project. Explain specifically how you will make these improvements. c) Identify two things about this project which are still unclear to you. d) Identify two insights that you have acquired in doing this project.

51 Page 51 Calculus III : Group Project 3 ( 9 points ) Due 11/09/2016 Choose one of Problems 1 and 2 (for 5 points), and one of Problems 3 and 4 (for 4 points). Extra problems can be worked for bonus. 1. (5 points) a) Calculate the centroid of a uniform density slab of material in the shape of an equilateral triangle of side WÞPlace one side of the triangle along the B axis so that the top vertex is in the first qudrant and the origin is at the left vertex. b) Calculate the radius of gyration of the equilateral triangle about an axis perpendicular to the plane of the triangle and passing through the origin. c) Calculate the radius of gyration of the equilateral triangle about an axis perpendicular to the plane of the triangle and passing through the center of mass.

52 Page 52 Calculus III : Group Project 3 ( 9 points ) Due 11/09/ (5 points) Consider a uniform density right circular cylinder of base radius + and height 2. a) Calculate the radius of gyration of the cylinder about an axis perpendicular to the axis of the cylinder and passing through the center of mass. b) Evaluate the limit of your answer to part a) as + tends to zero. c) Calculate the radius of gyration of the cylinder about an axis perpendicular to the axis of the cylinder and attached to the end of the cylinder in such a way that it passes through the center of the circle of radius. +

53 Page 53 Calculus III : Group Project 3 ( 9 points ) Due 11/09/ Sphere Game (4 points) A game is played by marking a point, A, on the surface of a spherical billiard ball. The ball is then shaken in a closed box and when the box is opened, the distance, P, from the point of tangency on the floor of the box to E is determined. This distance is the straight line distance along a segment passing from the point of tangency through the ball to point E. a) If the radius of the ball is +, what is the average of P over a large number (really an infinite number) of games? b) On any one play of the game what is the probability of getting a value of P larger than the average you calculated in part b). Bonus (4 points): c) On any one play of the game you win if our guess of Pis within? PÐ? P +Ñof the value that comes up. What would be the "best bet" for the value of P? Explain your answer.

54 Page 54 Calculus III : Group Project 3 ( 9 points ) Due 11/09/ Volume of a Hypersphere (4 points) An R dimensional hypersphere of radius + centered at the origin is the set of points ÐB ß B ß B ß ÞÞÞB Ñ in R dimensional Euclid space which satisfy the inequality B" # B# # B$ # ÞÞ. B # R Ÿ + # Þ a) Set up and evaluate the 4 D integral which gives the volume of a hypersphere in four dimensions. " # $ R ean 2 Bonus points: b) Set up and evaluate the 5 D integral which gives the volume of a hypersphere in five dimensions. Extra 2 Bonus points: c) Determine the recursion formula for the volume of a sphere in R" dimensions in terms of the volume of a sphere in dimensions. [ Hint : The formulas are different depending on whether R is odd or even.] R Extra-Extra 2 Bonus points: d) Solve the recursion formula to get an explicit formula for the volume of a sphere in dimensions. R

55 Page 55 Calculus III : Project 5 /40 _ Due 11/22/ (1 point) Integrate the scalar function # # / CD'B # # over that part of the surface of the cylinder C D œ * in the first octant. 2. (3 points) $ # # # # For positive + and 2, let E designate the region of d enclosed by the elliptic hyperboloid, B C D œ +, and the two planes, 2 2 Dœ # and Dœ #. Let F represent the orientable surface of E. a) Determine the volume of E. p p b) For the position vector field JœVœBs3 Cs4 Ds5 œ BßCßD, calculate the the flux out of the lateral surface of E. p p c) For the position vector field JœV, c alculate the the flux out of the top surface of E. p p d) For the position vector field JœV, c alculate the the flux out of the bottom surface of E. p p e) What is the total outward flux of JœVover F? '' p J 8.W s œ F f) Explain the relationship between your answer to part a) and your answer to part f).

56 Page 56 2 p 3. (" point) Integrate the scalar function 0ÐBß Cß DÑ œ C along the path V œ #> s # 3 > s4 > s5 from Ð!ß!ß!Ñ to Ð'ß *ß $ÑÞ 4. (2 points) Calculate the flow integral ' p J X.= s for the vector field p J along the path specified. ( p J X.= s p œ J.< p œ J B.B J C.C ) p a) J œ #Bs3 s # # $C 4 the fourth quadrant path from Ð&ß $Ñ to Ð)ß!Ñ along the curve B "!B C "' œ! p b) J œ BcosÐBÑs3 BsinÐCÑs4 ÈBCD s5 from Ð!ß!ß!Ñ to Ð111 ß ß Ñ along C œ B and D œ # # BC # 1 Þ 5. (5 points) For each closed curve and vector field stated calculate both the counterclockwise circulation ) G J p 8.=Þ s ) G J p X.= s and the outward flux a) G À The circle of radius + centered at Ð2ß5ÑÞ p p J œb s3 Cs4 ) J X s.= G ) G J p 8.= s

57 Page 57 3 b) G À The circle of radius + centered at Ð2ß5ÑÞ p p J œb s3 Cs4 ) J X s.= G ) G J p 8.= s c) G À The circle of radius + centered at Ð2ß5ÑÞ p p J œc s3 Bs4 ) J X s.= G ) G J p 8.= s d) G À The circle of radius + centered at Ð2ß5ÑÞ p p J œc s3 Bs4 ) J X s.= G ) G J p 8.= s # # B C #C e) G À The ellipse + #,# œ, Þ p p J œbc s # 3 Bs4 ) J X s.= G ) G J p 8.= s

58 Page (5 points) For each vector field indicate which have a non-zero circulation and which have a non-zero flux. a) b) c) d)

59 Page 59 5 e) f) g) h)

60 Page 60 6 i) j)

61 Page 61 7 Ä 7. (2 points) Consider the vector field: J œ ÐD CÑs3 ÐBDÑs4 ÐCBÑs5 œ D CßBDßCB Þ a) Calculate the magnitude of f Ä J b) Calculate the direction of f J Ä Go to the website, and run the Java 3DGrapher. From the Choose a graph to add menu select Vector Field. In the Function entry spaces labeled dx(x,y,z), dy(x,y,z) and dz(x,y,z) enter the BC,, and Dcomponents of the field. From the Choose a graph to add menu select Parametric In the Parametric entry spaces labeled x(t), y(t) and z(t) Curve. enter functions to generate the line BœCœD.

62 Page 62 8 From the Bounds menu check Show XYZ Frame and Show BoundBox and axes. In the Graph menu set tmin and tmax so that the line BœCœD runs from one corner of the view box to the opposite corner. c) Adjust the perspective with the mouse so that you are looking down the line BœCœD. If you prefer, you may use CalcPlot3D at Explain how what you see is consistent with your calculations in parts a) and b). 8. (3 points) a) Calculate the surface integral '' ÐB s3 s # # # C4Ñ 8.W s over the surface of the cylinder B C œ+ with!ÿdÿ2. W b) Does this answer make sense geometrically? Explain. '' c) Calculate the surface integral ÐBs3 Cs4 D s5 Ñ s8.w over the surface of the sphere of radius + centered at the origin W d) Does this answer make sense geometrically? Explain.

63 Page (3 points) # # lnðb C Ñ Let FÐBß CÑ œ #, a) Calculate the two dimesional gradient of F, ff # # 2 ` F ` F b) Calculate the 2-D Laplacian of F, f F œ f ff œ `B# `C# c) Let G be the circle of radius + centered at the origin. Calculate the following counterclockwise flux integral across G, ) G '' H f F 8.= s d) For the interior, H, of the circle G calculate the integral f f F.E e) Are the results of parts c) and d) inconsistent with the Divergence (Gauss's) Theorem? Explain. f) Consider any simple closed curve U which encloses G. What is the counterclockwise flux integral across U? ) U f F 8.= s 10. (3 points) Let FÐBßCßDÑœ ", É BCD # # # a) Calculate the three dimesional gradient of F, ff # # # 2 ` F ` F ` F b) Calculate the $ -D Laplacian of F, f F œ f ff œ `B# `C# `D#

64 Page c) Let G be the sphere of radius + centered at the origin. Calculate the flux integral across the surface of G, '' G f F 8.W s ''' d) For the interior, H, of the sphere G calculate f f F.Z H e) Are the results of parts c) and d) inconsistent with the Divergence (Gauss's) Theorem? Explain. f) Consider any simple closed surface U which encloses G. What is the flux integral across U? '' U f F 8.W s 11. (3 points) Ä Cs3 Bs4 The magnetic field about an infinite line of current along the Daxis varies like F ÐBßCÑ œ Þ BC # # Ä Ä a) Calculate the two dimesional curl of F f F b) Let G be the circle of radius + centered at the origin in the BC plane. Calculate the counterclockwise circulation around G c) For the interior, H, of the circle G calculate ) G Ä F X s.= '' H d) Are the results of parts b) and c) inconsistent with Stokes' Theorem? Explain. Ä Ðf F Ñ s 5.W f) Consider any simple closed curve U in the BC plane which encloses G. What is the counterclocwise circulation around U? ) U Ä F X s.=

65 Page (2 points) Ä Ä # # Suppose J is a radial vector field in the BC plane, i.e. J œ 0ÐB C ÑÐ Bs3 Cs4Ñ for some differentiable scalar function 0 a) Explain why such a field is called radial. Þ Ä b) Is J a conservative field? Explain. 13. (4 points) p C Consider the force field Jœ,Ð s BC,C,BC D 3 B s D 4 s D œ,b # 5Ñ D ß D ß D# for some constant,. a) Calculate the work done by this field in going from Ð"ß %ß #Ñ to Ð&ß "#ß %Ñ along the path p < œð"%>ñs3 Ð%)>Ñs4 Ð##> Ñ s5 for!ÿ>ÿ"þ b) Calculate the work done by this field in going from Ð"ß %ß #Ñ to Ð&ß "#ß %Ñ along a path made up of three line segments each parallel to the coordinate axes, starting with the path parallel to the B axis and ending with the path parallel to the Daxis. c) Explain the relationship between the answers to a) and b). d) Calculate the work done by this field in going from Ð"ß %ß #Ñ to Ð&ß "#ß %Ñ along any path. 14. (" point) Consider the force field JœÐBC p # $ C Ñ s $ 3 * Bs4. a) Is this a conservative force? Explain. b) Find the simple closed path in the BC plane along which the work done by J p in a counterclockwise circulation is a maximum.

66 Page (2 points) # # # # Let Lbe that portion of the sphere with boundary B C D œ+ and + # ŸDŸ+ Þ Let Frepresent the oriented surface of L + and G be the intersection of D œ # and the surface of the sphere 3 œ + p Þ For J œ BDs3 BDs4 CDs5 œ BDß BDß CD calculate the following: a) '' F p J 8.W s œ b) ) G Ä J X s.=

67 Page Self-Assessment: (2 bonus points) a) Describe three strengths in your perfomance on this project. Include why each is a strength. b) What are three things that could be improved about your performance on this project. Explain specifically how you will make these improvements. c) Identify two things about this project which are still unclear to you. d) Identify two insights that you have acquired in doing this project.

68 Page 68 Calculus III : Group Project 4 ( 9 points ) Due 12/01/2016 Choose one of Problems 1 and 2 (for 5 points), and one of Problems 3 and 4 (for 4 points). You may do extra problems for bonus. 1. (5 points) Differentiable (analytic) functions of a complex variable D œ B 3C can be written in the form 0ÐDÑ œ T ÐBß CÑ 3UÐBß CÑ, where w T and U are real functions with total differential in the real variables B and C Þ For 0 ÐD! Ñ to exist the following limit must exist for all w paths in the complex plane from D to D! Þ0 ÐD! Ñ œ lim 0ÐDÑ0ÐD! Ñ DÄD! DD Þ! w a) What conditions does the existence of 0ÐDÑ impose on the four partial derivatives `T `T `U `U! ß ß and Þ These equations are `B `C `B `C known as the Cauchy-Riemann equations. w # # b) What does the the existence of 0ÐDÑ! imply for ftand fu? (This result is important in the theory of two dimensional irrotational fluid flow.) c) The differential of Dß.D œ.b 3.CÞ So the line integral about a simple closed curve G in the complex plane is given by ) 0ÐDÑ.D œ ) ÐT ÐBß CÑ.B UÐBß CÑ.CÑ 3) ÐT ÐBß CÑ.C UÐBß CÑ.BÑ Þ From Green's theorem and the Cauchy-Riemann G G G equations what can you conclude about ) G 0ÐDÑ.D? This result is known as Cauchy's Theorem.

69 Page 69 Calculus III : Group Project 4 ( 9 points ) Due 12/01/ (5 points) Show by explicit calculation that the following $ D vector identities are valid. F and G represent $ D scalar functions and Ä E and Ä F represent $ D fields. a) f ff œf # F b) f f œ! Ä F Ä c) f Ðf E Ñœ! d) fðfgñ œ FfGGfF Ä Ä Ä Ä Ä Ä e) f ÐE F ÑœF f E E f F Ä Ä Ä f) f ÐFE ÑœFf E f F E Ä Ä Ä g) f ÐFE Ñœ Ff E f F E Ä Ä Ä Ä h) f Ðf E ÑœfÐf E Ñf E Here, f E # # = 2 `E Ä 2 `E Ä 2 `E Ä `B2 `C2 `D2 Þ

70 Page 70 Calculus III : Group Project 4 ( 9 points ) Due 12/01/ (4 points) `V Ä A unit vector? salong the direction of a general coordinate variable? can be calculated as follows :? s œ `? 2,? Ä where V œ Bs3 s4 s `V C D 5 and 2? œl Ä `? l Þ a) For cylindrical coordinates <ß ) ß D calculate in terms of s3, s4, s5, < ß ) and D the following three unit vectors : <s )s Ds b) For spherical coordinates 3 ß ) ß 9 calculate in terms of s3, s4, s5, 3 ß ) and 9 the following three unit vectors : 3s )s 9s c) In terms of general coordinates?" ß?# and? $ the gradient and divergence are given by the following expressions : $ $ Ä Ä Ä ff œ! " ` F "? s ; f E œ! ` 222 " # $ 2 `? `? Ð E where E œ E? s is the component of E along? s 3 3 " # $ Ñ œ" 3œ" For cylindrical coordinates determine the following expressions in terms of <ß ) ß D : ff f E Ä f 2 F œ For spherical coordinates determine the following expressions in terms of 3 ß ) and 9 : ff f E Ä f 2 F

71 Page 71 Calculus III : Group Project 4 ( 9 points ) Due 12/01/ ( 4 points) Maxwell's Equations (in free space) for the electromagnetic fields in rationalized MKS units are given by the following : f I Ä ; f I Ä Ä œ ; f F Ä! ; f F Ä œ Ä Ä 3 `F `I œ `> œ.!ðn %! `> Ñ % /! Ä Ä Ä Here I is the electric field, F is the magnetic field, > is time, 3 / is the electric charge density, N is the electric current density, %! "# F ( H is the permittivity of free space ()Þ)&% "! m),.! is the permeability of free space (% 1 "! m). The amount of electric charge Q in a region of space V is given by the volume integral : ''' 3.Z, while the total electric current (charge per unit of p time) flowing through a surface W is given by the surface integral '' N 8. s 5 Þ W a) Consider a charge density which is spherically symmetric about the origin and which vanishes for 3 (the radial distance from the + Ä Ä origin) +. Let Uœ% 1' # 3 Ð3Ñ 3. 3 What is the electric field at V where V?! /. l l+ V / b) Show that electric charge conservation is a consequence of Maxwell's equations. That is, show that the rate at which electric charge changes in a region of space is equal to the rate at which charge is flowing into or out of the region. c) Under what conditions can the electric field be written as ff where F is a scalar potential function? Ä d) The EMF (electro motive force) measured in volts is the line integral I X.= s around the closed path G Þ How is this EMF G related to what the magnetic field is doing? )

72 Page 72 Calculus III : Project 6 /40 _ Due 12/05/ (18 points) Solve the following first order differential equations subject to the stated conditions..0 a).b œlbl with 0Ð#Ñœ"Þ.C BC b).b œ BC with C œ # at B œ! Þ.C # c).b œbc C # cos ÐBÑ with CÐ1Ñœ#Þ w # d) BC $C œ B with CÐ &Ñ œ! Þ.0 e).b œ / $B $0ÐBÑ with 0Ð!Ñ œ $ Þ

73 Page 73 2.C f).b œ $C sin ÐBÑ with CÐ!Ñ œ!þ* Þ w g) C C œ lb"l with CÐ!Ñ œ! Þ Ð ÑÐ Ñ.C h) B" C œ/ B.B with CÐ!Ñœ! Þ! for B Ÿ ".C i).b #Cœ1ÐBÑœÖ with CÐ!Ñœ"Þ B" for B" 2. (3 points).c a) Solve.B #BCœB subject to CÐ!Ñœ+Þ

74 Page 74 3 b) b) Generate the direction field graph for this equation with a window from #Þ& to #Þ& in both the B and C directions. On this same graph plot the five functions CÐBÑ for +œ!þ&ß+œ!ß+œ!þ&ß+œ" and +œ"þ&þ In WinPlot use the 2-dim Window and from the Equa menu select Differential dy/dt. In the differential equation menuu, set x' equal.c to 1 and y' equal to.b. Select the " vectors" check box. Change the window using the command sequence View, Set corners. Initial values can be specified through the One/dy/dt trajectory menu. Just set x equal to zero (for this example) and y equal to + ( ignore the setting for t), then press the " draw" button with " both " checked. Repeat this procedure for each initial condition. Choosing the IVPs a different color from the slope field makes the display easier to interpret. c) Explain what happens at +œ!þ&þ 3. (5 points).c a) Given.B $C œ 'B & with CÐ!Ñ œ +! ß let?ðbñ be the linear function that CÐBÑ approaches as B Ä _. For what value of + is C œ?ðbñ?.c.c b) Given.B Cœ# with CÐ!Ñœ,ß determine lim CÐBÑand lim BÄ_ BÄ_.B Þ.C # CC c) Solve.B œ with CÐ"Ñ œ, Á "ß then determine C Explain whether or not this limit is the solution for, œ "Þ BB# lim Þ,Ä"

75 Page 75 4 # # d) Find the value of + that makes Ð+B %BCÑ.B Ð+B CÑ.C œ! an exact first order equation. For this value of + solve for the implicit curve 0ÐBßCÑ œ! that passes through the point Ð!ß#ÑÞ w 7 e) The Bernoulli equation C +ÐBÑC œ,ðbñc ß with 7 a constant different from either! or "ß can be reduced to a linear 5 equation with the transformation : Z œ C where 5 depends on 7Þ Determine how 5 depends on 7, then solve the following.c # # equation : BC.B œc B with CÐ"Ñœ%Þ 4. (2 points) For each family of curves 0ÐBßCÑ œ - determine the equation for the family of orthogonal trajectories. Sketch several curves from each family. # # a) 0ÐBßCÑœB C œ- b) 0ÐBßCÑœ/ BCœ- 10 y 10 y x x

76 Page (4 points) Linear Circuits: a) A resistor V Ð"!Þ! HÑ and an inductor P Ð!Þ#&! H) are hooked up in series to a power supply Z Ð#!Þ! volts Ñ which establishes a steady state current of Z V Ð#Þ! amperesñ. The power supply is then removed and the current decays in a manner described by : V3 P.3.> œ! Þ How long after the power supply is removed is the current down to!þ&! amperes? Note : " H œ " second. " H For arbitrary values of Vß Pand Z, how long after the power supply is removed is the current down to Z %V? b) A capacitor G Ð)Þ!.F) is charged to a potential difference Z Ð"#Þ! volts). The power supply is removed and the capacitor is allowed to discharge through a resistor V Ð"Þ! MHÑ. The charge U on the capacitor is given by U œ GZ and decays in a manner.u U described by : V.> G œ!. How long after the power supply is removed is the charge on the capacitor down to!þ#&! its starting value? Note : 1F " H œ" second Þ For arbitrary values of Vß G and Z (the power supply voltage), how long after the GZ power supply is removed is the charge down to %? 6. (2 points) Concentration Dilution a) A tank of volume Z has a chlorine concentration of -!. It is desired to reduce this to -0 by pumping in water with a chlorine concentration of -M Ð-! -0 -M Ñ. To prevent over flowing water is pumped out at the same rate. If the water is pumped in at a rate of < gallons per minute and if the water is so well mixed that at any time the chlorine concentration is uniform through out the tank, how long will it take to reach the desired concentration? b) How long would it take if "pure" water were used? Ð- œ!ñ M

77 Page (6 points) Population Growth. a) The most naive (and most frightening) model for population growth is the exponential growth model. This assumes that the rate of.r growth is directly proportional to the current population size. In symbols this is expressed as.> œ5r, where R is the number of individuals in the population at time > and 5 is a constant. Roughly, 5 is the number of new 'offspring' produced by each individual in.r a relevant unit of time. Solve.> œ5r, subject to the initial condition that RÐ!ÑœR! Þ b) The exponential model is obviously unrealistic for large times in that it leads to unbounded population growth. There is only so much matter in the universe! At some point limited food and living space limit the population size of any species. One way to model.r R this behavior mathematically is to modify the rate equation as follows :.> œ5rð" P Ñ, where P is the limiting population size. Initially R increases exponentially as before; however, as R approaches P the rate of growth slows to zero. Hence, R œ P is a horizontal asymptote of the solution. This rate equation is sometimes called the logistic equation. For the logistic equation model what value of R makes the rate of growth a maximum? What is this maximum rate of growth? How does this compare to the rate of growth of the exponential model for the same population size? Solve the logistic equation, subject to the initial condition that RÐ!Ñ œ R Þ! Evaluate lim >Ä_ RÐ>Ñ " For R! œ &!! ß P œ &!!ß!!! and 5 œ!þ"! year, according to the exponential model how long would it take the population to double to "!!!? To increase to $!!ß!!!? " For R! œ &!! ß P œ &!!ß!!! and 5 œ!þ"! year, according to the logistic model how long would it take the population to double to "!!!? To increase to $!!ß!!!?

78 Page 78 7 " For R! œ &!! ß P œ &!!ß!!! and 5 œ!þ"! year, make a careful graph of the solutions of both the exponential and logistic models. A graphing calculator or computer program would be helpful here.

79 Page 79 8 Self-Assessment: (2 bonus points) a) Describe three strengths in your perfomance on this project. Include why each is a strength. b) What are three things that could be improved about your performance on this project. Explain specifically how you will make these improvements. c) Identify two things about this project which are still unclear to you. d) Identify two insights that you have acquired in doing this project.

80 Page 80 Calculus III : Project 7 /40 _ Due 12/09/ (5 points) Perform the following operations on the given complex numbers and express the result in standard rectangular form (+,3ÑÞ a) & È %* œ b) Ð#""3ÑÐ&È $'Ñ œ c) Ð& %3ÑÐ$ È * Ñ œ d) #$3 È*È"' œ e) Ð% $3Ñ # œ f) Ð#3Ñ ( œ g) 3 ")#( œ h) Òcos( 1 3sin 1 ' ' Ó Ñ Ð Ñ "# œ i) sin Ð#3Ñ œ 1 j) sin Ð $ 3Ñ œ

81 Page (4 points) For each of the following complex numbers i) Express the result in standard rectangular form ii) Express the result in standard exponential form ( </ 3) Ñ iii) Graph the number a) Ð# 3ÑÐ$ 3Ñ œ ( Rectangular) œ ( Exponential) b) Ð$/ % ÑÐ#/ # Ñ œ ( Rectangular) œ ÐExponential)

82 Page 82 3 # c) ln Ð/ Ñœ ( Rectangular) œ ÐExponential) d) È3 œ ( Rectangular) œ ÐExponential) 3. (1 point) Consider the equation / / œ / results in what trigonometric identity? 3! 3" 3(! " ). Using Euler's identity three times and equating the real and imaginary parts of both sides