MATH 2443.009 - Calculus and Analytic Geometry IV - Spring 2013
TR 10:30-11:45 a.m., SEC M0204

Instructor: Nikola Petrov, 802 PHSC, (405)325-4316, npetrov AT math.ou.edu

Office Hours: Mon 9:30-10:30 a.m., Tue 2:45-3:45 p.m., or by appointment.

Course catalog description: Prerequisite: 2433. Vector calculus; functions of several variables; partial derivatives; gradients, extreme values and differentials of multivariate functions; multiple integrals; line and surface integrals (F, Sp, Su)

Text: J. Stewart, Calculus, 7th edition, Brooks/Cole, 2012.

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Homework:

• Homework 1 (problems given on January 15, 17), due January 24 (Thursday).
• Homework 2 (problems given on January 22, 24), due January 31 (Thursday).
• Homework 3 (problems given on January 29, 31), due February 7 (Thursday).
• Homework 4 (problems given on February 5, 7), due February 19 (Tuesday).
• Homework 5 (problems given on February 12, 19), due February 26 (Tuesday).
• Homework 6 (problems given on February 21, 26), due March 5 (Tuesday).
• Homework 7 (problems given on February 28, March 5), due March 12 (Tuesday).
• Homework 8 (problems given on March 7, 12), due March 26 (Tuesday).
• Homework 9 (problems given on March 14, 26), due April 2 (Tuesday).
• Homework 10 (problems given on April 2, 4), due April 11 (Thursday).
• Homework 11 (problems given on April 9, 11), due April 18 (Thursday).
• Homework 12 (problems given on April 16, 18), due April 30 (Tuesday).

Content of the lectures:

• Lecture 1 (Tue, Jan 15): Functions of several variables: functions of two variables, independent variables, dependent variable, domain, range, graph, level curves, examples; functions of three or more variables [Sec. 14.1]
  Homework: Exercises 14.1/4, 14, 18, 20, 36; FFT: 14.1/19, 21, 25, 32, 47, 59-64.
  [In the old (6th) edition these exercises have numbers 15.1/3, 12, 16, 18, 32; FFT: 15.1/17, 19, 23, 30, 43, 55-60.]
  Remark: The FFT ("Food For Thought") problems are to be solved like a regular homework problems, but do not have to be turned in.
• Lecture 2 (Thu, Jan 17): Limits and continuity: limits of functions of one and several variables, examples; continuity of functions of one and several variables, proving discontinuity of a function at a point by finding different values when approaching the point along different paths, examples [read Examples 1-3, 4-9 of Sec. 14.2 in the 7th edition, or, equivalently, Examples 1-3, 4-9 of Sec. 14.2 in the 6th edition]
  Homework: 14.2/5 (see Example 5), 9 (hint), 15 (see Example 3 and consider, e.g., the paths x=0 and y=2x²), 17 (use polar coordinates), 19, 32, 41.
  FFT: 14.2/13 (hint), 25 (hint), 33.
  The complete Homework 1 is due on January 24 (Thursday).
• Lecture 3 (Tue, Jan 22): Partial derivatives: definition of partial derivatives for functions of two variables, examples, practical rules for finding partial derivatives, Clairaut's Theorem, partial differential equations [pages 924-928, 930-933 of Sec. 14.3]
  Homework: 14.3/24, 26, 42, 52 (hint), 60, 64, 71, 75.
  FFT: 14.3/10, 41, 45, 73 (hint).
• Lecture 4 (Thu, Jan 24): Partial derivatives (cont.): definition of partial derivatives for functions of n variables; general form of solution of the wave equation u_tt(x,t)=c²u_xx(x,t): u(x,t)=f(x−ct)+g(x+ct) (challenge: prove this!) [pages 929-930 of Sec. 14.3]
  Tangent planes and linear approximations: equation of the tangent plane to a surface defined as a graph of a function, linear approximation (tangent plane approximation) of a function at a point, a function that has partial derivatives but no tangent plane [pages 939-943 of Sec. 14.4; skip Equations 5, 6, and Definition 7 on page 942]
  Remark: Reviewing pages 183-184 of Sec. 2.9 will be useful.
  Homework: 14.3/34, 40, 50 (see Example 4 on page 929), 78(a,d); 14.4/6, 14 (see Theorem 8 on page 942), 18, 22, 42.
  FFT: 14.3/39, 79; 14.4/11 (hint).
  The complete Homework 2 is due on January 31 (Thursday).
• Lecture 5 (Tue, Jan 29): Tangent planes and linear approximations (cont.): a function f(x,y) is differentiable at the point (a,b) if it has a tangent plane at (a,b,f(a,b)); the continuity of the partial derivatives of a function implies the differentiability of the function; differentiability implies continuity; increments Δx=dx and Δy=dy of the independent variables and increment Δz=f(a+Δx,b+Δy)−f(a,b) of the function value for z=f(x,y), differential dz=f_x(a,b)dx+f_y(a,b)dy of the function z=f(x,y) at the point (a,b) for given increments dx and dy of the independent variables, using differentials to estimate increments of functions, an example of the increase of the volume of a cylinder when its radius and height increase by small amounts; linear approximation, differentiability, and differentials of functions of more than two variables [pages 942-946 of Sec. 14.4]
  The chain rule: cases 1 and 2 of the chain rule [pages 948-950 of Sec. 14.5]
  Remark: Reviewing pages 148-152 of Sec. 2.5 will be useful.
  Homework: 14.4/19, 20 (just find the linear approximation, no graphing), 26, 30, 32, 33, 34; 14.5/2, 10, 12, 14, 16.
  FFT: 14.4/21 (hint), 31 (hint), 35 (hint); 14.5/5 (hint).
• Lecture 6 (Thu, Jan 31): The chain rule: implicit differentiation of a function of one variable and a function of more than one variables, examples [pages 952-954 of Sec. 14.5]
  Directional derivatives and the gradient vector: directional derivative of a function of two variables, expressing D_uf(x₀,y₀) in terms of the partial derivatives of f and the components of the unit vector u [pages 957-960 of Sec. 14.6]
  Homework: 14.5/18, 24, 30, 34, 41, 55; 14.6/5, 13 (v is not a unit vector!).
  FFT: 14.5/17 (hint), 39 (hint); 14.6/11 (hint).
  The complete Homework 3 is due on February 7 (Thursday).
• Lecture 7 (Tue, Feb 5): Directional derivatives and the gradient vector (cont.): the gradient vector, expressing the directional derivative as a scalar product of a unit vector and the gradient vector; maximizing the directional derivative; tangent lines to level curves; tangent planes to level surfaces; significance of the gradient vector, path of steepest descent/ascent [pages 960-966 of Sec. 14.6]
  Homework: 14.6/10, 20, 23 (hint), 28, 33 (hint), 37 (a,b,d), 43, 49.
  FFT: 14.6/18, 19 (hint), 56.
• Lecture 8 (Thu, Feb 7): Maximum and minimum values: local maxima/minima/extrema, absolute maxima/minima, vanishing of the partial derivatives at an extremum of a differentiable function, critical point, remarks about non-differentiable functions, second derivatives test, example - saddle point [pages 970-973 of Sec. 14.7]
  Homework: 14.7/2, 3 (hint), 11, 13 (hint), 19, additional problem.
  FFT: 14.7/1 (hint).
  The complete Homework 4 is due on February 19 (Tuesday).
• Lecture 9 (Tue, Feb 12): Maximum and minimum values (cont.) more examples, absolute maxima/minima, a boundary point of a closed bounded set, extreme value theorem for continuous functions of two variables, algorithm for finding the absolute minima and maxima of a function on a domain D (don't forget the values on the boundary of the domain D!) [pages 974-976 of Sec. 14.7]
  Homework: 14.7/31 (hint), 41 (hint), 43 (hint).
  FFT: 14.7/55.
  
• Lecture 10 (Thu, Feb 14): Exam 1 [on Sec. 14.1-14.7]
• Lecture 11 (Tue, Feb 19): Double integral over rectangles: areas and single integrals in Calculus II, volumes and double integrals, double Riemann sums, double integral over a rectangle, volumes under graphs of functions of two variables over rectangles, midpoint rule for double integrals, average value, linearity of double integrals, monotonicity of double integrals [Sec. 15.1]
  Homework: 15.1/5(a), 11, 12, 14, 18.
  FFT: 15.1/9(a) (hint), 9(b) (hint), 17 (hint).
  The complete Homework 5 is due on February 26 (Tuesday).
• Lecture 12 (Thu, Feb 21): Iterated integrals: the concept of an iterated integral, Fubini's theorem, examples [Sec. 15.2]
  Double integrals over general regions: definition of a double integral over a general region D (by continuing the function on a rectangle R containing D and integrating over the rectangle), type I regions [pages 1012-1013 of Sec. 15.3]
  Homework: 15.2/2, 8, 12, 17 (hint), 20, 25, 37.
  FFT: Chapter 14 Concept Check (page 991), Chapter 14 True-False Quiz (pages 991-992).
• Lecture 13 (Tue, Feb 26): Double integrals over general regions: integrals over a type I region; type II regions, integrals over a type II regions, computing integrals over general regions, examples [pages 1013-1016 of Sec. 15.3]
  Homework: 15.3/1, 14, 17 (hint), 22, 25 (hint), 46, 56 (only express D as a union of regions of type I or type II, do not solve the integral).
  FFT: 15.3/11, 12, 15.
  The complete Homework 6 is due on March 5 (Tuesday).
• Lecture 14 (Thu, Feb 28): Double integrals over general regions: more examples; properties of double integrals: linearity, monotonicity, additivity of domains, normalization [pages 1017-1019 of Sec. 15.3]
  Double integrals in polar coordinates: polar coordinates, areas of polar rectangles, change to polar coordinates in double integrals, Examples 1 and 2 on page 1024 [pages 1021-1024 of Sec. 15.4].
  Homework: 15.3/16, 48, 51, 58, 59, 64; 15.4/2, 4, 6, 7.
  FFT: 15.3/37, 45, 47 (hint), 49 (hint), 62 (hint); 15.4/1 (hint), 11 (hint).
• Lecture 15 (Tue, Mar 5): Double integrals in polar coordinates (cont.): discussion on change of variables in functions, more examples [pages 1024-1025 of Sec. 15.4]
  Surface area: expressing the surface area of a graph of a function of two variables as a double integral, comparison with the formula for length of the graph of a function of one variable [Sec. 15.6]
  Homework: 15.4/13 (hint), 17, 25 (hint), 29, 31, 39 (hint); 15.6/3 (hint), 9 (hint).
  FFT: 15.4/40; 15.6/12 (hint).
  The complete Homework 7 is due on March 12 (Tuesday).
• Lecture 16 (Thu, Mar 7): Triple integrals: definition, Fubini's Theorem for triple integrals, examples [pages 1041-1045 of Sec. 15.7]
  Homework: 15.7/5, 9, 18, 34, 36.
  FFT: 15.7/19 (hint), 23 (hint), 27 (hint), 35 (hint).
• Lecture 17 (Tue, Mar 12): Triple integrals (cont.): more examples, click here for a detailed solution of Problem 15.7/33.
  Triple integrals in cylindrical coordinates: cylindrical coorcinates in R³; computing triple integrals in cylindrical coordinates [Sec. 15.8]
  Homework: 15.7/53 (hint); 15.8/8, 16, 17 (hint), 29.
  FFT: 15.7/55(a); 15.8/10, 21 (hint).
  The complete Homework 8 is due on March 26 (Tuesday).
• Lecture 18 (Thu, Mar 14). Triple integrals in spherical coordinates: spherical coordinates, volume element in spherical coordinates, examples [Sec. 15.9]
  Homework: 15.9/8, 10, 14, 18, 19 (in spherical coordinates), 28, 40.
  FFT: Chapter 15 Concept Check (page 1073; skip question 10), Chapter 15 True-False Quiz (page 1073).
• Lecture 19 (Tue, Mar 26). Vector fields: vector fields on subsets of R² and R³, plots of vector fields, gradient vector fields, conservative vector fields, potential function of a conservative vector field, examples [Sec. 16.1]
  Homework: 16.1/2, 8, 11-14, 24.
  FFT: 16.1/15-18, 29-32.
  The complete Homework 9 (problems given on March 14, 26) is due on April 2 (Tuesday).
• Lecture 20 (Thu, Mar 28): Exam 2 [on Sec. 15.1-15.4, 15.6-15.9, 16.1]
• Lecture 21 (Tue, Apr 2): Line integrals: line integral (with respect to arc length) of a function along a curve C in 2 and more dimensions, piecewise-smooth curves, velocity v(t)=r'(t) and speed v(t)=|v(t)|, expressing the arc length as ds=v(t)dt; line integrals with respect to x and y [pages 1087-1093 of Sec. 16.2, including Example 6 on page 1093]
  Homework: 16.2/5, 7 (hint), 13, 14, 33 (hint, see also Example 3 on pages 1089-1090).
• Lecture 22 (Thu, Apr 4): Line integrals (cont.): behavior of integrals with respect to arc length and with respect to x and y when changing the direction of traversing the curve C; line integrals of vector fields, unit tangent vector T(t)=r'(t)/|r'(t)|; expressing work as an integral of F⋅T with respect to the arc length, as an integral of F(r(t))⋅r'(t) with respect to the parameter t, and as an integral of F(r)⋅dr [pages 1094-1096 of Sec. 16.2]
  The Fundamental Theorem for line integrals: the Fundamental Theorem for line integrals (with proof using the Chain Rule and the Fundamental Theorem of Calculus); independence of path - definition, vanishing of the line integral over any closed path is a necessary and sufficient condition for path independence (Theorem 3) [pages 1099-1101 of Sec. 16.3]
  Homework: 16.2/27, 29(a), 39 (hint), 41, 46; 16.3/1, 2, 11 (hint), 12, 15 (hint), 20.
  FFT: 16.2/21 (hint), 52.
  The complete Homework 10 (problems given on April 2, 4) is due on April 11 (Thursday).
• Lecture 23 (Tue, Apr 9): The Fundamental Theorem for line integrals (cont.): definitions of open domain and connected domain, path independence of a vector field over an open connected domain implies that the vector field is conservative (Theorem 4, with proof); for a conservative vector field F(x,y)=P(x,y)i+Q(x,y)j, the equality ∂P/∂y=∂Q/∂x holds (Theorem 5); simple curve, simply-connected domain, necessary and sufficient condition for a vector field in a simply-connected domain in R² to be conservative (Theorem 6); an example: the vector field F(x,y)=(e^y+2x)i+(xe^y+y)j (defined in R² which is an open and simply-connected domain) is conservative: F(x,y)=∇f(x,y) for f(x,y)=xe^y+x²+y²/2 (check that F(x,y) satisfies the condition P_y=Q_x); on the other hand, the vector field F(x,y)=(e^y+2x)i+(xe^y+y+x²sin(y))j does not satisfy P_y=Q_x, so there is no function f(x,y) such that F(x,y)=∇f(x,y) - try to find such a function as we did in class and see what goes wrong! [pages 1101-1105 of Section 16.3, including all examples]
  Reading assignment: Conservation of energy [pages 1105-1106 of Sec. 16.3]
  Homework: 16.3/7 (hint), 23, 28, 29 (hint), 35 (hint).
  FFT: 16.3/25 (hint), 30, 31-34.
• Lecture 24 (Thu, Apr 11): Green's Theorem: orientation of the boundary of a planar domain, Green's Theorem, proof for a domain that is both type-I and type-II, proofs that the theorem works for domains that are not type-I and type-II, and for domains with holes; examples, derivation of formulas for areas of planar domains [Sec. 16.4]
  Reading assignment: Read Example 5 from Sec. 16.4 and think about its connection with Problem 16.3/37 (assigned as a homework in Lecture 23) and Problem 16.4/27 (one of the FFT problems assigned in this lecture).
  Homework: 16.4/3 (hint), 7 (hint), 12, 19, additional problem.
  FFT: 16.4/17 (hint), 27.
  The complete Homework 11 (problems given on April 9, 11) is due on April 18 (Thursday).
• Lecture 25 (Tue, Apr 16): Curl and divergence: dot and cross product of vectors in R³, different methods for memorizing the expression for a×b, cyclic permutations; the gradient operator as a vector: ∇=i∂_x+j∂_y+k∂_z; definition of divergence of a vector field F, div(F)=∇⋅F; definition of curl of a vector field F(x,y,z) in R³, curl(F)=∇×F; examples, div(curl(F))=0 for any vector field F(x,y,z), curl(grad(f))=0, condition for conservativeness in terms of curl; Laplacian of a scalar function [pages 1115-1119 of Sec. 16.5]
  Parametric surfaces and their areas: parametric curves and parametric surfaces in R³, grid curves in a parameterized sur examples (plane, plane, sphere) [pages 1123-1126 of Sec. 16.6]
  Reading assignment: Look at Equation 4 on page 926 and Figure 1 on page 927, and reread "Interpretations of partial derivatives" on pages 927-928 of Sec. 14.3; look at Fig. 12 on page 1127 and Equations 4 and 5 on pages 1127-1128, and think about the connection with the definition and interpretation of partial derivatives from Sec. 14.3 that you just read.
  Homework: 16.5/3, 16 (hint: see Example 5 on pages 1104-1105), 19 (hint), 21 (hint, "irrotational" means "with zero curl"), 25, 30, 31 (hint); 16.6/1, 3 (hint), 23 (hint).
  FFT: 16.5/12; 16.6/13 (hint), 19 (hint).
• Lecture 26 (Thu, Apr 18): Parametric surfaces and their areas (cont.) tangent vectors r_u(u₀,v₀) and r_v(u₀,v₀) to the surface r(u,v) at the point r(u₀,v₀), normal vector r_u(u₀,v₀)×r_v(u₀,v₀) to the tangent plane of the surface r(u,v) at the point r(u₀,v₀), equation of the tangent plane, |a×b| as the area of the parallelogram spanned by the vectors a and b, area dS of an infinitesimal surface element dS=|r_u×r_v|dA, dA=(du)(dv); area of a parametric surface, area of a graph of a function, examples [pages 1127-1131 of Sec. 16.6]
  Homework: 16.6/33 (hint), 39 (hint), 45 (hint), 59(a,c) (hint).
  FFT: 16.6/13-18.
  The complete Homework 12 (problems given on April 16, 18) is due on April 30 (Tuesday).
• Lecture 27 (Tue, Apr 23): Surface integrals: motivation and definition of surface integrals, definition, surface integrals over graphs of functions z=g(x,y), physical applications: total mass; orientable surfaces, normal vectors to surfaces, outward unit normal vector to an orientable surface, surface integrals of vector fields, flux of a vector field, an example from physics: fluid flow through a surface; "natural" orientation of a surface S defined by a vector equation r(u,v) - given by the unit normal vector n=r_u×r_v/|r_u×r_v|, alternative expressions for surface integrals ∫_SF•dS [Sec. 16.7]
  Homework: 16.7/7, 9 (hint), 23 (hint).
  FFT: 16.7/38.
• Lecture 28 (Thu, Apr 25): Exam 3 [on Sec. 16.1-16.7]
• Lecture 29 (Tue, Apr 30): Stokes' Theorem: orientation of the boundary of an oriented surface consistent with the orientation of the surface, Stokes' Theorem, circulation of a vector field over a simple closed curve, physical meaning of the curl of a vector field [Sec. 16.8]
  Homework: 16.8/1 (hint), 3, 9, 15 (hint), 16, 20.
• Lecture 30 (Thu, May 2): The Divergence Theorem: the Divergence Theorem, examples [Sec. 16.9]
  Homework: 16.9/3, 7 (hint), 25 (hint), 27, 28.

Attendance: You are required to attend class on those days when an examination is being given; attendance during other class periods is also expected. You are fully responsible for the material covered in each class, whether or not you attend. Make-ups for missed exams will be given only if there is a compelling reason for the absence, which I know about beforehand and can document independently of your testimony (for example, via a note or a phone call from a doctor or a parent).

Homework: It is absolutely essential to solve a large number of problems on a regular basis!
Homework will be assigned regularly and will be posted on the this web-site. The homework will be due at the start of class on the due date. Each homework will consist of several problems, of which some pseudo-randomly chosen problems will be graded. Your lowest homework grade will be dropped. Giving just an answer to a problem is not worthy any credit - you have to write a complete solution which gives your step-by-step reasoning and is written in grammatically correct English. Although good exposition takes time and effort, writing your thoughts carefully will greatly increase your understanding and retention of the material.
You are allowed (and encouraged) to work in small groups. However, each of you will need to prepare individual solutions written in your own words - this is the only way to achieve real understanding!
Your homework should have your name clearly written on it, and should be stapled; the problems should be in the same order in which they are given in the assignment. No late homework will be accepted!

Exams: There will be three in-class midterms and a (comprehensive) final.
Tentative dates for the midterms are February 12 (Tue), March 14 (Thu), April 23 (Tue).
The final will be given from 8:00 to 10:00 a.m. on May 9 (Thursday).
All tests must be taken at the scheduled times, except in extraordinary circumstances.
Please do not arrange travel plans that prevent you from taking the exams at the scheduled times.

Grading: Your grade will be determined by your performance on the following coursework:

Useful links: the academic calendar, the class schedules.

Policy on W/I Grades : From January 29 (Tuesday) to March 29 (Friday), you can withdraw from the course with an automatic "W". Dropping after April 1 (Monday) requires a petition to the Dean. (Such petitions are not often granted. Furthermore, even if the petition is granted, I will give you a grade of "Withdrawn Failing" if you are indeed failing at the time of your petition.) Please check the dates in the academic calendar!

The grade of "I" (Incomplete) is not intended to serve as a benign substitute for the grade of "F". I only give the "I" grade if a student has completed the majority of the work in the course (for example everything except the final exam), the coursework cannot be completed because of compelling and verifiable problems beyond the student's control, and the student expresses a clear intention of making up the missed work as soon as possible.

Academic Misconduct: All cases of suspected academic misconduct will be referred to the Dean of the College of Arts and Sciences for prosecution under the University's Academic Misconduct Code. The penalties can be quite severe. Don't do it!
For details on the University's policies concerning academic integrity see the Student's Guide to Academic Integrity at the Academic Integrity web-site. For information on your rights to appeal charges of academic misconduct consult the Academic Misconduct Code. Students are also bound by the provisions of the OU Student Code.

Students With Disabilities: The University of Oklahoma is committed to providing reasonable accommodation for all students with disabilities. Students with disabilities who require accommodations in this course are requested to speak with the instructor as early in the semester as possible. Students with disabilities must be registered with the Office of Disability Services prior to receiving accommodations in this course. The Office of Disability Services is located in Goddard Health Center, Suite 166: phone 405-325-3852 or TDD only 405-325-4173.

Good to know:

• The greek_alphabet.
• Some useful notations.