MATH 2443.002 - Calculus and Analytic Geometry IV (Honors) - Fall 2009
TR 12:00-1:15 p.m., 119 PHSC

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

Office Hours: Tue 3:00-4:00 p.m., Wed 2:00-3:00 p.m., or by appointment.

Prerequisite: MATH 2433 (Calculus and Analytic Geometry III).

Course catalog description: 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, 6th edition, Brooks/Cole, 2007. The course will cover major parts of chapters 15-17.

Check out the OU Math Blog! It is REALLY interesting!

Check out the Problem of the Month !

The problem-solving sessions for the annual Putnam Competition will be held each Wednesday from 5 to 6 p.m. in 1025 PHSC. Everybody is welcome to attend! For more info contact Prof. John Albert at jalbert AT math.ou.edu or 325-3782.

Homework:

• Homework 1 (problems given on Aug 25, 27, Sep 1, 3), due Sep 8 (Tue).
• Homework 2 (problems given on Sep 8, 10, 15, 17), due Sep 22 (Tue).
• Homework 3 (problems given on Sep 22, 24, 29, Oct 1), due Oct 8 (Thu).
• Homework 4 (problems given on Oct 6, 8, 13), due Oct 15 (Thu).
• Homework 5 (problems given on Oct 15, 20, 27, 29), due Nov 3 (Tue).
• Homework 6 (problems given on Nov 3, 5, 10, 12), due Nov 17 (Tue).
• Homework 7 (problems given on Nov 17, 19, 24), due in class on Dec 1 (Tue).

Content of the lectures:

• Lecture 1 (Tue, Aug 25): 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. 15.1).
  Homework: Exercises 15.1/13 (hint), 16, 18, 27, 26, 42, 66.
• Lecture 2 (Thu, Aug 27): 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 (Sec. 15.2).
  Homework: 15.2/7 (hint: do not use the ε definition, look at page 911), 9 (hint), 15 (hint: see Example 3), 16, 19 (hint: see Example 5), 26, 30, 41.
• Lecture 3 (Tue, Sep 1): Partial derivatives: definition of partial derivatives for functions of two variables, examples, practical rules for finding partial derivatives, Clairaut's Theorem, definition of partial derivatives for functions of n variables, partial differential equations [challenge: prove that the function u(x,y)=f(x-ct)+g(x+ct) satisfies the wave equation, u_xx-(1/c²)u_tt=0] (Sec. 15.3).
  Homework: 15.3/24, 36, 40, 46, 50 (hint), 60, 62, 71.
• Lecture 4 (Thu, Sep 3): Tangent planes and linear approximations: equation of the tangent plane to a surface defined as a graph of a function, linear approximations (tangent plane approximations) of a function at a point, a function that has partial derivatives but no tangent plane, increments of the independent variables and of the function value, differentiable functions, continuity of partial derivatives imlpies differentiability, examples (pages 928-932 of Sec. 15.4).
  Homework: 15.4/6, 14, 18, 20 (don't graph f, but instead find the absolute error of the linear approximation, i.e., the absolute value of the difference between the exact and the approximate values), 22, 42, 43 (hint).
  Food for thought (i.e., problems for you to think about, but they are not to be turned in): 15.4/45, 46.
  The complete homework is due on Sep 8 (Tue).
• Lecture 5 (Tue, Sep 8): Tangent planes and linear approximations (cont.): differentiability implies continuity, differentials, using differentials to estimate increments, a detailed example of the increase of the volume of a cylinder when its radius and height increase by small amounts (Sec. 15.4).
  The chain rule: derivation of the chain rule for the derivative of f(g(t),h(t)) with respect to t, example (illustrating that the chain rule gives the same result as first substituting g(t) and h(t) into f, and then differentiating); chain rule for the partial derivatives of f(g(s,t),h(s,t)) with respect to s and t (first guess what it will look like, then read it from the book), the general version of the chain rule (pages 937-941 of Sec. 15.5).
  Homework: 15.4/28, 32, 36; 15.5/8, 14, 20, 24.
• Lecture 6 (Thu, Sep 10): The chain rule: implicit differentiation (Sec. 15.5). Directional derivatives and the gradient vector: directional derivative, 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 of a function of two variables, tangent planes to level surfaces of a function of three variables, meaning of the gradient vector (Sec. 15.6).
  Homework: 15.6/4, 10, 28, 32, 40, 48, 54, 58.
  Food for thought: 15.6/37, 59.
• Lecture 7 (Tue, Sep 15): 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, absolute minima and maxima (don't forget the values on the boundary of the domain D!) (Sec. 15.7).
  Homework: 15.7/4, 17, 19, 29, 31 (hint), 41 (hint), 43 (hint).
• Lecture 8 (Thu, Sep 17): Lagrange multipliers: Lagrange mutiplier method for finding the extrema of a function f(x,y) subjected to one constraint g(x,y)=k; Lagrange mutiplier method for finding the extrema of a function f(x,y,z) subjected to one constraint g(x,y,z)=k; Lagrange mutiplier method for finding the extrema of a function f(x,y,z) subjected to two constraints g(x,y,z)=k and h(x,y,z)=l; examples (Sec. 15.8).
  Homework: 15.7/55; 15.8/4 (the answer is obvious from the geometry), 12 (the answer is obvious from the symmetry), 16, 19, 21.
  The complete homework is due on Sep 22 (Tue).
• Lecture 9 (Tue, Sep 22): Double integral over rectangles: areas and single integrals, volumes and double integrals, double Riemann sums, double integral over a rectangle, midpoint rule for double integrals, average value, linearity of double integrals (Sec. 16.1).
  Homework: 16.1/8, 11, 12, 14, 18.
• Lecture 10 (Thu, Sep 24): Exam 1 (covers Sections 15.1-15.8).
  Homework: the homework in pdf format.
• Lecture 11 (Tue, Sep 29): Iterated integrals: the concept of an iterated integral, Fubini's theorem, examples (Sec. 16.2).
  Homework: 16.2/2, 5, 12, 15, 21, 25, 36.
• Lecture 12 (Thu, Oct 1): Double integrals over general regions: definition of a double integral over a general region (by continuing the function on a rectangle and integrating over the rectangle), type I and type II regions, computing integrals over general regions, examples; property of double integrals: linearity, monotonicity, additivity of domains, normalization (Sec. 16.3).
  Homework: 16.2/38; 16.3/6, 12, 18, 25, 42, 49, 52 (only express D as a union of regions of type I or type II).
• Lecture 13 (Tue, Oct 6): Double integrals in polar coordinates: polar coordinates, areas of polar rectangles, change to polar coordinates in double integrals, examples; a digression on computing the area of a circle if the formula for the circumference is known in two ways (cutting it like a pizza or cutting it like a "2-dimensional onion") (Sec. 16.4).
  Homework: 16.4/6, 9, 14, 15 (hint), 21, 29.
  The complete homework (problems given on Sep 22, 24, 29, Oct 1) is due on Oct 8 (Thu).
• Lecture 14 (Thu, Oct 8): Triple integrals: definition, Fubini's Theorem for triple integrals, examples (Sec. 16.6).
  Homework: 16.4/35, 37 [hints: (a) integrate the result of 16/36(c) by parts, (b) change variables in (a)]; 16.6/5, 9, 18, 34, 36.
  Solution of problem 16.6/33: page 1, page 2, page 3, page 4, page 5, page 6, page 7.
• Lecture 15 (Tue, Oct 13): Triple integrals in cylindrical coordinates: cylindrical coordinates, volume element in cylindrical coordinates, evaluating triple integrals with cylindrical coordinates; Jacobian of a change of variables, change of variables in multiple integrals, Jacobian of the change from Cartesian to polar coordinates in R², Jacobian of the change from Cartesian to cylindrical coordinates in R³ (Sec. 16.7, parts of Sec. 16.9).
  Homework: 16.7/8, 16, 22, 28; 16.9/3, 8 (in this problem, after finding the image of S, compute the Jacobian and write Equation 9 on page 1052 for this change of variables).
  The complete homework (problems given on Oct 6, 8, 13) is due on Oct 15 (Thu).
• Lecture 16 (Thu, Oct 15): Triple integrals in spherical coordinates: spherical coordinates, volume element in spherical coordinates, evaluating triple integrals with spherical coordinates, examples; change of variables in triple integrals, Jacobians of the change from Cartesian to cylindrical coordinates, and from Cartesian to spherical coordinates in R³ (Sec. 16.7, parts of Sec. 16.9). (Sec. 16.8).
  Homework: 16.8/8, 10, 14, 18, 19 (in spherical coordinates), 28, 40; 16.9/15.
  Food for thought: The true-false quiz on page 1057.
• Lecture 17 (Tue, Oct 20): Vector fields: vector fields on subsets of R² and R³, plots of vector fields, gradient vector fields, conservative vector fields, examples (Sec. 17.1).
  Homework: 17.1/2, 6, 11-14, 26, 36 (hint: see 17.1/35).
• Lecture 18 (Thu, Oct 22): Exam 2 (covers Sections 16.1-16.4, 16.6-16.8).
• Lecture 19 (Tue, Oct 27): 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, center of mass of a wire with a given density, line integrals with respect to x and y, parametrizing segments of a straight line, 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 (Sec. 17.2).
  Homework: 17.2/5, 7 (hint), 13, 14, 19, 33 (hint), 39 (hint).
• Lecture 20 (Thu, Oct 29): 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); 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) (pages 1082-1085 of Sec. 17.3).
  Homework: 17.2/30(a), 32(a), 41 (hint), 48; 17.3/2, 24, 26, 30, 32 (for problems 30 and 32, read what "simply connected" means on pages 1085-1086).
  The complete homework (problems given on Oct 15, 20, 27, 29) is due on Nov 3 (Tue).
• Lecture 21 (Tue, Nov 3): The Fundamental Theorem for line integrals (cont.): for a conservative vector field F(x,y)=P(x,y)i+Q(x,y)j, the equality ∂P/&party=∂Q/&partx 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); conservation of energy (pages 1085-1089 of Sec. 17.3).
  Homework: 17.3/8, 9, 14, 20, 22, 27 (hint), 34(a,b).
• Lecture 22 (Thu, Nov 5): Green's Theorem: orientation of the boundary of a planar domain, Green's Theorem, proof for a simple region, derivaiton of formulas for areas of planar domains, connection with the formula of the area under a curve (the way definite integrals are defined in Calculus II), applications (Sec. 17.4).
  Homework: 17.4/1, 4, 10, 11, 18, 22, 23, additional problem in pdf format.
• Lecture 23 (Tue, Nov 10): Curl and divergence: the gradient operator as a vector, definitions of curl and divergence, curl(grad(f))=0, and div(curl(F))=0, condition for conservativeness in terms of curl, vector forms of Greene's Theorem (Sec. 17.5).
  Homework: 17.5/3, 5, 10, 16, 20, 25, 30.
  Food for thought: 17.5/12.
• Lecture 24 (Thu, Nov 12): Parametric surfaces and their areas: parametric curves and parametric surfaces in R³, grid curves, examples (cylinder, plane, sphere), 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 (pages 1106-1111 of Sec. 17.6).
  Homework: 17.6/2, 6, 20, 24, 26 (hint), 30 (just find the parametric equations), 36 (just find the tangent plane and write its equations in a parametric form and as a linear equation as on page 834 of the book).
  Food for thought: 17.6/13-18.
  The complete homework (problems given on Nov 3, 5, 10, 12) is due on Nov 17 (Tue).
• Lecture 25 (Tue, Nov 17): Parametric surfaces and their areas (cont.): 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 (pages 1111-1117 of Sec. 17.6).
  Surface integrals: definition, surface integrals over graphs of functions z=g(x,y), orientable surfaces (pages 1117-1122 of Sec. 17.7).
  Homework: 17.6/40, 47 (hint), 55(a,c) (hint); 17.7/5 (hint), 9.
• Lecture 26 (Thu, Nov 19): Surface integrals (cont.): surface integrals of vector fields, flux of a vector field, flux of a vector field through a graph of a function z=g(x,y), examples from physics: fluid flow, heat flux (Sec. 17.7).
  Stokes Theorem: orientation of the boundary of an oriented surface consistent with the orientation of the surface, formulation of the Stokes Theorem (pages 1128-1129 of Sec. 17.8).
  Homework: 17.7/21, 25 (hint), 29; 17.8/1 (hint), 3, 9.
• Lecture 27 (Tue, Nov 24): Stokes Theorem (cont.): proof of Stokes Theorem, circulation of a vector field over a simple closed curve, physical meaning of the curl of a vector field (pages 1129-1134 Sec. 17.8).
  Homework: 17.8/13, 16, 20 (a typo in problem 20: the problems you have to use are 24 and 26 from Sec. 17.5).
  Food for thought:: 17.8/15 (here is the solution).
  The complete homework (problems given on Nov 17, 19, 24) is due IN CLASS on Dec 1 (Tue).
• Lecture 28 (Tue, Dec 1): The Divergence Theorem: statement of the theorem and remarks about its applications (flow of incompressible fluid, sources and sinks, Gauss's law in electricity, Maxwell's equations), beginning of proof of the Divergence Theorem (Sec. 17.9).
  Homework: 17.9/3, 9 hint), 17, 23, 25 hint), 27, 29.
  Food for thought: 17.9/19, 22.
• Lecture 29 (Thu, Dec 3): Exam 3 (covers Sections 17.1-17.8).

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. Periodically I will collect it to be graded (these days will be announced in advance). 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 and due every class period. You should be prepared to present any of the homework problems due on a given day. Periodically I will collect it to be graded (these days will be announced in advance).
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! Please write the problems in the same order in which they are given in the assignment.
All homework should be written on a 8.5"×11" paper with your name clearly written, and should be stapled. No late homework will be accepted!

Exams: There will be three in-class midterms and a (comprehensive) final.
Tentative dates for the midterms are September 24 (Thursday), October 27 (Tuesday), December 3 (Thursday).
The final is scheduled for December 14 (Monday), 1:30-3:30 p.m.
All tests must be taken at the scheduled times, except in extraordinary circumstances.
Please do not arrange travel plans that prevent you from taking any of the exams at the scheduled time.

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

Academic calendar for Fall 2009.

Course schedule for Fall 2009.

Policy on W/I Grades : Through October 2 (Friday), you can withdraw from the course with an automatic "W". In addition, from October 5 (Monday) to December 11 (Friday), you may withdraw and receive a "W" or "F" according to your standing in the class. Dropping after November 30 (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!

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 A Student's Guide to Academic Integrity. For information on your rights to appeal charges of academic misconduct consult the Rights and Responsibilities Under 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.