MATH 2443.180 - Calculus and Analytic Geometry IV - Summer 2012
MTWRF 11:45-12:50, 120 PHSC

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

Office Hours: Tue 9:30-10:30 a.m., Thu 1:00-2:00 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, 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!

Homework:

• Homework 1 (problems given on May 14, 15, 16), due May 18 (Friday).
• Homework 2 (problems given on May 17, 18, 21), due May 23 (Wednesday).
• Homework 3 (problems given on May 22, 23, 24), due May 29 (Tuesday).
• Homework 4 (problems given on May 25, 29, 31), due June 4 (Monday).
• Homework 5 (problems given on June 1, 4, 5), due June 7 (Thursday).
• Homework 6 (problems given on June 6, 7, 8), due June 12 (Tuesday).
• Homework 7 (problems given on June 11, 12, 14), due June 18 (Monday).
• Homework 8 (problems given on June 15, 18, 19), due June 21 (Thursday).
• Homework 9 (problems given on June 20, 21, 22), due June 26 (Tuesday).
• Homework 10 (problems given on June 25, 26, 27, 29, July 2, 3), due July 5 (Thursday).

Content of the lectures:

• Lecture 1 (Mon, May 14): 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 (Tue, May 15): 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, 35, 41.
• Lecture 3 (Wed, May 16): 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.
  The complete Homework 1 is due on May 18 (Friday).
• Lecture 4 (Thu, May 17): Partial derivatives (cont.): implicit differentiation of a function of one variable and a function of more than one variables, examples (page 919 of Sec. 15.3).
  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 (pages 928, 929 of Sec. 15.4).
  Homework: 15.3/28, 38, 42, 48, 58, 64, 72(c).
• Lecture 5 (Fri, May 18): 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; differentiability of a function implies continuity of the function; differentials, using differentials to estimate increments, an example of the increase of the volume of a cylinder when its radius and height increase by small amounts (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), 28, 32, 36, 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.

Lecture 6 (Mon, May 21): Tangent planes and linear approximations: more examples, discussion on the meaning of the differential; obtaining the expression for the differential from geometric considerations, example: differential of the volume of a cylinder (Sec. 15.4).
The chain rule: recalling the chain rule from Calculus I; Case I 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); Case II of the chain rule - for the partial derivatives of f(g(s,t),h(s,t)) with respect to s and t the general version of the chain rule (pages 937-941 of Sec. 15.5).
Homework: 15.5/4, 6, 8, 10, 14, 16, 20, 24.
The complete Homework 2 is due on May 23 (Wednesday).

• Lecture 7 (Tue, May 22): The chain rule: finding the the derivative of an implicitly defined function of one variable by implicit differentiation; please read from the book about finding the the derivative of an implicitly defined function of more than one variables by implicit differentiation (pages 941-943 of 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, examples (pages 946-952 of Sec. 15.6).
  Homework: 15.5/28, 32, 45; 15.6/6, 9, 13 (careful, the vector v is not a unit vector, so you have to normalize it as in Example 4).
• Lecture 8 (Wed, May 23): Directional derivatives and the gradient vector (cont.): 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 953-955 of Sec. 15.6).
  Homework: 15.6/20, 21, 28, 32, 40, 48, 54.
  Food for thought: 15.6/37, 59.

• Lecture 9 (Thu, May 24): 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 958-964 of Sec. 15.7).
  Homework: 15.7/4, 17, 19, additional problem.
  The complete Homework 3 is due on May 29 (Tuesday).
• Lecture 10 (Fri, May 25): Maximum and minimum values (cont.): more examples; a boundary point of a set, open and closed sets, bounded and unbounded sets, extreme value theorem for 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!); example of a degenerate case (in which the Second derivatives test does not work) - show that for the function f(x,y)=x⁴y³ the critical points are all points on the coordinate exes, and figure out if these points are local minima or local maxima or neither (pages 964-966 of Sec. 15.7).
  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 (pages 987-990 of Sec. 16.1).
  Homework: 15.7/29, 31, 41, 43; 16.1/11, 12, 14.
• Lecture 11 (Tue, May 29): Double integral over rectangles (cont.): midpoint rule for double integrals, average value, linearity of double integrals, monotonicity of double integrals (pages 991-994 of Sec. 16.1).
  Iterated integrals: the concept of an iterated integral, Fubini's theorem, examples (Sec. 16.2).
  Homework: 16.1/18; 16.2/2, 5, 9, 17, 21, 25, 35.
• Lecture 12 (Wed, May 30): Exam 1 (on Sections 15.1-15.7).
• Lecture 13 (Thu, May 31): 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 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, 13, 18, 25, 42, 49, 52 (only express D as a union of regions of type I or type II, do not solve the integral).
  The complete Homework 4 is due on June 4 (Monday).
• Lecture 14 (Fri, Jun 1): Double integrals in polar coordinates: polar coordinates, areas of polar rectangles, change to polar coordinates in double integrals, Examples 1 and 2 on page 1012 (pages 1010-1012 of Sec. 16.4).
  Homework: 16.4/9, 11, 13.
• Lecture 15 (Mon, Jun 4): Double integrals in polar coordinates (cont.): polar coordinates in more complicated regions, examples (pages 1013-1014 of Sec. 16.4).
  Triple integrals: definition, Fubini's Theorem for triple integrals, examples (pages 1026-1029 of Sec. 16.6).
  Homework: 16.4/6, 21, 29, 35; 16.6/5, 9, 18, 34.
• Lecture 16 (Tue, Jun 5): Triple integrals (cont.): more examples, Exercise 16.6/33 (page 1030 of Sec. 16.6).
  Homework: 16.6/36, 53.
  The complete Homework 5 is due on June 7 (Thursday).
• Lecture 17 (Wed, Jun 6): Triple integrals in cylindrical coordinates: cylindrical coordinates, volume element in cylindrical coordinates in R³ - comparison with area element in R² in polar coordinates; examples of computing triple integrals with cylindrical coordinates (Sec. 16.7).
  Homework: 16.7/8, 16, 22, 28.
• Lecture 18 (Thu, Jun 7): Triple integrals in spherical coordinates: spherical coordinates, volume element in spherical coordinates (pages 1041-1043 of Sec. 16.8).
  Homework: 16.6/52; 16.8/4, 8, 10, 14.
  Food for thought: The true-false quiz on page 1057.
• Lecture 19 (Fri, Jun 8): Triple integrals in spherical coordinates (cont.): examples (pages 1044-1045 of Sec. 16.8).
  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: 16.8/18, 19 (in spherical coordinates), 28, 40; 17.1/2, 11-14, 26, 36 (hint: see 17.1/35).
  The complete Homework 6 is due on June 12 (Tuesday).
• Lecture 20 (Mon, Jun 11). 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, behavior of integrals with respect to arc length and with respect to x and y when changing the direction of traversing the curve C (pages 1070-1076 of Sec. 17.2).
  Homework: 17.2/5, 7, 13, 14, 33.
• Lecture 21 (Tue, Jun 12): Line integrals (cont.): 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: 17.2/27, 29, 39.
• Lecture 22 (Wed, Jun 13): Exam 2 (on Sections 16.1-16.4, 16.6-16.8, 17.1).
• Lecture 23 (Thu, Jun 14): 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 1082-1084 of Sec. 17.3).
  Homework: 17.2/41, 48; 17.3/1, 2, 11, 12, 20, 21.
  The complete Homework 7 is due on June 18 (Monday).
• Lecture 24 (Fri, Jun 15): 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 an open simply-connected domain in R² to be conservative (Theorem 6) (pages 1084-1088 of Sec. 17.3).
  Homework: 17.3/4, 9, 14, 26, 27, 30, 32
• Lecture 25 (Mon, Jun 18): Green's Theorem: orientation of the boundary of a planar domain, Green's Theorem, proof for a simple region, derivation of formulas for areas of planar domains, examples (pages 1091-1094 of Sec. 17.4).
  Homework: 17.4/2, 10, 11, 18, additional problem.
• Lecture 26 (Tue, Jun 19): Green's Theorem (cont.): proofs that the theorem works for domains that are not type-I and type-II, and for domains with holes; solution and discussion of Example 5 on page 1095 (pages 1095-1096 of Sec. 17.4).
  Curl and divergence: the gradient operator as a vector, definitions of curl F(x,y,z), curl F as ∇×F (pages 1097-1098 of Sec. 17.5).
  Homework: 17.4/19, 27; 17.5/2(a), 8(a).
  The complete Homework 8 is due on June 21 (Thursday).
• Lecture 27 (Wed, Jun 20): Curl and divergence (cont.): cyclic permutations, different methods for memorizing the expression for curl F; curl(grad(f))=0, condition for conservativeness in terms of curl, example of a vector field that has zero curl but is not conservative (pages 1099-1100 of Sec. 17.5).
  Homework: 17.5/16, 20, 21, 31, additional problem.
• Lecture 28 (Thu, Jun 21): Curl and divergence (cont.): divergence of a vector field, examples, div(curl F)=0 for any vector field F(x,y,z), Laplacian of a scalar function (pages 1101-1102 of Sec. 17.5).
  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 (pages 1106-1111 of Sec. 17.6).
  Homework: 17.5/2(b), 8(b), 25, 30; 17.6/2, 6, 20, 26, 30 (just the parametric equations).
  Food for thought: 17.5/12, 17.6/13-18.
• Lecture 29 (Fri, Jun 22): A digression: orientable and non-orientable surfaces in R³, Moebius strip.
  Parametric surfaces and their areas (cont.): |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 1112-1114 of Sec. 17.6).
  Surface integrals: motivation and definition of surface integrals (pages 1117-1118 of Sec. 17.7).
  Homework: 17.6/35 (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), 41, 55(a,c); 17.7/5
  The complete Homework 9 is due on June 26 (Tuesday).
• Lecture 30 (Mon, Jun 25): Surface integrals: definition, surface integrals over graphs of functions z=g(x,y), physical applications: total mass, center of 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 (pages 1119-1123 of Sec. 17.7).
  Homework: 17.7/9, 21.
• Lecture 31 (Tue, Jun 26): Surface integrals (cont.): remarks on computing the potential function f(r) of a conservative vector field F(r), arbitrariness in choosing the additive constant C f(r) and connection with the arbitrariness with choosing the "zero level" of the potential energy U(r)=-f(r) in physics, physical examples of conservative (gravity) and non-conservative (friction) forces; "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; comparison of line integrals (from Section 17.2) and surface integrals (pages 1123-1125 of Sec. 17.7).
  Homework: 17.7/36, 40(a); 17.2/33 (this problem is from an old section).
• Lecture 32 (Wed, Jun 27). Stokes Theorem: orientation of the boundary of an oriented surface consistent with the orientation of the surface, formulation and beginning of the proof of the Stokes' Theorem (pages 1128-1130 of Sec. 17.8).
  Homework: 17.8/1, 3, 9.
• Lecture 33 (Thu, Jun 28): Exam 3 (on Sections 17.2-17.7).
• Lecture 34 (Fri, Jun 29): 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: 17.8/16, 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).
• Lecture 35 (Mon, Jul 2): A digression: on history of physics and math.
  Food for thought: The true-false quizzes on pages 980-981 and 1142.
• Lecture 36 (Tue, Jul 3): The Divergence Theorem: the Divergence Theorem (with proof); summary of different theorems related to the Fundamental Theorem of Calculus (Sec. 17.9, 17.10).
  Homework: 17.9/7, 25, 27, 28.
  The complete Homework 10 is due on July 5 (Thursday).
• Lecture 37 (Thu, Jul 5): Applications of multivariable calculus: derivation of the heat equation from the Energy Conservation Law, Fourier law of heat conduction, and Divergence Theorem.
• Lecture 38 (Fri, Jul 6): Final exam (cumulative).

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. Your homework should have your name clearly written on it, and should be stapled. Please write the problems in the same order in which they are given in the assignment. No late homework will be accepted!
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!

Exams: There will be three in-class midterms and a (comprehensive) final.
Tentative dates for the midterms are May 29 (Tuesday), June 13 (Wednesday), June 28 (Thursday).
The final will be given in class on July 6 (Friday).
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:

Useful links: the academic calendar, the class schedules.

Policy on W/I Grades : From May 18 (Friday) to June 11 (Monday), you can withdraw from the course with an automatic "W". Dropping after June 12 (Tuesday) 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.