MATH 241 F1H: Calculus III Honors
Spring 2012
Professor A.J. Hildebrand
http://www.math.illinois.edu/~hildebr/241/

Final Exam Results and Course Grades

You can access your score as usual at this link. The grade shown at the end of the online score display (including plusses and minuses) is the grade that has been reported as your official course grade; it is based strictly on the total number of accumulated points and is nonnegotiable. Cutoffs between letter grades were chosen to minimize hardships and close calls. As a result, nobody was within a few points of he next higher grade, and most gaps were substantially larger.

Course Policies, Exams, Grades

• Math 241 F1H Honors FAQ. This page answers questions about the "honors" nature of this course, and how it differs from regular Math 241 sections. BE SURE TO READ THIS FIRST.
• Course Information Sheet. General course information: Everything you need to know about this course: Instructor contact, text and syllabus, exam and homework information, grading policies, etc.
• Exams:
  • Sample exams. Links to sample exams from past Math 241 Honors classes. These should give you a good idea of what to expect.
  • Midterm Exam 1: Feb. 15 (in class)
    • General Information and Study Guide
    • Syllabus and Checklist
  • Midterm Exam 2: March 14 (in class)
    • General Information and Study Guide
    • Syllabus and Checklist
  • Midterm Exam 3: April 18 (in class)
    • General Information and Study Guide
    • Syllabus and Checklist
  • Final Exam: Wednesday, May 9, 7:00 pm - 10:00 pm, 143 Henry.
    • Final Exam Information and Syllabus.
• Link to Online Scores. Click on this link and log in with your NetID and password to access your scores. The display shows all the scores on all assignments and exams given out so far, and your total accumulated score. If a score is missing or incorrect, let me know right away.
• Add/drop deadlines: January 30 and March 9. The first date (January 30) is the last date you can add a course; if you want to switch to a non-honors section, you have to do so by that date since a section switch involves dropping one course and adding another. The second date (March 9) is the campus deadline for dropping a course. Engineering students need to get their Dean's approval if they drop a course after the 10th day of class.

Announcements

• U of I Undergraduate Math Contest, Saturday, 3/3/2012, 10 am - 1 pm, 145 Altgeld Hall. A math problems contest open to all U of I undergraduates. Registration not required; just show up! For more informationon visit the U of I Math Contests webpage.
• Open House Hours: Tuesdays/Thursdays, beginning at around 5:15 pm and Sundays, beginning at around 3:15 pm, in 159 Altgeld, beginning the second week of class. The Open House is an informal office hour and get-together for students in my classes, and the primary point of contact for my students. I will stay as long as necessary - typically at least an hour, and sometimes much longer. The Open House room, 159 Altgeld, is a small classroom located in the basement of Altgeld Hall on the south side of the building. We have exclusive use of this room during the above Open House hours.
• Homework: The first graded homework assignment will be due the second week of class (week of Monday, Jan. 23).
• Textbook and WebAssign access: Our text is Stewart, Calculus: Early Transcendentals, 6th Edition. Be sure to get the 6th Edition, not the newer 7th Edition (which has a white cover). Click on the above link for more information on the text and on setting up WebAssign, an online homework system with access to a variety of online materials. The use of WebAssign is optional, and I will not be using the homework feature of WebAssign, but you may want to check it out for the other features it provides.
• Illinois Geometry Lab (IGL). The IGL is a recently established center that provides an opportunity for undergraduates to participate in research projects. For more information, click on the above link, email igl@math.uiuc.edu, or visit the IGL open house on January 17, 5 pm, in 102 Altgeld Hall.

Class Diary

• Wednesday, 5/2.
  • Class: Derivation of the limit formula for divergence. More generally, derivation of a formula expressing an integrand function at a given point as a limit of integrals over balls centered at this point.
  • Read: The limit formula for divergence derived in class can be found at the end of Section 16.9.
• Tuesday, 5/1:
  • Class: Recap of the Fundamental Theorems. Emphasis on connections and analogies between these theorems and on the general form of the theorems: Integral of f over delta D equals the integral of delta f over D.
• Monday, 4/30:
  • Class: A real-world illustration of flux: Calculating the amount of rain collected in a box moving at a given speed. Do you get more wet or less wet if you move faster?
    Interpretation of divergence as flux density.
  • Homework: HW 15 (not graded)..
• Thursday, 4/26:
  • Class: Stokes' Theorem. Statement, examples, and analogies to FTC, FTLI, and Green's Theorem. "Independence of surface" property.
  • Read: Section 16.8
• Wednesday, 4/25:
  • Class: More on vector surface integrals. Flux interpretation. Applications of the Divergence Theorem to inverse square fields.
• Tuesday, 4/24:
  • Class: Examples of surface integral computations of various types.
  • Read: Sections 16.7 and 16.9.
• Monday, 4/23:
  • Class: Surfaces and surface integrals, continued. Motivation for the conversion factor |r_u x r_v|. Some special cases.
  • Handout: Surfaces and Surface Integrals..
  • Read: Section 16.6.
  • Homework: HW 14, due Monday, 4/30.
    Honors HW 5, due Wednesday, 5/2.
    
• Thursday, 4/19:
  • Class: Overview of surfaces, surface integrals, and Gauss' Theorem. Analogies to curves, line integrals, and Green's Theorem. Parameter representation of surfaces. Standard parametrizations for sphere and plane.
• Wednesday, 4/18: Midterm Exam 3.
• Tuesday, 4/17:
  • Class: Curl test and mixed partials test. Green's Theorem. Vector/curl version and explicit coordinate form.
  • Read: Section 16.4.
  • Handout: Green's Theorem..
• Monday, 4/16:
  • Class: Grad, div, and curl. Definitions and same examples. Combinations: Curl/grad and div/curl. Curl test for conservative fields. Potential of an inverse square field.
  • Read: Section 16.5.
  • Homework: HW 13, due Monday, 4/23.
    
• Thursday, 4/12:
  • Class: Recap of Fundamental Theorem for Line Integrals and related properties: Conservative fields, existence of potential/anti-gradient; independence of path; loop integral zero; mixed partial test. Proof of the FTLI via chain rule for derivatives in Rn.
• Wednesday, 4/11:
  • Class: Conservative vector fields and potentials ("anti-gradients"). "Mixed partials" test for "conservativeness". Finding potentials by "reverse partial differentiation".
  • Read: Section 16.3. In particular, read up on the "mixed partials" test for conservative fields (Theorems 5 and 6 and Examples 2 and 3) and the technique for finding potentials via "inverse partial differentiation" (Examples 4 and 5).
• Tuesday, 4/10:
  • Class: Line integrals continued. Connections between the different versions of line integrals. Examples of line integral computations using different methods. Fundmamental Theorem for Line Integrals.
  • Read: Sections 16.2 and 16.3.
  • Handout: Line integrals.
• Monday, 4/9:
  • Class: Section 16.2, line integrals. Definition and interpretation of line integrals of scalar functions and line integrals of vector fields.
  • Read: Section 16.2.
  • Homework: HW 12, due Monday, 4/16.
    
• Thursday, 4/5:
  • Class: Started Chapter 16. Overview and goals of this chapter. Vector fields, examples, and formal definition.
  • Read: Section 16.1.
• Wednesday, 4/4:
  • Class: Linear algebra excursion, continued. Determinants. Recursive computation of n by n determinants. Algebraic properties of determinants. Volume interpretation. Determinants of diagonal and triangular matrices.
  • Read/do: Tomorrow's quiz will be on the linear algebra material covered this week. In preparation, review this material, using your classnotes and the Linear Algebra handout as guide.
• Tuesday, 4/3:
  • Class: Excursion into linear algebra. Geometric properties of linear functions from Rn to Rn. Conversion factor for areas and volumes. Determinants. (To be continued tomorrow.)
  • Handout: Linear Algebra, II.
• Monday, 4/2:
  • Class: Applications of change of variables formula.
  • Read: Section 15.9 (Change of variables in multiple integrals).
  • Homework: HW 11, due Monday, 4/9.
    Honors HW 4, due Monday, 4/16.
    
• Thursday, 3/29:
  • Class: Change of variables formula for multiple integrals. The Jacobian determinant as the "correct" conversion factor. Connection with matrix derivative of the transformation function.
• Wednesday, 3/28:
  • Class: More triple integral practice. Volume of icecream cones of various shapes.
  • Read: Finish reading Sections 15.7 and 15.8. There will a short quiz tomorrow on these sections (in particular, on the various conversion formulas for cylindrical and spherical coordinates).
• Tuesday, 3/27:
  • Class: Triple integral practice. Integrals over spherical regions.
• Monday, 3/26:
  • Class: Triple integrals. Cylindrical and spherical coordinates.
  • Read: Section 15.6 on triple integrals in rectangular coordinates (focus on Examples 2 and 4 in this section). Also, get familiar with cylindrical and spherical coordinates that are introduced at the beginning of Sections 15.7 and 15.8. You should know the conversion formulas between rectangular, cylindrical, and spherical coordinates, and the equations of simple surfaces (such as cones, spheres, half-planes) in cylindrical and spherical coordinates.
  • Do: HW 10.
• Thursday, 3/15:
  • Class: Fun with n-dimensional integrals: the volume of n-dimensional cubes, n-dimensional tetrahedra, and n-dimensional spheres.
  • Do/read: If you have not done so, be sure to read/review sections 15.4 and 15.5, which were the subject of this week's regular HW assignment (HW 9). The applications you need to know are those covered in HW 9. You can skip over the part dealing with moments of inertia (p. 983/984), but you should be familar with applications to mass and center of mass, and with probability applications of the type covered in Tuesday's class and in HW 9.
    After the break, I will go straight to triple integrals (Sections 15.6/15.7/15.8).
  • Homework: Honors HW Assignment 3, due Monday, 4/2.
• Wednesday, 3/14: Midterm Exam 2.
• Tuesday, 3/13:
  • Class: Application of double integrals to probability.
• Monday, 3/12:
  • Class: Application of double integrals in polar coordinates: Volume of a sphere, sphere cap, and "sphere with hole". Evaluation of Gaussian integral by converting to a double integral and changing to polar coordinates. (The remarkable trick involved here is described in Exercise 36 of 15.4.)
  • Homework: HW 9.
• Thursday, 3/8:
  • Class: Double integrals in polar coordinates. Applications of double integrals.
  • Do/read:Finish HW 8 and Honors HW 2, which are both due on Monday. Get ready for Wednesday's midterm exam, using the Exam 2 Study Guide and the Exam 2 Syllabus and Checklist as study guide.
• Wednesday, 3/7:
  • Class: Computation of double integrals. Fubini's theorem. Double integrals over rectangles and over general regions. Reversing order of integration.
  • Read: Read sections 15.2 and 15.3 in preparation of tomorrow's quiz.
• Tuesday, 3/6:
  • Class: Wrapped up excursion into derivatives in Rn. Illustration of Taylor's formula in a concrete situation. Connection with differentials, linear approximation, and tangent planes.
    Started Chapter 15 on multiple integrals. Review of the one-variable case. The integral as an area and as an antiderivative. Generalization of the area interpretation to double integrals.
  • Homework: HW 8.
• Monday, 3/5:
  • Class: Excursion: Derivatives in Rⁿ, part II. Formal definition of differentiability and derivatives of functions from R^m to Rⁿ via Taylor's formula. Limit formulas for derivatives of vector functions and directional
  • Handout: Differentiability and Derivatives in Rn, Part II.
  • Read/do: No new assignment, but use the time to review Chapter 14 in preparation for the upcoming midterm.
• Thursday, 3/1:
  • Class: Maxima, minima, and saddle points. First and second derivative tests for functions of one and two variables. Motivation for first/second derivative tests via Taylor's formula.
• Wednesday, 2/29:
  • Class: More application of the Lagrange multiplier method. Maximizing the Cobb-Doublas production function. The meaning of the Lagrange multiplier. Proof of Cauchy-Schwarz inequality via Lagrange multipliers.
  • Read: Section 14.7. Focus on the first part of this section dealing with critical points and the second derivative test (p. 923 - 926, and Examples 1-4). (For other examples such as Examples 5 and 6, the Lagrange multiplier method is the more natural approach, and you should be able to do these examples with Lagrange multipliers. The same goes for many of the exercises in this section. In fact, most of the problems on p. 932 come up again in 14.8, this time as Lagrange multiplier exercises.)
• Tuesday, 2/28:
  • Class: Applications of Lagrange multiplier method.
  • Homework: HW Assignment 7.
• Monday, 2/27:
  • Class: Started Section 14.8, optimization with constraints and the Lagrange multiplier method. Motivation of the method and a first example.
  • Homework: Honors HW Assignment 2, due Monday, 3/12.
• Thursday, 2/23:
  • Class: Wrapped up the discussion of gradients, directional derivatives, and tangent planes.
• Wednesday, 2/22:
  • Class: Directional derivatives, gradients, level curves/surfaces.
  • Do/read: Read Section 14.6.
• Tuesday, 2/21:
  • Class: From Linear Algebra back to derivatives. The derivative matrix. Connections with other derivative concepts. Matrix form of chain rule.
    Handout: Derivatives in Rn, I. The Derivative Matrix
  • Homework: HW Assignment 6.
• Monday, 2/20:
  • Class: Linear algebra excursion, continued. Some examples. Checking whether a function is linear. Finding the matrix A associated with a linear function. Formula for A in terms of partial derivatives.
  • Do: Finish HW 5 (due Wednesday).
• Thursday, 2/16:
  • Class: Excursion into linear algebra. Linear combinations. Linear functions, definition and key properties. Key Fact: The linear functions from Rm to Rn are exactly the functions of the form f(x) = A x, where A is an n x m matrix.
    Handout: Linear Algebra
  • Homework: HW Assignment 5.
  • Read/do: Read Section 14.5 and get started with the new HW assignment.
• Wednesday, 2/15: Midterm Exam 1.
• Monday, 2/13:
  • Class: Section 14.4. Differentials, motivation, and applications (e.g., error estimates).
  • Do: No additional assignment (the next HW won't be due until at least Tuesday or Wedensday next week), but use the time to prepare for Wednesday's midterm! See the Exam Study Guide linked above.
• Thursday, 2/9:
  • Class: Partial derivatives, continued. Geometric interpretations. Partial derivative calculations in ideal gas law.
  • Read/do: HW 4 is due Monday. Read Section 14.3 if you have not done so; the midterm will cover the material through this section (14.3) and HW 4.
• Wednesday, 2/8:
  • Class: Level curves and level surfaces. Started 14.3. Partial derivatives, definition and notation. Clairaut's Theorem on mixed partials.
• Tuesday, 2/7:
  • Class: Started Chapter 14. Multivariable functions, overview. Interpretation as "black box" with vector input and scalar output. Domain and graph.
  • Homework: HW Assignment 4.
  • Read: Read Section 14.1. This section lots of computer-generated sketches of domains, graphs, and level curves to help develop a good intution for these concepts. The section also has a number of examples similar to this week's homework problems.
• Monday, 2/6:
  • Class: Kepler's Laws. Proof of the First and Second Laws.
  • Do/read: Since we are at the end of Chapter 13 and the first midterm is approaching soon, this is a good time to begin reviewing the material covered so far. An excellent starting point for such an exam review are the end-of-chapter sections titled "Concept checks": p. 849-850 and p. 812-813.
• Thursday, 2/2:
  • Class: Representation of acceleration in TNB frame, normal and tangential components of acceleration, derivation of formulas.
  • Read/do: Read Section 13.4. Focus on the first part (Examples 1 - 4), and the discussion of tangential and normal components of acceleration (p. 842-844). You can skip over the projectile problems in Examples 5 and 6.
    Also, get started on the homework on 13.2-13.4; don't wait till the last day!
• Wednesday, 2/1:
  • Class: Normal and osculating planes. Curvature.
  • Read/do: Finish reading 13.3. You can skip formula (11) and Example 5 (curvature formula for plane curve), but the rest of this section is important and relevant for this week's homework, for tomorrow's quiz, and for the next midterm (in two weeks). In particular, study the examples illustrating the computation of the various quantities (arclength, curvature, T, N, B, normal and osculating planes) in concrete situations such as the helix. HW 3 has a number of problems of this type.
• Tuesday, 1/31:
  • Class: More on derivatives of vector functions. Geometric interpretation. Limit definition. Vector function of constant magnitude.
    The TNB frame.
  • Homework: HW Assignment 3.
  • Read: Section 13.3. In particular, read the first part, on arclength, and the last part of this section, dealing with the TNB frame. In tomorrow's class, we will cover the remaining topic of this section, curvature.
• Monday, 1/30:
  • Class: Started Chapter 13. (Section 12.6 will be covered later.) Vector functions, overview. Derivatives of vector functions, componentwise definition, algebraic properties, examples.
  • Read: 13.1 and 13.2. In 13.1 you can skip over theoretical aspects such as limits and continuity (we will cover this later). However, 13.2 is an important section, with lots of good applications and problems. I will spend another class hour on this section, and there will be around ten problems from 13.2 on the next HW assignment
  • Homework: Honors HW Assignment 1, due Thursday next week.
• Thursday, 1/26:
  • Class: Lines and planes: big picture, underlying ideas, and connections between the various types of equations.
  • Read/do: HW 2 is due Tuesday. Get started if you have not done so. Also, finish reading 12.5. We will start with Chapter 13 next week.
• Wednesday, 1/25:
  • Class: Quiz 1. Problems/proofs from the handout on n-dimensional spaces and the Cauchy-Schwarz and Triangle Inequalities. Application: The GPA Problem.
  • Read/do: If you have not done so, finish HW 1 (due tomorrow). Also, read Section 12.5 lines and planes) in preparation for tomorrow's class (and HW 2).
• Tuesday, 1/24:
  • Class: Finished discussion of cross product, algebraic definition via determinants, why cross product makes only sense in 3 dimensions.
    Started excursion into n-dimensional space (one of the occasional ventures beyond the standard material). Definition of basic operations (addition, scalar multiplications, dot product, norm) for n-dimensional vectors. Statement of Cauchy-Schwarz and Triangle Inequalities.
  • Handout: n-dimensional space, Cauchy-Schwarz and Triangle Inequalities
  • Homework: HW Assignment 2 (due Tuesday next week).
  • Read/do: If you have not done so, be sure to finish studying 12.4. There will be a short quiz tomorrow on the properties and formulas involving dot and cross products (Sections 12.3 and 12.4). Also, work on HW 1 (due Thursday)
• Monday, 1/23:
  • Class: The cross product. Geometric definition, properties, and applications.
  • Do: Read 12.4 in preparation for tomorrow's class. In particular, familiarize yourself with 3 x 3 determinants, which are needed for the computation of cross products.
• Thursday, 1/19:
  • Class: More on the dot product. Algebraic properties of the dot product, and using these properties in formal proofs. Orthogonal (perpendicular) and parallel vectors, formal definition. Scalar and vector projections.
  • Handout: HW Assignment 1.
  • Do: Finish reading 12.3 if you have not done so. Get started on the HW Assignment.
• Wednesday, 1/18:
  • Class: Recap of 12.2. Basic arithmetic operations with vectors and their algebraic and geometric descriptions. Representation of arbitrary vectors as linear combinations of basis vectors. Examples of using vector methods to prove geometric properties.
    Started 12.3. Products of two vectors, overview. The dot product and the cross product. Definition of dot product and geometric interpretation in terms of magnitude and angles.
  • Do: Read 12.3 in preparation for tomorrow's class.
• Tuesday, 1/17:
  • Class: General course information; handed out the Course Information Sheet and the Honors FAQ. Overview of the calculus sequence. One variable calculus versus multivariable calculus. The n-dimensional space Rⁿ. Vectors.
  • Do: In preparation for tomorrow's class, read Sections 12.1 and 12.2.