MATH 241 F1H: Calculus III (Honors version), Fall 2007
Professor A.J. Hildebrand
http://www.math.uiuc.edu/~hildebr/241/

Final Exam Results and Course Grades

You can access your score on the final exam, your total score (rescaled in case of excused scores), and your grade, as usual under 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.

If you are still in town and want to see your final exam, stop by my office, 241 Illini Hall, Monday, 12/17, 12 pm - 1 pm. (I'll be leaving town on Tuesday.) Otherwise, relax and enjoy the break!

Exams, Grades, etc.

• Course Information Sheet. The sheet handed out at the beginning of the semester. Syllabus, grading policies, office hours, etc.
• Exams
  • Final Exam, Saturday, December 15, 1:30 pm - 4:30 pm.
    Final Exam Information Sheet.
    Final Exam Solutions.
  • Midterm Exam 3, Wednesday, November 28, 2007.
    Exam 3 Review Sheet. Exam 3 Solutions.
  • Midterm Exam 2, Wednesday, October 24, 2007.
    Exam 2 Review Sheet. Exam 2 Solutions.
  • Midterm Exam 1, Wednesday, September 26, 2007.
    Exam 1 Review Sheet. Exam 1 Solutions.
• Grading information:
  • Online access to scores. Log in with your NetID and password to access your scores on each hw assignment and exam, and your current total score. If there is a discrepancy between the scores in the computer display and the score on the assignment/exam, let me know right away.
  • Prefinal grade information:
    • Prefinal points total: As of 11/29/2007, the maximal total number of points is 618, based on 11 homework assignments (18 points each) and 3 midterm exams (140 points each). This takes into account all grade components except for the final exam, which will contributed an additional 300 or so points.
    • Drop scores: The lowest of the 12 hw scores (indicated by a double asterisk (**)) has been dropped, so only the 11 highest hw scores are counted towards the total. Note that there is no dropped exam score; all three exam scores count.
    • Excused scores: An "excused" score on a hw assignment or exam is indicated by a single asterisk (*) in the score display. If you got an excused score, then the assignment/exam in question is not included in the points total, i.e., it is treated as if it had never happened. For those with excused scores the total score was appropriately rescaled; in the online score display the original and rescaled scores are indicated by a notation such as "250/500=309/618".
    • Current grade: As a rough guideline, based on the current (pre-final) total of 618 points, I'd place the cutoff between the A and the B range at around 520, the cutoff between the B and the C range at around 420; A score in the 600s would be a solid A+, a score in the mid to high 500s a straight A, and a score in the low 500s an A- or B+.
  • Course grade: The Final Exam, the only remaining grade component, will be worth roughly 300 points, so the maximal total number of points at the end of the semester will be around 900 points. Course grades will be based strictly on the accumulated points total (rescaled in case there are excused scores) after the final exam scores have been added in.

    I don't use a predetermined curve to assign letter grades corresponding to these points totals, but rather look for gaps in the ordered list of points totals, and try to place the dividing lines between letter grade ranges (A, A-, B+, etc.) in those gaps. This allows for more flexibility, makes for fairer grading and helps avoid situations where a very small difference in the points total would make a difference in the course grade. With a small class like this one there will likely be enough large gaps in which to place grade cut-offs so such situations should not arise.

Announcements

• [11/29/07] Final exam information: The Final Exam will be Saturday, December 15, 1:30 pm - 4:30 pm. As was mentioned before, you must take the final exam at this officially assigned slot. The exam will be in the regular classroom. It will be cumulative and about 2 - 2.5 times as long as a one hour midterm exam. I expect to have 4 - 6 problems on each of the three midterm topics, and one or two problems on the material covered after the third midterm. (I don't have practice exams on the latter material. For practice exams on the midterm materials, see the links above.)
• [11/5/07] Midterm 3 date: The third midterm will be Wednesday, November 28. This seems to be the most sensible date. On the one hand, it is far enough after Thanksgiving break so that travel delays coming back from the break won't be a problem. On the other hand, it avoids the exam rush during the final week of class when many classes have their last midterm. The exam will cover Chapter 14 (including the postponed section 13.8. An Exam Review Sheet with a detailed syllabus is available.
• [10/9/07] Midterm 2 date: By a 10 to 1 margin, the next midterm date was voted to be is Wednesday, October 24. The exam at the regular class hour and in the usual room, i.e., 2:00 pm - 2:50 pm, in 142 Henry. The exam will be on the left-over sections of Chapters 11 and 12 (11.8, 12.5, 12.9, 12.10) and essentially all of Chapter 13. I'll post a more detailed syllabus by early next week.
• [9/18/07] Exam 1 Review Sheet. Click on this link for an Exam Review Sheet with general information about the exam, a detailed syllabus, and links to old exams.
• [9/13/07] Midterm 1 date: The date, decided by popular vote, is Wednesday, September 26. The exam at the regular class hour and in the usual room, i.e., 2:00 pm - 2:50 pm, in 142 Henry. The exam will on Chapters 11 and 12 through 12.8, excluding the deferred sections 11.7, 11.8, and 12.5. This corresponds roughly to the material covered in class through the week of 9/17.
• [8/20/07] Getting into this course:. Over the past few weeks, I have received a number of emails from students who wanted to get into this class, but were unable to register. Unfortunately, there is nothing I can do about this. This is a special honors section of Math 241, and enrollment is therefore strictly limited and requires departmental approval as well as meeting certain prerequisites. If you are trying to get in, you have to (1) meet the requirements, (2) get the appropriate departmental approval, and (3) there has to be a space available in at least one of the two honors sections (currently this section is full); send email to advising@math.uiuc.edu for more information. Note that I cannot grant the "departmental approval"; for that you must see one of the Undergraduate Advisors in 313 Altgeld.

Homework Solutions

• HW 12 Solutions (Section 14.7).
• HW 11 Solutions (Sections 14.5, 14.6).
• HW 10 Solutions (Sections 14.3, 14.4).
• HW 9 Solutions (Sections 14.1, 14.2).
• HW Solutions, Section 13.9.
  (Because of the proximity to the exam, this assignment was not collected and graded.)
• HW 8 Solutions (Sections 13.6, 13.7).
  Additional solutions: 13.6. 13.7.
• HW 7 Solutions (Sections 11.8, 13.4, 13.5).
  Additional solutions: 11.8. 13.4. 13.5.
• HW 6 Solutions (Sections 13.1, 13.2, 13.3).
  Additional solutions: 13.1. 13.2. 13.3.
• HW 5 Solutions (Sections 12.5, 12.9, 12.10).
  Additional solutions: 12.5. 12.9. 12.10.
• HW 4 Solutions (Sections 12.7, 12.8).
  Additional solutions: 12.7. 12.8.
• HW 3 Solutions (Sections 12.4, 12.6, 12.7).
  Additional solutions: 12.4 12.6 12.7.
• HW 2 Solutions (Sections 11.4, 11.5, 11.6).
  Additional solutions: 11.4 11.5 11.6.
• HW 1 Solutions (Sections 11.1, 11.2, 11.3).
  Additional solutions: 11.1 11.2 11.3.

Class Diary

• Thursday, 12/6: Open House. I will be available in the usual classroom for questions, but don't plan on covering new material. (I will schedule additional open house sessions next week.)
• Monday, 12/3 - Wednesday, 12/5:
  • Handout: Differentiability in n-dimensional spaces.
  • Class: Differentiability of functions f from Rn to Rm. Formal definition and motivation via Taylor's formula. Derivative matrix Df(a). Representation in terms of partial derivatives. Connection with familiar derivative concepts: Ordinary one-variable derivative, gradient, derivative of vector function, Jacobian. Matrix form of chain rule; examples and special cases. Examples of Taylor's formula.
• Thursday, 11/29:
  • Class: Wrapped up the discussion of limits with some more epsilon-delta proofs, and returned graded exam and exam solutions.
• Wednesday, 11/28: Midterm Exam 3.
• Tuesday, 11/27:
  • Class: Handout on functions, limits, continuity, in n-dimensional spaces. Examples of epsilon-delta proofs of limits, proving non-existence of limits.
  • Read: 12.3. This section, which was skipped over earlier in the semester, contains the formal (epsilon-delta) definitions of limits and continuity for functions of several variables.
  • Do: 12.3: 21, 29, 41, 43, 51. These (along with the class examples) are typical of the problems that you should be able to handle.
• Monday, 11/26:
  • Class: Overview of topics to be covered in remainder of semester. n-dimensional spaces, functions on n-dimensional spaces, norm, formal (epsilon-delta) definition of limit.
• Thursday, 11/15:
  • Class: A brief discussion of the mathematics underlying the integral theorems of calculus (Green, Stokes, Divergence, and Fundamental Theorem of Calculus), and a unifying theorem that encompasses all of these integral theorems.
• Wednesday, 11/14:
  • Class: Finished 14.7 with some further examples illustrating Stokes Theorem.
• Tuesday, 11/13:
  • Class: 14.7, continued. Stokes Theorem. Interpretation of curl.
  • Do: Graded HW Assignment 12. Due Thursday, 11/15. This is a short assignment, with an earlier due date, because of the upcoming exam. NOTE (11/14/07): I have dropped the last two problems (those from 14.M) from the assignment; I'll probably do these (or at least one) in class.
• Monday, 11/12:
  • Class: Started 14.7, the final section in Chapter 14. Independence of path in 3 dimensions. Conservative/irrotational fields. Potential/antiderivative of a vector field.
• Thursday, 11/8:
  • Class: Review of surface integrals. Intepretation as flux. Sources (positive flux) and sinks (negative flux). Generalized divergence theorem for regions with holes. Application to inverse square fields.
  • Do: Graded HW Assignment 11.
• Wednesday, 11/7:
  • Class: Surface integrals, continued. Divergence theorem (14.6). More examples. Comparison of direct computation of a surface integral and computation via the divergence theorem.
  • Do: 14.6: 3*, 5*, 8*, 10*.
• Tuesday, 11/6:
  • Class: Surface integrals, continued. Unit normal vector to surface. Representation of conversion factor in terms of Jacobians. Change of variables formula, revisited.
    Examples: Parametrization of spherical surfaces. Computation of surface area, mass, centroid.
  • Do: 13.8: 2*, 3*, 4*, 9*, 11*
    14.5: 1*,4*, 6*, 16*, 19*, 28*
• Monday, 11/5:
  • Class: Surfaces, Surface Area, Surface Integrals (13.8/14.5/14.6). Overview of different types of surface integrals. Parametrized surfaces. Surface area element. Derivation of conversion factor.
• Thursday, 11/1:
  • Class: Review of line integrals.
  • Do: Graded HW Assignment 10.
• Wednesday, 10/31:
  • Class: Review: Indendence of path property. Relevance of simply connected regions (i.e., regions without holes). Handout: Different forms of Green's Theorem. Applications of Green's Theorem: Calculation of some line integrals via Green's Theorem. Finding areas via the Green's Theorem Area Formula.
  • Read: Example 4, as this illustrates an application of a modified version of Green's theorem when the underlying region has a hole.
  • Do: 14.4: 3*, 4*, 13*, 17*, 21*, 22*, 27*, 29*.
• Tuesday, 10/30:
  • Class: Wrapped up 14.3. Finding a potential for a conservative field. Independence of path.
• Monday, 10/29:
  • Class: Started 14.3. Fundamental theorem for line integrals. Conservative vector fields, gradient fields Potential. Test for conservative fields. (To be continued tomorrow.)
  • Do: 14.3: 6*, 8*, 16*, 23*, 28*, 30*, 36*.
• Thursday, 10/25:
  • Class: Finished 14.2. Line integrals with respect to coordinate variables. Work integral.
  • Do: Graded HW Assignment 9, due Monday, October 29.
• Wednesday, 10/24: Midterm Exam 2.
• Tuesday, 10/23:
  • Class: Finished 14.1 and started 14.2. Line integrals. Parametrized curves. Smooth and piecewise smooth curves. Parametric version of line integral. (To be continued on Thursday.)
  • Do: 14.2:1*, 7*, 9*, 16*, 19*, 20*, 35*, 36*, 42*.
• Monday, 10/22:
  • Class: Continued with 14.1. The "del" (inverted triangle) operator. Divergence and curl of vector field. Examples.
  • Do: 14.1:15*, 19*, 35*, 36*, 39*, 40*. (These problems will be part of the next graded assignment.)
• Thursday, 10/18:
  • Class: Started Chapter 14 (Vector Calculus).
    14.1: Vector fields. Definition of a vector field, and examples. (To be continued Monday.)
• Wednesay, 10/17:
  • Class: Finished 13.9 (change of variables for multiple integrals).
  • Do: 13.9: 1, 3, 13, 14; 13.M: 53. (Disregard 13.M:54, which I originally also assigned, as it leads to very nasty computations.) Solutions will be posted above, under Homework Solutions.
• Tuesday, 10/16:
  • Class: Wrapped up 13.6/13.7. Started 13.9 (change of variables for multiple integrals). (13.8 will be deferred till Chapter 14.)
  • Do: Graded HW Assignment 8, due Thursday, October 18. Note the early due date. This will be the last graded hw assignment before the exam. (I will assign, but not collect, problems from 13.9, and make solutions available before the exam.)
• Monday, 10/15:
  • Class: Sections 13.6/13.7, continued, with more examples: A tetrahedron integral, and icecream cones of various shapes.
• Thursday, 10/11:
  • Class: Started Sections 13.6/13.7 on triple integrals. This is the most important and the most difficult part of Chapter 13, and I plan to spend another 1 - 2 class hours just working triple integral problems.
  • Do: 13.6: 5*, 9*, 48*.
    13.7: 1*, 7*, 8*, 9*, 16*, 19*, 21*, 25*, 33*, 37*, 39*
• Wednesday, 10/10:
  • Class: 11.8. Cylindrical and spherical coordinates.
  • Read: Section 11.8 (especially, the pictures). The discussion of longitudes, latitudes, great circles, etc. near the end is quite interesting, but isn't suitable for exam/hw problems, so you can safely skip it.
  • Do: Graded HW Assignment 7.
• Tuesday, 10/9:
  • Class: More examples of applications of double integrals. Average value of a function of two variables.
  • Read: 13.5. You can skip the parts on Pappus' Theorems (both -first and second), and the formulas for kinetic energy and radius of gyration.
    11.8. The most interesting part (and the most relevant one in terms of real-world applications) part of this section is the discussion of longitudes, latitudes, great circles, etc. near the end.
  • Do: 13.5: 7*, 15*, 26*, 31*, 33*
• Monday, 10/8:
  • Class: 13.4: Double integrals in polar coordinates: More examples
    13.5: Applications of double integrals: Mass, center of mass, centroid, moments of inertia (to be continued), probability
  • Do: 13.5: 7*, 15*, 26*, 31*, 33*
• Thursday, 10/4:
  • Class: 13.4: Double integrals in polar coordinates.
  • Do: 13.4: 9*, 13*, 17*, 27*, 29*, 34*, 36*, 41* (this will be part of next week's graded hw assignment).
• Wednesday, 10/3:
  • Class: 13.3: More double integral examples. Areas and volumes.
  • Read: 13.3.
  • Homework 6. Graded HW assignment 6, due Monday, 10/8.
• Tuesday, 10/2:
  • Class: 13.2. Double integrals over general regions.
• Monday, 10/1:
  • Class: Started Chapter 13 (Multiple Integrals). Review of concept of an integral in the one variable case, and extension to two variable case. Iterated integrals.
  • Read: Sections 13.1/13.2. (You can skim over the more theoretical material in 13.1.)
  • Do: 13.1: 13, 19, 21*, 29*, 33*.
    13.2: 11, 13, 15*, 19*, 31*, 33*.
• Thursday, 9/27:
  • Class: Returned the graded exam and passed out exam solutions. (Grading/curving information will be posted shortly.)
    Section 12.9. Examples of applications of the Lagrange multiplier method. Application to economics: maximizing production function subject to budget constraint. Interpretation of Lagrange multiplier in this context. Application to mathematics: Proving mathematical inequalities (Cauchy-Schwartz and arithmetic geometric mean inequalities).
  • Read: Section 12.9. You can skip over the final part (on the case of multiple constraints).
  • Homework 5: Graded HW assignment 5, due Monday, 10/1.
• Wednesday, 9/26: Midterm Exam 1.
• Tuesday, 9/25:
  • Class: Started 12.9. Optimization with constraints and the Lagrange multiplier method.
  • Do: 12.9: 1, 7, 13*, 19*, 20*, 21*, 35*, 63*.
    12.M: 36*
• Monday, 9/24:
  • Class: 12.5/12.10 (Maxima/minima of functions of several variables, optimization without constraints). First and second derivative tests for functions f(x,y). Critical points. Classification as max, min, saddle points.
    Motivation of second derivative test via Taylor's formula.
  • Do: 12.5: 3, 9, 13*, 19*, 54* (see 53 for a concrete version of this problem).
    12.10: 7, 17*, 26*.
    12.M: 51* ("least squares" method)
  • Read: 12.5 and 12.10. Note that many of the problems and examples in 12.5 (e.g., Problems 29 - 42) are of a type where the Lagrange multiplier from 12.9 can be applied and, in fact, is the more natural method since those problems involve optimization with constraints.
• Thursday, 9/20:
  • Class: Wrapped up 12.8. Tangent planes to surfaces. Comparison of the methods from 12.6 and 12.8.
    Started 12.5/12.10 (Maxima/minima of functions of several variables, optimization without constraints). Overview and review of one-variable case. First and second derivative test.
  • Homework: Graded HW assignment 4, due Monday, 9/24. (This was handed out Wednesday.)
• Wednesday, 9/19:
  • Class: Continued 12.8 (directional derivatives and gradients). The directional derivative via the chain rule. Geometric interpretation of the gradient. Connection with level curves and level surfaces. Computation of tangent planes via gradients.
  • Read: Section 12.8, but you can skip the limit definition of the directional derivative (p. 909) and Example 7 (which involves the intersection of two surfaces).
  • Do: 12.8: 1, 9, 11, 15, 21, 31*, 33*, 47*, 49*, 61*. 12.M:22
  • Graded Homework Assignment: HW 4, due Monday, 9/24.
• Tuesday, 9/18:
  • Class: Derivation of implicit differentiation formula. Started 12.8 (directional derivatives and gradients). Motivation and definition of the directional derivative. Connection with gradient.
• Monday, 9/17:
  • Class: Yet another motivation for the "derivative" of a multivariable function (as the "coefficient" of the second term in Taylor's formula). The gradient vector. More from 12.7: Matrix form of the chain rule. Implicit differentiation.
  • Read: Remainder of Section 12.7 (covering implicit differentiation and the matrix form of the chain rule).
  • Do: 12.7: 19*, 20*, 25*, 58*, 59*, 61*.
• Thursday, 9/13:
  • Class: Finished 12.6. Differentials and tangent planes. Geometric interpretation. Applications to error estimates and propagation of errors.
    Started 12.7 on the multivariable chain rule.
  • Read: Section 12.7 through p. 902. The remainder of this section, implicit partial differentiation and the matrix form of the chain rule, will be covered on Monday.
  • Do: 12.7: 3, 5, 13, 34*, 38*, 40*, 49* 12.M (Misc. Exercises, p. 989): 19*
  • Homework: Graded HW assignment 3, due Monday, 9/17.
• Wednesday, 9/12:
  • Class: The ideal gas law (Problem 12.4:63).
    [12.5 will be covered right before 12.10]
    12.6: Differentials. Tangent planes. Linear approximation. Application to Error estimates.
    
  • Read: Section 12.6 through Example 5. You can skip the last part of this section, on "differentiability".
  • Do: 12.6: 1, 7, 15*, 17*, 34*, 35, 38*.
• Tuesday, 9/11:
  • Class: Recap of the basic concepts from 12.2: graph of a function of two variables, level curve, contour curve, level surface.
    Overview of derivative-like concepts for multivariable functions: partial derivative, differential, directional derivative, gradient.
    12.4: Partial derivative. Definition, notations (subscript notation and "delta" notation). Examples. First and second partials. Rate of change interpretation. Geometric interpretation (see the pictures in the book). Partial differential equations.
    
  • Read: Section 12.4. This is an easy section.
  • Do: 12.4: 3, 5, 7, 31, 35*, 55*, 56*, 62*, 63*, 65*, 71*.
• Monday, 9/10:
  • Class: Started Chapter 12. (11.7 and 11.8 will be deferred till later.) Recap/overview of the concept of a function. "Black box" interpretation. Classification of functions according to type of input and output (scalar or vector/multiple scalars).
    Functions of several variables. Examples. Domain (covered lightly), graph, level curves/surfaces, contour curves.
  • Read: Sections 12.1 and 12.2. 12.1 is a half-page introduction to Chapter 12. 12.2 is largely a picture section that is not very suitable to presenting in class, so I won't spend much class time on it. Make sure to read this section; in particular, just look at the pictures and try to understand the basic concepts (graph, level curve/surface, contour curve) from these pictures.
• Thursday, 9/6:
  • Class: Wrapped up 11.6. More on curvature: Definition in terms of unit tangent vector. Formulas for curvature.
    More on acceleration: Decomposition into tangential and normal components, aT and aN. Interpretation of these components, and examples. Derivation of formulas for aT and aN.
  • Read: Section 11.6. You can skip the discussion of curvature for plane curves (p. 820 middle to p. *23, top), and you do not need to memorize the (rather complicated) formulas (12), (13) for this case. You can also skip the final part of the section (applications to Kepler's laws and planetary motion), since I didn't get around to this material in class. The remainder of this section, however, is important. For K, aT, and aN, you need to know two formulas each, one involving r' and r'', and the other involving v. (There is a third set of formulas for these quantities, involving the arclength s (e.g., K=dT/ds); however, for calculations these are not particularly useful, and you do not need to memorize these.)
  • Preread: Sections 12.1 and 12.2. Preread these two sections. 12.1 is a half-page introduction to Chapter 12. 12.2 is largely a picture section that is not very suitable to presenting in class, so I won't spend much class time on it. If nothing else, just look at the pictures and try to understand the basic concepts (graph, level curve, contour curve) from these pictures.
  • Homework: Graded HW assignment 2, due Monday, 9/9.
    CORRECTION: In the yellow sheet handed out, replace 11.5:55 by 11.5:50. (Both were in the original assignment, but I removed 11.5:55 since it's the same problem as 11.5:40.) In the pdf file linked above this has been corrected.
• Wednesday, 9/5:
  • Class: Started 11.6 (Curvature and acceleration).
    Arc length formula. Unit tangent vector T, principal unit normal vector N.
    Curvature K. Best-approximating ("osculating") circle to a curve.
    [To be continued.]
  • Read: 11.6. You can skim through the discussion of curvature for plane curves (p. 820 middle to p. *23, top), and you do not need to memorize the (rather complicated) formulas (12), (13) for this case.
    The final part, on Kepler's laws and planetary motion, will be covered lightly (if at all), but it won't be on exams.
  • Do: 11.6: 1; 33*, 38*, 43* (all of these three involve the same curve), 46* (see Example 5), 66*, 67*. 11.M (Miscellaneous problem section, p. 847-849): 23*, 24*.
• Tuesday, 9/4:
  • Class: Wrapped up 11.5. Properties of derivatives of vector functions (see p. 808). Product rule for dot and cross products. Application: computation of derivative of |r(t)|.
    Motion of projectiles: a simple example, with initial position (0,0), and initial velocity (100,100). Computation of v(t) and r(t) at any time t. Maximal height reached, horizontal distance at impact, and speed at impact.
  • Read: 11.5. In particular, study Examples 9 and 10. The first is like the example worked out in class, but with general initial position and general initial velocity. The second is an example with an additional acceleration component that requires working in 3 dimensions; Problem 65 from the HW assignment for this section is of the same type.
  • Do: 11.5; 5, 8, 15, 17, 23, 27*, 35*, 40*, 42*, 45*, 50*, 55*, 62*, 65*. (Starred problems will be part of Graded HW 2, due Monday.)
• Thursday, 8/30:
  • Class: Started 11.5 (Vector functions and vector calculus). Recap of concept of a function; "black box" interpretation. A vector function is a function that takes a scalar input t, and produces as output a vector r(t). Geometric interpretation as motion of a particle.
    Derivatives of vector functions: algebraic definition (coordinatewise differentiation); limit definition of derivative. Geometric interpretation as tangent vector to curve r(t). Physics connection: velocity, acceleration, force (Newton's law).
    [I will spend at least another half hour of class on this section, covering differentiation rules, and (due to popular demand!) motions of projectiles.]
  • Read: 11.5 (entire section).
  • Do: 11.5; 5, 8, 15, 17, 23, 27*, 35*, 40*, 42*. (Starred problems will be in Graded HW 2. More problems will be added next time)
• Wednesday, 8/29:
  • Class: 11.4. Lines and planes in space. Vector and parameter equations for line (skip symmetric equations). Vector, scalar, linear equations for plane. Finding equation of a line or a plane given types of information (e.g., point and two vectors in the plane, etc.). Possible configurations (perpendicular, parallel, intersecting, skew, etc.) of two lines, a line and a plane, and two planes, and methods to differentiate between the various cases.
  • Read: 11.4. Everything except the parts involving symmetric equations (e.g., the last part of p. 802).
  • HW: 11.4: 3, 7, 9, 15*, 17*, 23, 29, 31, 33*, 35, 37, 39*, 54*, 55*. (Ignore questions asking for symmetric equations. Problems with an asterisk will be included in next week's graded HW.)
• Tuesday, 8/28:
  • Class: Wrapped up 11.3. Properties of dot and cross products; scalar triple product. Applications of cross product: Finding a vector perpendicular to two given vectors. Computing areas of parallelograms and triangles. Computing volumes of parallelepipeds. Testing three vectors (or four points) for coplanarity.
    Food for thought: Is the cross product associative?
  • HW: Graded HW assignment 1, due Thursday, 8/30.
• Monday, 8/27:
  • Class: Finished 11.2. Review of dot product, and its main properties. Dot product of parallel and perpendicular vectors. Testing for perpendicularity and parallelity of two vectors. Computation of angles between two vectors. Direction angles. Projection (component) of one vector onto another.
    Started 11.3: Algebraic definition of cross product and geometric interpretation. Some simple examples. (To be continued tomorrow. Some food for thought: Can the dot and cross products be extended (in some natural way) to n dimensions?)
  • Read: 11.3 except for the very last paragraph (on "torque"). If you are not familiar with 3 by 3 determinants, read up on it (see Examples 1 and 2 in 11.3).
  • HW: 11.3: 1, 7, 13*, 15*, 17*, 19*, 36* (Problems marked by an asterisk are part of the graded assignment, to be turned in Thursday, 8/30. The final list of problems will be handed out tomorrow.)
• Thursday, 8/23:
  • Class: Wrapped up 11.1 and started 11.2. Recap of basic vector concepts, in the 3-dimensional setting. Arithmetic with vectors: addition, multiplication of a vector by scalar. Products between two vectors, overview; the dot product (covered in 11.2) and the cross product (covered in 11.3). Started with the dot product: algebraic definition, geometric definition, examples, basic properties.
  • Read: 11.2 except for Examples 1,2 (p. 782) and application to "work" (p.787 bottom through the end of the section). In particular, read up on topics that I didn't get to: Dot products of perpendicular vectors. Computing angles via dot products. Direction angles. Component (or scalar projection) of one vector along another.
  • Do (non-graded hw): 11.2 Problems 3, 15, 39, 41, 43, 45, 49.
  • Graded HW Preview: 11.1: 47, 51, 53, 54; 11.2: 59, 62, 68. (I will add more problems on Monday. The HW will be due Thursday, 8/30.)
• Wednesday, 8/22:
  • Class: General course information; handed out the Course Information Sheet.
    Overview of the calculus sequence. One variable calculus versus multivariable calculus.
    Started 11.1 (Vectors in the plane). Concept of a vector. Geometric representation (as an arrow in the plane) and algebraic interpretation (as a tuple with two components). Notation for vectors (boldface in printed text, arrow notation when writing by hand). Magnitude/length and direction. Unit vector. Unit basis vectors.
  • Read: 11.1, except for the last two examples (p. 778). In particular, read up on the following, which I didn't get to in class: Position vector. Vectors versus scalars. Algebraic operations on vectors (addition, negative of a vector, subtraction, multiplication by scalar), and geometric interpretation of these operations.
  • Do (non-graded hw): 11.1 Problems 1, 5, 9, 17, 21, 33. (These are all routine, and rather boring, but shouldn't take more than a minute or two each. More interesting are the problems near the end of the section (from 47 onwards); some of these will be included in the first graded hw assignment.)