MATH 241 F1H: Calculus III Honors
Spring 2014
Professor A.J. Hildebrand
http://www.math.illinois.edu/~hildebr/241/
Final Exam Results and Course Grades
Final Exam Statistics: The final exam was worth 170 points, plus
up to 5 bonus points. The highest scores were 166, 159, 158, and the
median score 138/170.
Course grade:
You can access your final exam score, and your total score and course
grade, as usual
at
this link. The letter grade shown at the end of the online score display
(including plusses and minuses) is your official course grade.
This grade is based on the total number of accumulated points,
with cutoffs chosen so as to minimize hardships and close calls.
As a result, nobody was within 10 points of the next higher grade.
Viewing your final exam:
Because of departmental policies I have to keep final exams for one
year and thus cannot return the exam to you. However, if you want to
see your final, I will hold a special Open House hour Monday, May 12,
5 pm  6 pm, in 145 Altgeld. Alternatively,
send me an email (ajh@illinois.edu) to set up another time.
Final words:
This has been a great class to teach, and I was pleased with how it
turned out in the end. I tried to make it a challenging, but also
rewarding experience, and I hope you did get something out of this class
beyond the material in the syllabus. If you have questions about other math
courses and opportunities in mathematics, I'd be happy to discuss this with
you. Good luck on your remaining finals, and all the best on the rest of
your academic career here at Illinois!
Have a wonderful summer!
Announcements

[1/23]: Open House Hours:
Sundays, 2pm, and Wednesdays, 7 pm, in 159 Altgeld Hall or an adjacent
room.

[1/22]: WebAssign.
The first WebAssign homework, WA1, is now available. The assignment is on Sections
12.112.3 and is due at midnight on Sunday, Jan. 26. To register for
WebAssign, use the
WebAssign UIUC login page.
Note: If you switched sections, you may need to refresh your web browser
(or exit the browser), so that the new section shows up.
General Information, Course Policies, Exams, Grades, etc.
 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.

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.
 Exam Information
 Midterm Exam 1: Wednesday, Feb. 19
 Midterm Exam 2: Wednesday, March 12
 Midterm Exam 3: Wednesday, April 16.
 Final Exam: Friday, May 9, 1:30 pm  4:30 pm.

Sample exams. Links to sample exams from past Math 241 Honors
classes. These should give you a good idea of the type and level of
problems to expect.
 Add/drop deadlines: February 3 and March 14. The first date
(February 3) is the last date you can add a course; if you want to switch to a
nonhonors section, you have to do so by that date since a section
switch involves dropping one course and adding another.
The second date (March 14) 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.

WebAssign login
for UIUC students.
Log in with your NetID and AD password to access Webassign.

Link to Online Scores.
Click on this link and log in with your NetID and password to access
your scores. The display shows the scores on written (nononline) homework
assignments (hw1, hw2, etc), honors homework assignments (project1, etc),
quizzes, and exams given out so far,
and your total accumulated score from these components.
(For the scores on online homework see the Webassign site; the accummulated
Webassign score will be added in at the end of the semester.)
If a score is missing or incorrect, let me know right away.

Emergency Information
Assignments and Due Dates
Class Diary
 Wednesday, 5/7: Exam Q and A.
 Tuesday, 5/6: Exam Q and A.
 Monday, 5/5:
 Thursday, 5/1:
 Class:
Irrotational/incompressible vector fields, revisited.
More on harmonic functions.
 Wednesday, 4/30:
 Class:
Harmonic functions, continued.
 Do:
Work this week's homework.
 Tuesday, 4/29:
 Class:
Curl, revisited. Limit formula for curl (see end of section 16.8).
Harmonic functions. Definition and mean value property.
 Do:
Read final part of Section 16.8 (p. 1126).
Work this week's homework.
 Monday, 4/28:
 Class:
Green's theorem, revisited. Vector version of Green's theorem. Green's
theorem for regions with "holes".
Divergence, revisited. Interpretation as flux density and
limit formula for div F (see end of section 16.9).
Handout: Stokes's Theorem and Green's Theorem
Revisited.
 Do:
Read final part of Section 16.9 (p. 1133).
Work this week's homework.
 Thursday, 4/24:
 Class:
Stokes' Theorem. Recap of the Big Theorems of Vector Calculus:
FTLI, Green, Gauss/Divergence
 Do:
Read Section 16.8, and
finish this week's homework.
 Wednesday, 4/23:
 Class:
The Divergence Theorem.
 Do:
Read Section 16.9.
Start working on this week's homework.
 Tuesday, 4/22:
 Class:
Vector surface integrals, continued. Relations between the various formulas
for surface integrals. More examples.
 Do:
Read Sections 16.7.
 Monday, 4/21:
 Thursday, 4/17:
 Class:
Returned the midterm.
Finished Section 16.5 on curl, div, and grad.
Curl test for existence of antigradient (i.e., for conservative fields)
and its relation to the mixed partials test.
Div test for existence of anticurl.
Started Section 16.6 on surfaces. Representation via parameter function
r(u,v) and as graph of function z=f(x,y). Surface area and
conversion factor for area element.
 Do: Read 16.5 and 16.6, and work the homewok on these sections,
WA 12 and HW 12.
 Wednesday, 4/16: Midterm Exam 3.
 Tuesday, 4/15:
 Class:
More on grad, div, and curl. Irrotational (curl zero) and incompressible
(divergence zero) fields. Combinations: Curl of grad and div of curl.
 Do: Prepare for Wednesday's exam!
 Monday, 4/14:
 Class:
Exam Q and A.
Grad, div, and curl. Definitions and same examples.
 Do: Prepare for Wednesday's exam!
 Thursday, 4/10:
 Class:
Green's Theorem. Computing line integrals and areas via Green's Theorem.
 Do: Read Section 16.4 (you can skip the final part, on extensions
of Green's Theorem.) Finish WA 11/HW 11 (due Sunday/Monday).
 Wednesday, 4/9:
 Class:
Overview of the Fundamental Theorem for Line Integrals and related
properties: Conservative fields, existence of potential/antigradient;
independence of path; loop integral zero; mixed partial test. Proof of
FTLI for gradient fields via chain rule for derivatives in Rn.
 Do: Read Section 16.3. Work on WA/HW 11.
 Tuesday, 4/8:
 Class:
Line integrals continued. The Fundamental Theorem for Line Integrals.
Independence of path and loop integral zero properties.
 Do: Read Section 16.3.
 Monday, 4/7:
 Thursday, 4/3:
 Class:
Started Chapter 16. Recap of vector functions and curves in
R^{2} and R^{3} and the arclength formula.
Line integrals as generalizations of arclength integrals.
Interpretations of line integrals.
 Do: Read Sections 16.1 and 16.2.
 Wednesday, 4/2:
 Class:
From Linear Algebra back to Calculus:
General change of variables formula for ndimensional integrals, and
some examples.
Handout:
Linear Algebra, II.
 Do: Finish reading 15.10 and work this week's homework (WA 10
and HW 10).
 Tuesday, 4/1:
 Class:
Finished linear algebra excursion. Determinants and their geometric
interpretation as volume conversion factors for linear maps. Computation of
n by n determinants.
Application of this theory to change of variables in multiple integrals: The
"correct" conversion factor is the determinant of the "Jacobian matrix",
the derivative matrix of the underlying transformantion of variables.

Do: Read 15.10 and do the problems from this section in this week's
homework (WA 10 and HW 10).
 Monday, 3/31:
 Class:
Excursion into linear algebra, part II. Geometric properties of linear
functions from R^{2} to R^{2}.
Conversion factor for areas and volumes.
(To be continued tomorrow.)

Do: Read 15.10 (Jacobian determinants,
change of variables in ndimensional integrals).
 Week of March 24:
SPRING BREAK. NO CLASS  HAVE A GOOD BREAK!
 Thursday, 3/20:
 Class:
Fun with multiple integrals.
The Monte Carlo method of computing probabilities, areas, volumes, and
integrals. Intersecting cylinder problem.
 Wednesday, 3/19:
 Class:
More triple integral practice. Volume of icecream cones of various shapes
in cylindrical and spherical coordinates.

Do:
Read 15.8 and 15.9, and start
working the HW problems on these sections.
 Tuesday, 3/18:
 Class:
Triple integral practice: Computing volumes of spherical regions using
cylindrical and spherical coordinates.

Do:
Read Sections 15.7, 15.8, and 15.9 on triples integrals in rectangular,
cylindrical, and spherical coordinates. The most important of these are
the latter sections dealing with integration in spherical and cylindrical
coordinates.
 Monday, 3/17:
 Class:
Applications of double and triple integrals to probability.

Do: Read the final part of 15.5, on applications to probability.
 Thursday, 3/13:
 Class:
Double integrals in polar coordinates. Applications of double integrals.

Do:Read Chapter 15 through 15.5 and do the online and written
homework WA 8 and HW 8.
 Wednesday, 3/12: Midterm Exam 2
 Tuesday, 3/11:
 Class:
Computation of double integrals. Fubini's theorem. Double integrals over
rectangles and over general regions. Reversing order of integration.

Do: Prepare for tomorrow's exam!
 Monday, 3/10:
 Class:
Started Chapter 15 on multiple integrals. Review of the
onevariable case: Riemann sums, the integral as area and as antiderivative.
Extending these ideas to integrals of twovariable functions f(x,y).
The double integral as volume under graph of f(x,y).

Do: Read Sections 15.1 and 15.2.
 Thursday, 3/6:
 Class:
Applications of the Lagrange Multiplier Method: Proof of CauchySchwarz
inequality via Lagrange multipliers.

Do: Finish this week's homework (WA 7 and HW 7).
Prepare for the midterm. Test yourself by going over each of the concepts,
formulas and theorems, and computational tasks, listed in the exam syllabus,
one at a time. If you are not sure about a topic, review the class notes
on it, look it up in the book, and work some relevant examples.
 Wednesday, 3/5:
 Class:
Wrapped up the discussion of differentiability in R^{n}.
Illustration of Taylor's formula in some concrete cases.
Started 14.8, on the Lagrange multiplier method.
Formulation of the general "optimization with constraint"
problem. Motivation of the gradient equation and the Lagrange multiplier.

Do: Read 14.8 through p. 961 (skip the part on multiple
constraints), and do the Webassign problems on this section.
 Tuesday, 3/4:
 Monday, 3/3:
 Class:
Matrix form of chain rule. Examples and special cases.
 Thursday, 2/27:
 Class:
From Linear Algebra back to derivatives. The derivative matrix.
Special cases and connections with other derivative concepts.
Handout: The Derivative Matrix
 Do: Read 14.7 (through 949, Examples 14).
You need to know how to find critical points, and how to
test a critical point for max, min, and saddle points.
Examples 14 in 14.7 and the last two problems on WA 6 illustrate this.
Finish WA 6 and HW 6.
 Wednesday, 2/26:
 Class:
Excursion into linear algebra. Linear functions, definition and key
properties. Matrix connection: The linear functions from R^{m} to
R^{n} are exactly the functions of the form f(x) = A x, where A is
an n x m matrix.
Handout: Linear Algebra

Do: Finish the problems from 14.6 in HW 6 and WA 6.
Review matrix multiplication and do the matrix multiplication
exercises (Problem 2) in HW 6.
 Tuesday, 2/25:
 Class:
Wrapped up Section 14.6 on gradients, level curves, and level surfaces.
Connection with tangent planes.

Do:
Start HW (written and online) on 14.6.
 Monday, 2/24:
 Class:
Section 14.6. Directional derivatives, gradients, and level curves.

Do:
Read 14.6.
 Thursday, 2/20:
 Class:
Chain rule. Partial derivative versus total derivative.
Implicit differentiation.

Do:
Read 14.5. Finish this week's homework (WA 5 and HW 5), due Sunday/Monday.
 Wednesday, 2/19: Midterm Exam 1
 Tuesday, 2/18:
 Class:
More on linear approximations, tangent planes, and differentials.
Applications to error estimates.

Do:
Prepare for midterm!
 Monday, 2/17:
 Class:
Exam Q and A. Section 14.4. Tangent planes, linear approximation, and
differentials, motivation and applications.

Do:
Prepare for midterm!
 Thursday, 2/13:
 Class:
Wrapped up 14.3. Higher order partial derivatives, Clairot's theorem.
Partial differential equations.
Interpretations/meaning of partial derivatives.
Partial derivative calculations in ideal gas law.

Do:
Finish WA 4 and HW 4 (due Sunday/Monday).
 Wednesday, 2/12:
 Class:
Domain, graph, level curves/surfaces/sets, formal definitions.
Started 14.3 on partial derivatives,
(Section 14.2 on limits and continuity will be covered later.)

Do:
Read Section 14.3.
 Tuesday, 2/11:
 Class:
More on tangential and normal components of acceleration. Derivation of
dot/cross product formulas for these quantities.
Started Chapter 14. Multivariable functions, overview and Big Picture.
Interpretation of function as "black box" with vector input and scalar output.
Visualizing multivariable functions via graphs, level curves and level surfaces.

Do:
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 sections titled
"Concept checks" and "True/false questions" at the end of Chapters 12 and 13.
Also read Section 14.1. This is mostly a picture sections with lots of
great computergenerated sketches of domains, graphs, and level curves to
help develop a good intuition for these concepts.
 Monday, 2/10:
 Class:
Computations of derivatives of T,N,B vectors.
Representation of acceleration in TNB frame, normal and tangential
components of acceleration.

Do:
Read Section 13.4. Focus on the first part (Examples 1  4), and the
discussion of tangential and normal components of acceleration.
You can skip over the projectile problems in Examples 5 and 6.
 Thursday, 2/6:
 Class:
Tangent, normal, and binormal vectors. The TNB frame. Normal and osculating
planes. Curvature.

Do: Finish WA 3 (due Sunday midnight) and HW 3 (due in class Monday).
Read 13.3. You can skip formula (11) and Example 5 (curvature
formula for plane curve), but the rest of this section is important material,
and relevant for this week's homework, Monday's quiz, and for the
midterm (scheduled for Wednesday, Feb. 19). 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. Problems of this type come up HW 3 and WA 3 have a number of problems
of this type.
 Wednesday, 2/5:
 Class:
More on derivatives of vector functions: Limit definition. Geometric
interpretation. Application to physics: velocity, acceleration.
interpretation.

Do: Start working HW 3; in particular, the first group of problems
require the same type of stepbystep argument as the examples worked in
Tuesday's class, using properties of derivatives and vector/dot products,
Also read the first part of 13.4 (Examples 1  4); this is quite
straightforward material.
 Tuesday, 2/4:
 Class:
Derivatives of vector functions, algebraic properties and rules, geometric
interpretation. Limit definition. Vector function of constant magnitude.

Do: Read Section 13.2.
 Monday, 2/3:
 Class:
Quiz 2.
12.6: Quadratic Equations: Overview and context. (We will cover this section
lightly, and mainly through WebAssign homework.)
Started Chapter 13 on vector functions. Overview and big picture.

Do:
Do the WebAssign problems for 12.6, and read through this section as needed.
 Thursday, 1/30:
 Class:
Some problems and proofs from yesterdays handout on Vectors in ndimensional
spaces and the CauchySchwarz and Triangle Inequalities.

Do: Finish WA 2 (due Sunday midnight) and HW 2 (due Monday in class
if you have not done so). If you are done with these assignments, you can
start thinking about the Honors homework ...
 Wednesday, 1/29:
 Class:
Started excursion into ndimensional space (one of the occasional ventures
beyond the standard material).
Addition, scalar multiplication,
dot product, and norm of ndimensional vectors, in the "obvious" way.
Why the cross product only makes sense in dimension 3. The CauchySchwarz
inequality and some applications.
 Handout: Vectors in ndimensional space,
CauchySchwarz and Triangle Inequalities

Do: Written HW 1 is due tomorrow! If you are finished with this
assignment, start working on HW 2.
 Tuesday, 1/28:
 Class:
Section 12.5: Lines and planes: Motivation, ideas, and big picture.
Connections between the various equations for lines and planes.

Do: Read Section 12.5. In particular, familiarize yourself with
the different ways to compute equations of lines and planes given various sets
of data (e.g., point and normal vector, point and two vectors in the plane,
three points in plane, etc.), and with computing intersections
among lines and/or planes, and checking if they are parallel or
perpendicular. Conceptually, these are very simple and straightforward tasks,
and we will not devote much class time to this, but you need to know the
relevant formulas and practice applying these in concrete situations.

Monday, 1/27:
 Class:
Quiz 1.
Section 12.4, the cross product. Geometric and algebraic definitions,
and applications. Algebraic properties of dot and cross products.
Triple products of various kinds, which combinations of dot/cross make sense,
and the triple scalar product identity.

Do: Read 12.4.
In particular, make sure to read up on 3 x 3 determinants and the
algebraic computation of cross products via determinants.
Start working on Written HW 1,
due Thursday, and WebAssign Homework, WA 2, due Sunday.

Thursday, 1/23:
 Class:
12.3: The dot product. Geometric and algebraic definitions, and applications:
Angle between vectors. Orthogonal (perpendicular) and parallel vectors.
Scalar and vector projections. Direction angles. Work.

Do: Finish reading 12.3 if you have not done so.
The formulas from this section will be on Monday's quiz.
Work on the WebAssign homework WA1 (due by midnight Sunday).
 Wednesday, 1/22:
 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 (12.3), the
cross product (12.4). Why the "naive" (componentwise) definition of a product
does not give anything useful.

Do: Preread 12.3 in preparation for tomorrow's class.
 Tuesday, 1/21:
 Class:
General course information; handed out the
Course Information Sheet
and the Honors FAQ.
The ndimensional space R^{n}.
Vectors in R^{n}.
 Do:
In preparation for tomorrow's class, read Sections 12.1 and 12.2.
Last modified: Mon 12 May 2014 01:40:29 PM CDT
A.J. Hildebrand