Calculus III: 2012 –13
Ladue Horton Watkins High School
Instructor: Dr. John Pais
Overview:
This course is an introduction to the study of curves and surfaces in three dimensional Euclidean
space. For the first time in their mathematical development students will acquire the tools
necessary to represent and analyze both the motion of particles and the forces acting on them in
the proper geometrical setting. In addition, this course will not only develop high level
mathematics skills, but also emphasize problem solving techniques, examine the necessity of
mathematics as it relates to career goals, enable students to communicate mathematically, and
illustrate the connection to real-world application.
Learning takes place through many types of activities we engage in during each ninety-minute
period we meet. While mastery of formal objectives may be measured through tests, quizzes,
and projects, other important skills developed in class are not so easily measured in traditional
assessments. Students who attend with the intent to learn will construct knowledge both
formally and informally. When the entire group comes to the classroom prepared to learn, an
environment conducive to growth is created.
Course Description:
Calculus III is a continuation of the material covered in AP Calculus BC. Topics covered include
vectors and curves in two and three dimensions, quadric surfaces, partial derivatives, extrema
(maxima and minima), Lagrange multipliers, vector fields in two and three dimensions, double
and triple integrals, Green’s Theorem, Stokes Theorem, Divergence Theorem, and differential
equations. Graphing calculators and MAPLE® software are used throughout the course. It is
recommended that students have a grade of B or better in AP Calculus BC before enrolling in the
course.
Methods of Instruction:
Class time is spent primarily in an interactive lecture/discussion/practice problem-solving format
which includes question and answer sessions, class discussion, interactive visual-ization, guided
practice, note taking, and seat work.
Classroom Expectations:
1. Be in your assigned seat, prepared and ready to work, when the bell rings.
2. Talk when it is appropriate - do not interrupt someone else who is speaking.
3. Follow directions the first time they are given.
4. Always respect other people, property, and yourself.
5. Cell phones should be turned off during the school day. Students should not listen to music
during class.
Grades:
Grades are determined on total points earned. Points are earned through tests, quizzes, warmups, homework checks, homework quizzes, projects, and in-class activities. This is a yearlong
course and so a final exam is given at the end of each semester worth twenty percent of the
semester grade.
Grading Scale:
H 97 - 100%
B 83 - 86% C- 70 - 72% F Below 60%
A 93 - 96%
B- 80 - 82% D+ 67 - 69%
A- 90 - 92%
C+ 77 - 79%
B+ 87 - 89%
C
D 63 - 66%
73 - 76% D- 60 - 62%
Homework:
In order to receive credit for a homework check, the assignment should be complete, the
problems written out, and all the necessary work shown. If the student does not know how to do
a problem, something should still be written for the problem to show that the problem was
attempted. All work should be done neatly and kept in each student’s math notebook.
Incomplete homework will receive half credit or less.
Homework will also be checked through homework quizzes. Unannounced homework quizzes
will be given frequently, so it is very important to keep up with daily homework.
Materials for Class and Website:
Each class day students should bring their math notebook or folder, pencils or pens, paper,
assignments, and a calculator. Course materials and activities will be posted on (linked to) the
class website located at http://drpcourses.blogspot.com/.
It is a requirement of the course that the website be checked often, since all course
information will be posted there.
Attendance/Tardies:
The school policy will be followed regarding absences and tardies (see your student planner).
Please remember that, according to district policy, absences not cleared within twenty four hours
of the absence are unexcused. Unexcused absences will result in a zero for the assignments and
activities for that day.
Makeup Work Due to Absence:
A one week deadline is given to makeup all missed assignments and tests. Tests may be made
up during Academic Lab. If assignments, quizzes, and tests are not completed within one week
of an absence, students will receive a zero. If the absence has been an extended absence due to
special circumstances, please see me and we’ll make appropriate arrangements. Please
remember that, according to district policy, you will not be allowed credit for any work due or
assigned on the day of an unexcused absence.
Communication:
I look forward to an exciting and successful school year! At any time if you have any questions
or concerns, please ask me. I am usually available in the math office for help before or after
school and during Academic Lab. In addition, the best way to reach me at school is via e-mail
jpais@ladueschools.net .
Resources (Textbook - Stewart):
Auroux, Denis. Multivariable Calculus. Mathematics 18.02, MITOPENCOURSEWARE,
Massachusetts Institute of Technology, Fall 2007. Web. 23 July 2010.
Fleisch, Daniel. A Student's Guide to Maxwell's Equation. New York, NY: Cambridge
University Press, 2008.
Marsden, Jerrold E., Tromba, Anthony J. Vector Calculus, 5th Edition. New York, NY: W. H.
Freeman and Company, 2003.
Murray, Daniel A. Differential and Integral Calculus. New York, NY: Longmans, Green, and
Company, 1908.
O'Neill, Barrett. Elementary Differential Geometry, Revised 2nd Edition. Burlington, MA:
Academic Press Elsevier, Inc., 2006.
Stewart, James. Multivariable Calculus, 6E. Belmont, CA: Brooks/Cole, 2008.
Detailed Syllabus with Active Links to Resources
Unit 1.1: Vectors and 3D Space Geometry - 3D Coordinates
Local Objective



Plot points in 3D coordinate systems.
Perform algebraic operations and transformations in 3D coordinate systems.
Use graphing software to visualize 3D points and transformations of these points.
Formative Assessment - Online interactive quizzes and tutorials correlated to textbook
(Stewart) chapters and sections. Click here: Interactive Quiz 13.1 This assessment may be
used either as an homework quiz or as a small group quiz.
Summative Assessment - Students are assessed over the entire unit.
Click here: Chapter 13 Test Questions 1-5 apply to this objective.
Learning Activity
PowerPoint slides addressing the current objective. Click here: Chapter 13 Section 1
Homework
Stewart Chapter 13.1: 7-15 (odd), 19, 21, 31
Unit 1.2: Vectors and 3D Space Geometry - 3D Vectors
Local Objective



Plot and compute with 3D vectors.
Use the standard basis to represent vectors.
Create a vector from two points.
Formative Assessment - Online interactive quizzes and tutorials correlated to textbook
(Stewart) chapters and sections. Click here: Interactive Quiz 13.2
Summative Assessment - Students are assessed over the entire unit.
Click here: Chapter 13 Test Questions 6-10 apply to this objective.
Learning Activity
PowerPoint slides addressing the current objective. Click here: Chapter 13 Section 2
Homework
Stewart Chapter 13.2: 13-23 (odd), 31, 35, 41
Unit 1.3: Vectors and 3D Space Geometry - Dot Product
Local Objective




Compute the dot product of two 3D vectors and use the related theorems.
Relate the magnitude of a vector to the dot product of the vector with itself.
Use the magnitude of a non-zero vector to create a unit vector with the same
direction.
Find the angle between two vectors.
Formative Assessment - Online interactive quizzes and tutorials correlated to textbook
(Stewart) chapters and sections. Click here: Interactive Quiz 13.3
Summative Assessment - Students are assessed over the entire unit.
Click here: Chapter 13 Test Questions 11-13 apply to this objective.
Learning Activity
PowerPoint slides addressing the current objective. Click here: Chapter 13 Section 3
Homework
Stewart Chapter 13.3: 23-29 (odd), 37, 41, 45, 49, 51, 53
Unit 1.4: Vectors and 3D Space Geometry - Cross Product
Local Objective




Compute the cross product of two 3D vectors and use the related theorems.
Use the algebraic properties of the dot product in combination with the cross
product.
Relate the magnitude of the cross product to the area of the parallelogram made by
the two vectors.
Use the cross product to find the angle between two vectors.
Formative Assessment - Online interactive quizzes and tutorials correlated to textbook
(Stewart) chapters and sections. Click here: Interactive Quiz 13.4
Summative Assessment - Students are assessed over the entire unit.
Click here: Chapter 13 Test Questions 14-17 apply to this objective.
Learning Activity
PowerPoint slides addressing the current objective. Click here: Chapter 13 Section 4
Homework
Stewart Chapter 13.4: 5, 7, 11, 19, 29, 33, 37, 43, 49
Unit 1.5: Vectors and 3D Space Geometry - Lines and Planes
Local Objective


Use the vector definitions of lines and planes.
Formulate and solve geometric problems involving lines and planes using vectors.
Formative Assessment - Online interactive quizzes and tutorials correlated to textbook
(Stewart) chapters and sections. Click here: Interactive Quiz 13.5
Summative Assessment - Students are assessed over the entire unit.
Click here: Chapter 13 Test Questions 14-17 apply to this objective.
Learning Activity
PowerPoint slides addressing the current objective. Click here: Chapter 13 Section 5
Homework
Stewart Chapter 13.5: 5, 7, 11, 17, 23, 29, 43, 53, 55
Unit 1.6: Vectors and 3D Space Geometry - Cylinders and Quadric Surfaces
Local Objective

Formulate and solve geometric problems involving lines, planes, cylinders, and
quadric surfaces.

Plot lines, planes, cylinders, quadric surfaces, and figures constructed from these.
Formative Assessment - Online interactive quizzes and tutorials correlated to textbook
(Stewart) chapters and sections. Click here: Interactive Quiz 13.6
Summative Assessment - Students are assessed over the entire unit.
Click here: Chapter 13 Test Questions 11-15 and 17 apply to this objective.
Learning Activity
PowerPoint slides addressing the current objective. Click here: Chapter 13 Section 6
Homework
Stewart Chapter 13.6: 3-17 (odd), 33, 41, 43
Unit 1 Test
Unit 2.1: Space Curves - 2D and 3D Space Curves
Local Objective


Write parametric and vector equations of space curves.
Use technology to graph space curves.
Formative Assessment - Online interactive quizzes and tutorials correlated to textbook
(Stewart) chapters and sections. Click here: Interactive Quiz 14.1
Summative Assessment - Students are assessed over the entire unit.
Click here: Chapter 14 Test Questions 1-3 apply to this objective.
Learning Activity
PowerPoint slides addressing the current objective. Click here: Chapter 14 Section 1
Homework
Stewart Chapter 14.1: 1-19 (odd), 27, 41
Unit 2.2: Space Curves - Derivatives and Integrals
Local Objective



Compute componentwise limits, derivatives, and integrals of space curves.
Use technology to graph space curves.
Use space curves to model particle motion.
Formative Assessment - Online interactive quizzes and tutorials correlated to textbook
(Stewart) chapters and sections. Click here: Interactive Quiz 14.2
Summative Assessment - Students are assessed over the entire unit.
Click here: Chapter 14 Test Questions 4-5, 8-9 apply to this objective.
Learning Activity
PowerPoint slides addressing the current objective. Click here: Chapter 14 Section 2
Homework
Stewart Chapter 14.2: 5-37 (odd)
Unit 2.3: Space Curves - 2D Arc Length and Curvature
Local Objective


Compute the arc length of curves in the plane.
Compute the classical curvature and radius of curvature of curves in the plane.
Learning Activity
Material from the supplementary textbook (Murray) provides students with perspective on how
mathematicians thought about curvature over 100 years ago! Interestingly, this point of view is a
natural continuation of the material learned in their previous calculus course. This classical
perspective is in contrast to the modern differential geometric perspective they will learn in
Calculus III. Click here: (Murray) Articles 95-105
Homework
Murray: Art. 95, p. 6, 1-3; Art. 96, p. 10, 1-2; Art. 99, p. 13, 1; Art. 100, p. 14, 1-2; Art. 101, p.
18, 3; Art. 103, p. 23, 3; Art. 104, p. 29, 2; Art. 105, p. 31, 1.
Unit 2.4: Space Curves - 3D Arc Length and Curvature
Local Objective


Compute the arc length of 3D space curves.
Compute the classical curvature and radius of curvature of 3D space curves.
Formative Assessment - Online interactive quizzes and tutorials correlated to textbook
(Stewart) chapters and sections. Click here: Interactive Quiz 14.3
Summative Assessment - Students are assessed over the entire unit.
Click here: Chapter 14 Test Questions 10-12 apply to this objective.
Learning Activity
PowerPoint slides addressing the current objective. Click here: Chapter 14 Section 3
Homework
Stewart Chapter 14.3: 1-11 (odd)
Unit 2.5: Space Curves - 2D and 3D Motion
Local Objective


Use space curves to model 2D and 3D motion.
Interpret the appropriate derivatives as velocity and acceleration.
Formative Assessment - Online interactive quizzes and tutorials correlated to textbook
(Stewart) chapters and sections. Click here: Interactive Quiz 14.4
Summative Assessment - Students are assessed over the entire unit.
Click here: Chapter 14 Test Questions 13-20 apply to this objective.
Learning Activity
PowerPoint slides addressing the current objective. Click here: Chapter 14 Section 4
Homework
Stewart Chapter 14.4: 3-15 (odd), 33-37 (odd), 41
Unit 2.6: Space Curves - Arc Length Reparameterization
Local Objective


Use various function to reparameterize a space curve.
Use unit speed reparameterizations to simplify analysis of 3D space curves.
Formative Assessment - This quiz is correlated to the corresponding material drawn from a
supplementary textbook (O'Neill). Click here: Arc Length Reparametrization Quiz
Summative Assessment - Students are assessed over the entire unit.
Click here: Frenet Frames Test Questions 1-4 apply to this objective.
Learning Activity
Material for this unit is drawn from a supplementary textbook (O'Neill).
Click here: (O'Neill) Chapter 2 Section 2
Homework
Stewart Chapter 14.3: 13-14, more TBD
Unit 2.7: Space Curves - Frenet Frame Fields
Local Objective


Create moving Frenet Apparatus to represent intrinsic geometry of a space curve.
Use unit speed reparameterization to simplify computation of Frenet Apparatus.
Formative Assessment - This quiz is correlated to the corresponding material drawn from a
supplementary textbook (O'Neill). Click here: Introduction to Frenet Frames Quiz
Summative Assessment - Students are assessed over the entire unit.
Click here: Frenet Frames Test Questions 5-8 apply to this objective.
Learning Activity
Material for this unit is drawn from a supplementary textbook (O'Neill).
Click here: (O'Neill) Chapter 2 Section 3A
Homework
Stewart Chapter 14.3: 17-20, more TBD
Unit 2.8: Space Curves - Curvature, Torsion, and the Frenet Apparatus
Local Objective


Find the rate of change of the Frenet Apparatus using the curvature and torsion
of the space curve.
Relate the intrinsic geometry of a space curve to its curvature and torsion.
Formative Assessment - This quiz is correlated to the corresponding material drawn from a
supplementary textbook (O'Neill).
Click here: Curvature, Torsion, and Frenet Apparatus Quiz
Summative Assessment - Students are assessed over the entire unit.
Click here: Frenet Frames Test Questions 9-13 apply to this objective.
Learning Activity
Material for this unit is drawn from a supplementary textbook (O'Neill).
Click here: (O'Neill) Chapter 2 Section 3B
Homework
Exercises 3.1-3.3 in the O’Neill notes above. Also, think about how to prove Theorem 3.3.
Unit 2 Test
Unit 3.1: Partial Derivatives - Functions of Several Variables
Local Objective


Construct and compute with functions from n-dim real space to m-dim real space.
Interpret geometrical representation of functions from n-dim real space to m-dim
real space.
Formative Assessment - Online interactive quizzes and tutorials correlated to textbook
(Stewart) chapters and sections. Click here: Interactive Quiz 15.1
Summative Assessment - Students are assessed over the entire unit.
Click here: Chapter 15 Test Questions 1 and 3 apply to this objective.
Learning Activity
PowerPoint slides addressing the current objective. Click here: Chapter 15 Section 1
Homework
Stewart Chapter 15.1: 1, 7-19 (odd), 39-47 (odd), 61-65 (odd)
Unit 3.2: Partial Derivatives - Limits and Continuity
Local Objective


Define and compute limits for functions from n-dim real space to m-dim real space.
Define and test for continuity of functions from n-dim real space to m-dim real
space.
Formative Assessment - Online interactive quizzes and tutorials correlated to textbook
(Stewart) chapters and sections. Click here: Interactive Quiz 15.2
Summative Assessment - Students are assessed over the entire unit.
Click here: Chapter 15 Test Questions 1, 2, and 10 apply to this objective.
Learning Activity
PowerPoint slides addressing the current objective. Click here: Chapter 15 Section 2
Homework
Stewart Chapter 15.2: 5-18 (odd), 29-37 (odd)
Unit 3.3: Partial Derivatives - Definition and Computation
Local Objective


Define and compute partial derivatives for functions from n-dim real space to mdim real space.
Define and use partial derivative rules to compute partial derivatives.
Formative Assessment - Online interactive quizzes and tutorials correlated to textbook
(Stewart) chapters and sections. Click here: Interactive Quiz 15.3
Summative Assessment - Students are assessed over the entire unit.
Click here: Chapter 15 Test Questions 4, 5, 8, and 14 apply to this objective.
Learning Activity
PowerPoint slides addressing the current objective. Click here: Chapter 15 Section 3
Homework
Stewart Chapter 15.3: 1, 15-66 (multiples of 3)
Unit 3.4: Partial Derivatives - Tangent Planes
Local Objective


Use partial derivatives of a function from 𝑹𝟐 𝒕𝒐 𝑹 to define the tangent plane of an
implicit surface in 𝑹𝟑 .
Use differential notation to compute tangent plane approximation of an implicit
surface at a given point.
Formative Assessment - Online interactive quizzes and tutorials correlated to textbook
(Stewart) chapters and sections. Click here: Interactive Quiz 15.4
Summative Assessment - Students are assessed over the entire unit.
Click here: Chapter 15 Test Questions 6, 7, and 15 apply to this objective.
Learning Activity
PowerPoint slides addressing the current objective. Click here: Chapter 15 Section 4
Homework
Stewart Chapter 15.4: 1-17 (odd)
Unit 3.5: Partial Derivatives - The Chain Rule
Local Objective


Use various forms of the chain rule for a function from 𝑹𝒏 𝒕𝒐 𝑹𝒎 .
Use the chain rule to compute (partial) implicit derivatives.
Formative Assessment - Online interactive quizzes and tutorials correlated to textbook
(Stewart) chapters and sections. Click here: Interactive Quiz 15.5
Summative Assessment - Students are assessed over the entire unit.
Click here: Chapter 15 Test Questions 9 and 14 apply to this objective.
Learning Activity
PowerPoint slides addressing the current objective. Click here: Chapter 15 Section 5
Homework
Stewart Chapter 15.5: 1-14 (all), 17-25 (odd)
Unit 3.6: Partial Derivatives - Directional Derivatives
Local Objective



Define and compute the directional derivative of functions from 𝑹𝟐 𝒕𝒐 𝑹 and 𝑹𝟑 𝒕𝒐 𝑹.
Use the directional derivative to find the direction and maximum rate of change
of functions from 𝑹𝟐 𝒕𝒐 𝑹 and 𝑹𝟑 𝒕𝒐 𝑹.
Use the gradient operator to define and compute the directional derivative.
Formative Assessment - Online interactive quizzes and tutorials correlated to textbook
(Stewart) chapters and sections. Click here: Interactive Quiz 15.6
Summative Assessment - Students are assessed over the entire unit.
Click here: Chapter 15 Test Questions 11-13 apply to this objective.
Learning Activity
PowerPoint slides addressing the current objective. Click here: Chapter 15 Section 6
Homework
Stewart Chapter 15.6: 5-25 (odd), 33, 37, 39-43 (odd)
Unit 3.7: Partial Derivatives - Maxima and Minima
Local Objective



Use partial derivatives to find local extrema (maxima, minima) in a given direction.
Use partial derivatives to find critical points of an implicitly defined surface.
Use the second partial derivative test (Hessian determinant) to analyze the geometry
of an implicitly defined surface in terms of local extrema and saddle points.
Formative Assessment - Online interactive quizzes and tutorials correlated to textbook
(Stewart) chapters and sections. Click here: Interactive Quiz 15.7
Summative Assessment - Students are assessed over the entire unit.
Click here: Chapter 15 Test Questions 16-19 apply to this objective.
Learning Activity
PowerPoint slides addressing the current objective. Click here: Chapter 15 Section 7
Homework
Stewart Chapter 15.7: 5-19 (odd), 29-35 (odd)
Unit 3.8: Partial Derivatives - Lagrange Multipliers
Local Objective


Use Lagrange multipliers to find maxima and minima of a function with respect to a
given constraint.
Apply Lagrange multipliers to solve a variety of interesting problems taken from
geometry, engineering, science, and economics.
Formative Assessment - Online interactive quizzes and tutorials correlated to textbook
(Stewart) chapters and sections. Click here: Interactive Quiz 15.8
Summative Assessment - Students are assessed over the entire unit.
Click here: Chapter 15 Test Questions 19 and 20 apply to this objective.
Learning Activity
PowerPoint slides addressing the current objective. Click here: Chapter 15 Section 8
Homework
Stewart Chapter 15.8: 3-21 (odd)
Unit 3 Test
Unit 4.1: Vector Fields - 2D Mappings and Plots
Local Objective


Define a 2D vector field F from 𝑹𝟐 𝒕𝒐 𝑹𝟐 and plot it as a vector attached to each
point.
Given the plot of a 2D vector field, find or match an appropriate function that
represents the geometry of the field.
Formative Assessment - Online interactive quizzes and tutorials correlated to textbook
(Stewart) chapters and sections.
Click here: Interactive Quiz 17.1 This assessment may be used either as an homework quiz or
as a small group quiz.
Summative Assessment - Students are assessed over the entire unit.
Click here: Vector Fields Visual Representation Quiz Questions 1-6 and 11-14 apply to this
objective.
Learning Activity
PowerPoint slides addressing the current objective. Click here: Chapter 17 Section 1
Homework
Stewart Chapter 17.1: 1-6 (all), 11-14 (all)
Unit 4.2: Vector Fields - 3D Mappings and Plots
Local Objective


Define a 3D vector field F from 𝑹𝟑 𝒕𝒐 𝑹𝟑 and plot it as a vector attached to each
point.
Given the plot of a 3D vector field, find or match an appropriate function that
represents the geometry of the field.
Formative Assessment - Online interactive quizzes and tutorials correlated to textbook
(Stewart) chapters and sections. Click here: Interactive Quiz 17.1 (Intentionally used for both
2D and 3D formative assessment.)
Summative Assessment - Students are assessed over the entire unit.
Click here: Vector Fields Visual Representation Quiz Questions 7-10 and 15-18 apply to this
objective.
Learning Activity
PowerPoint slides addressing the current objective. Click here: Chapter 17 Section 1
Homework
Stewart Chapter 17.1: 7-10 (all), 15-18 (all)
Unit 4.3: Vector Fields - 2D Gradient, Divergence, and Curl
Local Objective




Define a 2D vector field as the gradient of a scalar function f from 𝑹𝟐 𝒕𝒐 𝑹.
Recognize when a 2D vector field F is or is not the gradient of a scalar function f.
Use the gradient operator to define the divergence of a vector field.
Define the curl of a 2D vector field F.
Formative Assessment - Online interactive quizzes and tutorials correlated to textbook
(Stewart) chapters and sections. Click here: Interactive Quiz 17.5 (Restrict 3D exercises to
first two components to get 2D exercises.)
Summative Assessment - Students are assessed over the entire unit.
Click here: Chapter 17 Test Questions 1, 4, 6, 7, and 8 apply to this objective.
Learning Activity
PowerPoint slides addressing the current objective. Click here: Chapter 17 Section 5 (Restrict
3D examples to first two components to get 2D examples.)
Homework
Stewart Chapter 17.1: 21-26 (all), more TBD
Unit 4.4: Vector Fields - 3D Gradient, Divergence, and Curl
Local Objective




Define 3D vector field as the gradient of a scalar function f from F from 𝑹𝟑 𝒕𝒐 𝑹.
Recognize when a 3D vector field F is or is not the gradient of a scalar function f.
Use the gradient operator to define the divergence and curl of a vector field.
Recognize how the curl of a 2D vector field F can be viewed as the curl of a 3D
vector field with the z component function equal to zero.
Formative Assessment - Online interactive quizzes and tutorials correlated to textbook
(Stewart) chapters and sections. Click here: Interactive Quiz 17.5 (Intentionally used for both
2D and 3D formative assessment.)
Summative Assessment - Students are assessed over the entire unit.
Click here: Chapter 17 Test Questions 5, 10, 12, 13, 17, and 20 apply to this objective.
Learning Activity
PowerPoint slides addressing the current objective. Click here: Chapter 17 Section 5
(Intentionally used for both 2D and 3D learning activity.)
Homework
Stewart Chapter 17.5: 1-8 (all), 12, 19-22 (all)
Unit 4.5: Vector Fields - Algebraic Properties of Gradient, Divergence, and Curl
Local Objective




State the basic properties of the gradient, divergence, and curl operators and their
combinations.
Recognize that the curl of a gradient vector field is always zero.
Recognize that the divergence of a curl vector field is always zero.
Define the Laplacian operator using the gradient operator.
Formative Assessment - This quiz is correlated to the corresponding material drawn from a
supplementary textbook (Marsden). Click here: Algebraic Properties of Vector Fields Quiz
Summative Assessment - Students are assessed over the entire unit.
Click here: Vector Fields Test Questions 7-11, 16-17, and 19 apply to this objective.
Learning Activity
The supplementary textbook (Marsden) provides PowerPoint presentations on each unit.
Click here: (Marsden) Chapter 4 Section 3 and here: (Marsden) Chapter 4 Section 4
(Intentionally used for both Algebraic and Geometric learning activities.)
Homework
Stewart Chapter 17.5: 23-32 (all), 39
Unit 4.6: Vector Fields - Geometric Properties of Gradient, Divergence, and Curl
Local Objective


Interpret the divergence of a vector field in terms of the expansion or contraction of
the field geometry.
Interpret the curl of a vector field in terms of the rotation (small paddle wheel)
about an axis at each point.
Formative Assessment - This quiz is correlated to the corresponding material drawn from a
supplementary textbook (Marsden). Click here: Geometric Properties of Vector Fields Quiz.
Summative Assessment - Students are assessed over the entire unit.
Click here: Vector Fields Test Questions 12-13 and 27 apply to this objective.
Learning Activity
The supplementary textbook (Marsden) provides PowerPoint presentations on each unit.
Click here: (Marsden) Chapter 4 Section 3 and here: (Marsden) Chapter 4 Section 4
(Intentionally used for both Algebraic and Geometric learning activities.)
Homework
Marsden TBD
Unit 4.7: Vector Fields - Physical Interpretation of Gradient, Divergence, and Curl
Local Objective




Apply the gradient, divergence, and curl appropriately in physical applications.
Use the properties of the gradient to determine temperature gradients.
Use the properties of the gradient to show that conservative fields, e.g., gravitational
fields, are gradients of scalar functions.
Use the properties of the divergence and curl operators to represent, interpret, and
use Maxwell's Equations for electromagnetic fields.
Learning Activity
The supplementary textbook (Fleisch) provides an excellent online collection of podcasts,
problems, and solutions, which corresponds quite nicely to the current learning activity since it is
designed for the student to gain experience using mathematics (already learned) in physical
applications. At this stage of vector calculus the student is prepared to address only those
application problems involving the differential form of Maxwell's Equations, as indicated below.
Click here: A Student's Guide to Maxwell's Equations, and here: Problems 1.11-1.15, and
here: Problem 2.6, and here: Problems 4.6-4.10.
Homework
Stewart Chapter 17.5: 37, 38, more TBD
Unit 4 Test
Final Exam Review
First Semester Final Exam
Unit 5.1: Multiple Integrals - Double Integrals
Local Objective


Define the double integral of a function f from 𝑹𝟐 𝒕𝒐 𝑹 as the volume over a
rectangular region in the plane.
Compute the double integral as a double Riemann sum over a rectangular region.
Formative Assessment - Online interactive quizzes and tutorials correlated to textbook
(Stewart) chapters and sections. Click here: Interactive Quiz 16.1
Summative Assessment - Students are assessed over the entire unit.
Click here: Chapter 16 Test Questions 1 and 2 apply to this objective.
Learning Activity
PowerPoint slides addressing the current objective. Click here: Chapter 16 Section 1
Homework
Stewart Chapter 16.1: 1-7 (odd)
Unit 5.2: Multiple Integrals - Iterated Integrals
Local Objective



Define an iterated double integral using an iterated Riemann integral.
Use Fubini's theorem to show that iterated double integral (in either order) is
equivalent to the double integral defined over a general region.
Compute double integral over general region using various iterated integrals.
Formative Assessment - Online interactive quizzes and tutorials correlated to textbook
(Stewart) chapters and sections. Click here: Interactive Quiz 16.2
Summative Assessment - Students are assessed over the entire unit.
Click here: Chapter 16 Test Questions 3-5 apply to this objective.
Learning Activity
PowerPoint slides addressing the current objective. Click here: Chapter 16 Section 2
Homework
Stewart Chapter 16.2: 1, 3-27 (multiples of 3)
Unit 5.3: Multiple Integrals - Double Integrals Over a General Region
Local Objective


Extend definition of double integral to the volume over a general region in the
plane.
Compute double integral over general region using a double Riemann sum over a
rectangular region containing the general region.
Formative Assessment - Online interactive quizzes and tutorials correlated to textbook
(Stewart) chapters and sections. Click here: Interactive Quiz 16.3
Summative Assessment - Students are assessed over the entire unit.
Click here: Chapter 16 Test Questions 6, 7, and 9 apply to this objective.
Learning Activity
PowerPoint slides addressing the current objective. Click here: Chapter 16 Section 3
Homework
Stewart Chapter 16.3: 1, 3-27 (multiples of 3)
Unit 5.4: Multiple Integrals - Double Integrals in Polar Coordinates
Local Objective


Define an iterated double integral using an iterated Riemann integral in polar
coordinates.
Use Fubini's theorem to show that iterated double integral (in either order) in polar
coordinates is equivalent to the double integral defined over a general region in

polar coordinates.
Compute double integral over general region using various iterated integrals in
polar coordinates.
Formative Assessment - Online interactive quizzes and tutorials correlated to textbook
(Stewart) chapters and sections. Click here: Interactive Quiz 16.4
Summative Assessment - Students are assessed over the entire unit.
Click here: Chapter 16 Test Questions 8, 10, and 16 apply to this objective.
Learning Activity
PowerPoint slides addressing the current objective. Click here: Chapter 16 Section 4
Homework
Stewart Chapter 16.4: 5-27 (odd)
Unit 5.5: Multiple Integrals - Applications of Double Integrals
Local Objective
Use double integrals to compute total mass, total charge, center of mass, and moment of
inertia.
Formative Assessment - Online interactive quizzes and tutorials correlated to textbook
(Stewart) chapters and sections.
Click here: Interactive Quiz 16.5 and here: Interactive Quiz 16.6 This assessment may be
used either as an homework quiz or as a small group quiz.
Summative Assessment - Students are assessed over the entire unit.
Click here: Chapter 16 Test Questions 11-16 apply to this objective.
Learning Activity
PowerPoint slides addressing the current objective. Click here: Chapter 16 Section 5
Homework
Stewart Chapter 16.5: 1-19 (odd)
Unit 5.1-5.5 Test
Unit 5.6: Multiple Integrals - Triple Integrals
Local Objective







Define the triple integral of a function f from 𝑹𝟑 𝒕𝒐 𝑹 as the hyper-volume over a
rectangular box in F from 𝑹𝟑 (a general region in 𝑹𝟑 contained in a rectangular
box).
Compute the triple integral as a triple Riemann sum over a rectangular box in 𝑹𝟑 (a
general region in 𝑹𝟑 contained in a rectangular box).
Define an iterated triple integral using an iterated triple Riemann integral.
Use Fubini's theorem to show that iterated triple integral (in any order) is
equivalent to the triple integral defined over a general region in 𝑹𝟑 .
Compute triple integral over general region in 𝑹𝟑 using various iterated triple
integrals.
Interpret hyper-volume of triple integral with 𝒇(𝒙, 𝒚, 𝒛) = 𝟏 as volume of
a general region in 𝑹𝟑 .
Use triple integrals to compute center of mass and moments of inertia in 𝑹𝟑 .
Formative Assessment - Online interactive quizzes and tutorials correlated to textbook
(Stewart) chapters and sections. Click here: Interactive Quiz 16.7
Summative Assessment - Students are assessed over the entire unit.
Click here: Chapter 16 Test Questions 1 and 2 apply to this objective.
Learning Activity
PowerPoint slides addressing the current objective. Click here: Chapter 16 Section 6 (Note that
this section number is different from the quiz number due to different editions of the textbook.)
Homework
Stewart Chapter 16.6: 1-23 (odd), 27, 33
Unit 5.7: Multiple Integrals - Triple Integrals in Cylindrical Coordinates
Local Objective


Change back and forth from rectangular coordinates to cylindrical coordinates.
Identify geometrical settings that are natural for cylindrical coordinates.

Formulate and compute triple integrals expressed in cylindrical coordinates.
Formative Assessment - Online interactive quizzes and tutorials correlated to textbook
(Stewart) chapters and sections. Click here: Interactive Quiz 16.8
Summative Assessment - Students are assessed over the entire unit.
Click here: Chapter 13 Test Questions 18 and 19, and here: Chapter 16 Test Question 15, all
apply to this objective. (Note that some of the topics in the textbook have been moved from
Chapter 13 to Chapter 16.)
Learning Activity
PowerPoint slides addressing the current objective. Click here: Chapter 16 Section 7 (Note that
this section number is different from the quiz number due to different editions of the textbook.)
Homework
Stewart Chapter 16.7: 1-21 (odd), 27
Unit 5.8: Multiple Integrals - Triple Integrals in Spherical Coordinates
Local Objective



Change back and forth from rectangular coordinates to spherical coordinates.
Identify geometrical settings that are natural for spherical coordinates.
Formulate and compute triple integrals expressed in spherical coordinates.
Formative Assessment - Online interactive quizzes and tutorials correlated to textbook
(Stewart) chapters and sections. Click here: Interactive Quiz 16.8 (Intentionally used for both
cylindrical and spherical coordinates.)
Summative Assessment - Students are assessed over the entire unit.
Click here: Chapter 13 Test Questions 19-20, and here: Chapter 16 Test Questions 13-15, all
apply to this objective. (Note that some of the topics in the textbook have been moved from
Chapter 13 to Chapter 16.)
Learning Activity
PowerPoint slides addressing the current objective. Click here: Chapter 16 Section 8
Homework
Stewart Chapter 16.8: 1-25 (odd)
Unit 5.9: Multiple Integrals - Change of Variables
Local Objective




Define a change of variable in 𝑹𝟐 (𝑹𝟑 ) as a transformation T from 𝑹𝟐 𝒕𝒐 𝑹𝟐
(𝑹𝟑 𝒕𝒐 𝑹𝟑 ) such that T is a 1-1 continuously differentiable function.
Use the Jacobian matrix determinant corresponding to a change of variable
transformation T to rewrite and compute double and triple integrals.
Interpret the change from rectangular coordinates to polar coordinates in double
integrals as a change of variable using an appropriate Jacobian matrix determinant.
Interpret the change from rectangular coordinates to cylindrical or
spherical coordinates in triple integrals as a change of variable using an appropriate
Jacobian matrix determinant.
Formative Assessment - Online interactive quizzes and tutorials correlated to textbook
(Stewart) chapters and sections. Click here: Interactive Quiz 16.9
Summative Assessment - Students are assessed over the entire unit.
Click here: Chapter 16 Test Questions 11, 16, and 18-20 apply to this objective.
Learning Activity
PowerPoint slides addressing the current objective. Click here: Chapter 16 Section 9
Homework
Stewart Chapter 16.9: 1-6 (all), 7-23 (odd)
Unit 5.6-5.9 Test
Unit 6.1: Vector Calculus - Line Integrals
Local Objective

Define path (line) integral along a space curve in 𝑹𝟐 (𝑹𝟑 ).



Interpret path (line) integral as a generalization of an arc length integral.
Define the work done along a curve in terms of a path (line) integral.
Compute path (line) integrals using various techniques of integration.
Formative Assessment - Online interactive quizzes and tutorials correlated to textbook
(Stewart) chapters and sections. Click here: Interactive Quiz 17.2
Summative Assessment - Students are assessed over the entire unit.
Click here: Chapter 17 Test Questions 1-3 apply to this objective.
Learning Activity
PowerPoint slides addressing the current objective. Click here: Chapter 17 Section 2
Homework
Stewart Chapter 17.2: 1-30 (multiples of 3)
Unit 6.2: Vector Calculus - Fundamental Theorem of Line Integrals
Local Objective




State the Fundamental Theorem of line integrals using the gradient operator and
the dot product.
Interpret the Fundamental Theorem of line integrals as a generalization of the
Fundamental Theorem of Calculus.
Use the Fundamental Theorem of line integrals to compute path (line) integrals of
vector fields that are gradients of scalar fields (conservative vector fields) and
recognize the path independence.
For vector fields that represent physical forces, interpret path integrals as the work
done along the path.
Formative Assessment - Online interactive quizzes and tutorials correlated to textbook
(Stewart) chapters and sections. Click here: Interactive Quiz 17.3
Summative Assessment - Students are assessed over the entire unit.
Click here: Chapter 17 Test Questions 4-8 apply to this objective.
Learning Activity
PowerPoint slides addressing the current objective. Click here: Chapter 17 Section 3
Homework
Stewart Chapter 17.3: 1-21 (odd)
Unit 6.3: Vector Calculus - Green's Theorem
Local Objective




State Green's Theorem relating the path (line) integral around a simple closed curve
to the double integral over the enclosed region.
Interpret Green's Theorem as a generalization of the Fundamental Theorem of
Calculus for double integrals.
Use Green's Theorem to simplify the computation of a difficult path (line) integral
using a double integral.
Use Green's Theorem to simplify the computation of a difficult double integral using
a path (line) integral.
Formative Assessment - Online interactive quizzes and tutorials correlated to textbook
(Stewart) chapters and sections. Click here: Interactive Quiz 17.4
Summative Assessment - Students are assessed over the entire unit.
Click here: Chapter 17 Test Questions 9-11 apply to this objective.
Learning Activity
PowerPoint slides addressing the current objective. Click here: Chapter 17 Section 4
Homework
Stewart Chapter 17.4: 3-19 (odd)
Unit 6.4: Vector Calculus - Second Version of Green's Theorem
Local Objective


Restate Green's Theorem in terms of the curl and divergence operators.
Apply this form of Green's Theorem to flows of (incompressible) vector fields.
Formative Assessment - Online interactive quizzes and tutorials correlated to textbook
(Stewart) chapters and sections. Click here: Interactive Quiz 17.5
Summative Assessment - Students are assessed over the entire unit.
Click here: Chapter 17 Test Questions 12-14 apply to this objective.
Learning Activity
PowerPoint slides addressing the current objective. Click here: Chapter 17 Section 5
Homework
Stewart Chapter 17.5: 2-28 (even-review), 33-35 (all)
Unit 6.1-6.4 Test
Unit 6.5: Vector Calculus - Parametric Surfaces
Local Objective




Write the parametric equations of a surface in 𝑹𝟑 using a smooth mapping from
𝑹𝟐 𝒕𝒐 𝑹𝟑 .
Interpret the parameterization of a surface geometrically as a function that maps a
2D (flat) region of the plane to a (curved) surface in 3D space.
Use technology to visualize parameterized surfaces.
Use double integrals to compute the area of parameterized surfaces.
Formative Assessment - Online interactive quizzes and tutorials correlated to textbook
(Stewart) chapters and sections. Click here: Interactive Quiz 17.6
Summative Assessment - Students are assessed over the entire unit.
Click here: Chapter 17 Test Questions 9 and 14-16 apply to this objective.
Learning Activity
PowerPoint slides addressing the current objective. Click here: Chapter 17 Section 6
Homework
Stewart Chapter 17.6A: 1, 3-27 (multiples of 3)
Stewart Chapter 17.6B: 33-47 (odd)
Unit 6.6: Vector Calculus - Surface Integrals
Local Objective



Define a surface integral for a scalar field f mapping 𝑹𝟑 𝒕𝒐 𝑹, where the surface S is
contained in the domain of f and S is parameterized.
Compute surface integrals using appropriate parameterizations and double
integrals.
Compute the surface integral of a vector field F over a surface S using the normal
component of F with respect to S.
Formative Assessment - Online interactive quizzes and tutorials correlated to textbook
(Stewart) chapters and sections. Click here: Interactive Quiz 17.7
Summative Assessment - Students are assessed over the entire unit.
Click here: Chapter 17 Test Questions 16 and 18 apply to this objective.
Learning Activity
PowerPoint slides addressing the current objective. Click here: Chapter 17 Section 7
Homework
Stewart Chapter 17.7: 3-27 (multiples of 3)
Unit 6.7: Vector Calculus - Stokes Theorem
Local Objective




State Stokes Theorem for a smooth vector field F on 𝑹𝟑 , which relates the path (line)
integral of the tangential component of F around a simple closed boundary curve C
of a surface S to the surface integral of the normal component of the curl of F over
the enclosed surface S.
Interpret Stokes Theorem as a generalization of Green's Theorem.
Use Stokes Theorem to simplify the computation of a difficult path (line) integral for
the vector field F.
Use Stokes Theorem to simplify the computation of a difficult surface integral of the
flux of the vector field F through the surface.

Define the circulation of a vector field F about a closed curve and use Stokes
Theorem to relate it to the magnitude of the normal component of the curl of F.
Formative Assessment - Online interactive quizzes and tutorials correlated to textbook
(Stewart) chapters and sections. Click here: Interactive Quiz 17.8
Summative Assessment - Students are assessed over the entire unit.
Click here: Chapter 17 Test Questions 10 and 16 apply to this objective.
Learning Activity
PowerPoint slides addressing the current objective. Click here: Chapter 17 Section 8
Homework
Stewart Chapter 17.8: 3-27 (multiples of 3)
Unit 6.8: Vector Calculus - Divergence Theorem
Local Objective




State the Divergence Theorem for a smooth vector field F on 𝑹𝟑 , which relates
the surface integral of the normal component of F over the surface S, e.g., the
boundary surface of a region E of 𝑹𝟑 , to the triple integral (volume integral) of the
divergence of F over E.
Interpret the Divergence Theorem as a generalization of Green's Theorem.
Use the Divergence Theorem to simplify the computation of a difficult surface
integral for the vector field F.
Use the Divergence Theorem to simplify the computation of a difficult volume
integral for the vector field F.
Formative Assessment - Online interactive quizzes and tutorials correlated to textbook
(Stewart) chapters and sections. Click here: Interactive Quiz 17.9
Summative Assessment - Students are assessed over the entire unit.
Click here: Chapter 17 Test Questions 17 and 20 apply to this objective.
Learning Activity
PowerPoint slides addressing the current objective. Click here: Chapter 17 Section 9
Homework
Stewart Chapter 17.9: 3-27 (multiples of 3)
Unit 6.9: Vector Calculus - Physical Applications of Path, Surface, and Volume Integrals
Local Objective


Apply path, surface, and volume integrals and their related theorems to problems in
fluid dynamics and electrodynamics.
Use surface and volume integrals and their related theorems to state integral forms
of Maxwell's Equations for electromagnetic fields.
Learning Activity
The supplementary textbook (Marsden) provides PowerPoint presentations on each unit.
Click here: (Marsden) Chapter 8 Section 5
Homework
TBD
Unit 6.5-6.9 Test
Unit 6.10: Vector Calculus - Introduction to Differential Forms
Local Objective



Re-interpret differentials and their products in terms of 1-forms, 2-forms, and 3forms.
Compute exterior products of differential forms.
Restate Stokes Theorem in terms of differential forms.
Learning Activity
The supplementary textbook (Marsden) provides PowerPoint presentations on each unit.
Click here: (Marsden) Chapter 8 Section 6
Homework
TBD
Unit 6.11: Vector Calculus - Introduction to the Gauss-Bonnet Theorem
Local Objective




Define the shape operator and Gaussian curvature of a smooth, orientable patch in
𝑹𝟑 .
Compute the shape operator and Gaussian curvature of basic geometrical shapes in
𝑹𝟑 .
Redefine Gaussian curvature in terms of 2-forms for geometrical (metric) surfaces
in 𝑹𝟑 .
State the Gauss-Bonnet Theorem which relates the total Gaussian Curvature of a
compact, orientable, geometrical (metric) surface to its Euler characteristic with
respect to any rectangular decomposition of the surface.
Learning Activity
The supplementary textbook (Marsden) provides PowerPoint presentations on each unit.
Click here: (Marsden) Chapter 7 Section 7
Homework
TBD
Unit 7.1: Second-Order Differential Equations - Second-Order Linear Equations
Local Objective


Construct the solution of a linear homogeneous differential equation by finding a
basis for the solution space of the corresponding operator polynomial.
Solve linear homogeneous differential equations satisfying various initial and
boundary conditions.
Formative Assessment - Online interactive quizzes and tutorials correlated to textbook
(Stewart) chapters and sections. Click here: Interactive Quiz 18.1
Summative Assessment - Students are assessed over the entire unit.
Click here: Chapter 18 Test Questions 1-2 and 4-9 apply to this objective.
Learning Activity
PowerPoint slides addressing the current objective. Click here: Chapter 18 Section 1
Homework
Stewart Chapter 18.1: 3-30 (multiples of 3)
Unit 7.2: Second-Order Differential Equations - Non-Homogeneous Linear Equations
Local Objective



Use the method of undetermined coefficients to solve linear inhomogeneous
differential equations satisfying various initial and boundary conditions.
Use the method of variation of parameters to solve linear
inhomogeneous differential equations satisfying various initial and boundary
conditions.
Interpret both methods (Method of Undetermined Coefficients and Method of
Variation of Parameters) in terms of finding a basis for the solution space of an
appropriately chosen operator polynomial.
Formative Assessment - Online interactive quizzes and tutorials correlated to textbook
(Stewart) chapters and sections. Click here: Interactive Quiz 18.2
Summative Assessment - Students are assessed over the entire unit.
Click here: Chapter 18 Test Questions 3 and 10-14 apply to this objective.
Learning Activity
PowerPoint slides addressing the current objective. Click here: Chapter 18 Section 2
Homework
Stewart Chapter 18.2: 3-27 (multiples of 3)
Unit 7.3: Second-Order Differential Equations - Physical Applications
Local Objective

Apply the methods used to solve linear differential equation to represent and solve
physical problems involving simple harmonic motion, including those with various
forms of damped vibration.
Formative Assessment - Online interactive quizzes and tutorials correlated to textbook
(Stewart) chapters and sections. Click here: Interactive Quiz 18.3
Summative Assessment - Students are assessed over the entire unit.
Click here: Chapter 18 Test Questions 15-18 apply to this objective.
Learning Activity
PowerPoint slides addressing the current objective. Click here: Chapter 18 Section 3
Homework
Stewart Chapter 18.3: 1-17 (odd)
Unit 7.4: Second-Order Differential Equations - Series Solutions
Local Objective


Use power series methods to solve linear differential equations.
Extend power series methods to solve nonlinear differential equations.
Formative Assessment - Online interactive quizzes and tutorials correlated to textbook
(Stewart) chapters and sections. Click here: Interactive Quiz 18.4
Summative Assessment - Students are assessed over the entire unit.
Click here: Chapter 18 Test Questions 19-20 apply to this objective.
Learning Activity
PowerPoint slides addressing the current objective. Click here: Chapter 18 Section 4
Homework
Stewart Chapter 18.4: 1-11 (odd)
Final Exam Review
Second Semester Final Exam