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 emphasiz
e problem solving techniques, examine the necessity of
mathematics as it relates to career goals, enable students to communicate mathematically, and
illustrate the connec
tion 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 trip
le 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 enr
olling 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 ar
e 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 throug
h tests, quizzes, warm

ups, 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

1
00% B 83

86% C

70

72% F Below 60%
A 93

96% B

80

82% D+ 67

69%
A

90

92% C+ 77

79% D 63

66%
B+ 87

89% C 73

76%
D

60

62%
Homework:
In order to receive credi
t 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
attempt
ed. 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 (li
nked 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
in
formation will be posted there.
Attendance/Ta
rdies:
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 com
pleted 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 w
ill not be allowed credit for any work due or
assigned on t
he 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
(T
extbook

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 3
D coordinate systems.
Perform algebraic oper
ations and transformations in 3
D 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
PowerP
oint
slides address
ing
the current objective
.
Click here:
Chapter 13 Section 1
Homewor
k
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
PowerP
oint slides address
ing
the current objective.
Click here:
Chapter
13 Section 2
Homework
Stewart Chapter 13.2
:
13

23 (odd), 31, 35, 4
1
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 an
d tu
torials correlated to textbook
(Stewart)
c
hapters 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
PowerP
oint slides address
ing
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 Produc
t
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
PowerP
oint slides address
ing
the current objective.
Click here:
Chapter 13 Section 4
Homework
Stewart Chapter 13.4: 5, 7
,
11,
19, 2
9
, 3
3, 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
PowerP
oint slides address
ing
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

Studen
ts are assessed over the entire unit.
Click here:
Chapter 13 Test
Questions 11

15 and 17 apply to this objective.
Learning Activity
PowerP
oint slides address
ing
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
PowerP
oint slides address
ing
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
PowerP
oint slides address
ing
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
PowerP
oint slides addres
s
ing
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
PowerP
oint slides address
ing
the current objective.
Click here:
Chapter 14 Section 4
Homework
Stewart Chapter 14.4: 3

15 (odd)
, 33

37 (odd), 41
Unit 2
.6
: Spac
e Curves

Arc Length Reparamete
rization
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 textb
ook (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 en
tire 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
PowerP
oint slides address
ing
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
PowerP
oint slides address
ing
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
m

dim 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
PowerP
oint slides address
ing
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
to define the tangent
plane of an
implicit surface in
.
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
PowerP
oint slides address
ing
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 va
rious forms of the chain rule for
a function fro
m
.
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
PowerP
oint slides address
ing
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
and
.
Use
the directional derivative to find the direction and maximum r
ate of change
of
functions
from
and
.
Use the gradient operator to define and compute the directional derivative.
Formative Assessment

Online interactive quizzes and t
u
torials correlated to textbook
(Stewart)
c
hapters 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
PowerP
oint slides address
ing
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
(Stew
art)
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
PowerP
oint slides address
ing
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

Studen
ts are assessed over the entire unit.
Click here:
Chapter 15 Test
Questions 19 and 20 apply to this objective.
Learning Activity
PowerP
oint slides address
ing
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
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 Fiel
ds Visual Representation Quiz
Questions 1

6 and 11

14 apply to this
objective.
Learning Activity
PowerP
oint slides address
ing
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
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
PowerP
oint slides address
ing
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 gradi
ent of a scalar function f from
.
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 vect
or 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 Te
st
Questions 1, 4, 6, 7, and 8 apply to this objective.
Learning Activity
PowerP
oint slides address
ing
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
.
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 section
s.
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
PowerP
oint sli
des address
ing
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 corres
ponding material drawn from a
supplementary textbook (Marsden).
Click here:
Algebraic Properties of Vector Fields Quiz
Summative Assessment

Students ar
e 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 te
xtbook (Marsden) provides PowerP
oint 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 te
xtbook (Marsden) provides PowerP
oint 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 prope
rties 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 add
ress only those
application problems involving the differ
ential form of Maxwell's Equati
ons, 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
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
PowerP
oint slides address
ing
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
PowerP
oint slides address
ing 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
PowerP
oint slides address
ing
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 int
egral 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
PowerP
oint slides address
ing
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
u
sed 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
PowerP
oint slides address
ing
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
as the hyper

volume over a
rectangular
box in
F from
(a general region in
contained in a rectangular
box).
Compute
the triple
integral as a
triple Riemann sum over a rectangular
box in
(a
general region in
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
.
Compute
triple integral over general region in
using various iterated triple
integrals.
Interpret hyper

v
olume of triple integral with
(
)
as volume
of
a
general
region
in
.
Use triple integrals to compute center of mass and moments of inertia in
.
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
PowerP
oint slides address
ing
the current objective.
Click here:
Chapter 16 Section 6
(Note that
this
section number is different fro
m 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
PowerP
oint slides address
ing
the current objective.
Click here:
Chapter 16 Section 7
(Note that
this section number is diffe
rent fro
m 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

Onlin
e interactive quizzes and tu
torials correlated to textbook
(Stewart)
c
hapters 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
Q
uestions 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
PowerP
oint slides address
ing
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
(
) as a transformation T from
(
) such that T is a 1

1 continuously differ
enti
able 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 J
acobian 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 q
uizzes 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
PowerP
oint slides
address
ing
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
(
)
.
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
PowerP
oint slides address
ing
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 Theor
e
m of line integrals using the gradient operator and
the dot product.
Interpret
the Fundamental Theor
e
m of line integrals as a generalization of the
Fundamental Theorem of Calculus.
Use the Fundamental Theor
e
m of line integrals to compute path (line) integrals of
vector fields that are gradients of scalar fields (conservativ
e 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
(Stew
art)
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
PowerP
oint slides address
ing
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 di
fficult 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
PowerP
oint slides address
ing
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
PowerP
oint slides address
ing
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
using a smooth mapping from
.
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
PowerP
oint slides address
ing
the current objective.
Click here:
Chapter 17 Section 6
Homework
Stewart Chapter 17.6
A
:
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
,
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 tu
torials correlated to textbook
(Stewart)
c
hapters 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
PowerP
oint slides address
ing
the current objective.
Click here:
Chapter 17 Section 7
Homework
Stewart Chapter 17.7: 3

27 (multiples of 3)
Unit 6
.7
: Vect
or Calculus

Stokes
Theorem
Local
Objective
State Stoke
s
Theorem for a smooth vector field
F on
,
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 ge
neralization 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 abo
ut a closed curve and use Stoke
s
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

Studen
ts are assessed over the entire unit.
Click here:
Chapter 17 Test
Questions 10 and 16 apply to this objective.
Learning Activity
PowerP
oint slides address
ing
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
,
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
, 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
PowerP
oint slides address
ing
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 te
xtbook (Marsden) provi
des PowerP
oint 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 3

forms.
Compute exterior products of differential forms.
Restate Stokes
Theorem in terms of differential forms.
Learning
Activity
The supplementary te
xtbook (Marsden) provides PowerP
oint 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
.
Compute the shape operator and Gaussian curvature of basic geometrical
shapes in
.
Redefine Gaussian curvature in terms of 2

forms for geometrical (metric) surfaces
in
.
State the Gauss

Bonnet Theorem which relates the total Gaussian Curvature of a
compact, orientable, geometrical (metric) surface to its Euler characteristic wi
th
respect to any rectangular decomposition of the surface.
Learning Activity
The supplementary te
xtbook (Marsden) provides PowerP
oint 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 Assess
ment

Online interactive quizzes and tutorials correlated to textbook
(Stewart)
chapters and sections.
Click here:
Interactive Quiz 18.1
Summative Assessment

St
udents are assessed over the entire unit.
Click here:
Chapter 18 Test
Questions 1

2 and 4

9 apply to this objective.
Learning Activity
PowerP
oint slides address
ing
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
variatio
n 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
PowerP
oint slides address
ing
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.
Clic
k here:
Chapter 18 Test
Questions 15

18 apply to this objective.
Learning Activity
PowerP
oint slides address
ing
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 her
e:
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
PowerP
oint slides address
ing
the current objective.
Click here:
Chapter 18 Section 4
Homework
Stewart Chapter 18.4: 1

11 (odd)
Final Exam Review
Second
Semester Final Exam
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