€¦ · web viewcalculus iii: 2012 –13. ladue horton watkins high school. instructor: dr. john...

Calculus III: 2012 –13Ladue 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, 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 - 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% D 63 - 66%

B+ 87 - 89% C 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.

http://drpcourses.blogspot.com/

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 [email protected] .

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.

mailto:[email protected]

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


Homework


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.


Learning Activity


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


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.


Learning Activity


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


Homework


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.


Learning Activity


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


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.


Learning Activity


Homework


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.


Learning Activity


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


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


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.


Summative Assessment - Students are assessed over the entire unit. Click here: Chapter 15 Test Questions 1 and 3 apply to this objective.

Learning Activity


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.


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


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.


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


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 ¿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.



Learning Activity


Homework


Unit 3.5: Partial Derivatives - The Chain Rule

Local Objective

Use various forms of the chain rule for a function from Rn ¿ Rm. Use the chain rule to compute (partial) implicit derivatives.



Learning Activity


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 ¿R and R3 ¿ R. Use the directional derivative to find the direction and maximum rate of change

of functions from R2 ¿R and R3 ¿ R. Use the gradient operator to define and compute the directional derivative.



Learning Activity


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.



Learning Activity


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.



Learning Activity


Homework


Unit 3 Test

Unit 4.1: Vector Fields - 2D Mappings and Plots

Local Objective

Define a 2D vector field F from R2 ¿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


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 ¿ 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


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 ¿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 ¿ 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 ¿R as the volume over a rectangular region in the plane.

Compute the double integral as a double Riemann sum over a rectangular region.



Learning Activity


Homework


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.



Learning Activity


Homework


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.



Learning Activity


Homework


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.



Learning Activity


Homework


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.


Learning Activity


Homework


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 ¿ 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 R3contained 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 R3using 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.



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.


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


Homework


Unit 5.9: Multiple Integrals - Change of Variables

Local Objective

Define a change of variable inR2(R3) as a transformation T from R2 ¿R2 (R3 ¿ 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.


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


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.



Learning Activity


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.


Summative Assessment - Students are assessed over the entire unit.

Click here: Chapter 17 Test Questions 4-8 apply to this objective.

Learning Activity


Homework


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.



Learning Activity


Homework


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.



Learning Activity


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 ¿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.


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


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 ¿ 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.



Learning Activity


Homework


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.



Learning Activity


Homework


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.



Learning Activity


Homework


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 3-forms.

Compute exterior products of differential forms.

Restate Stokes Theorem in terms of differential forms.

Learning Activity


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


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.


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


Homework


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.


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


Homework


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.



Learning Activity


Homework


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.



Learning Activity


Homework


Final Exam Review

Second Semester Final Exam

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