Concordia University – Portland

I.            Course Identification:

1. Course Number and Title: Mth 213: Multivariable Calculus
2. Department: Math-Science

II            Goals:  Students successfully completing MTH 213 Multivariable Calculus will    have:

A.   A thorough understanding of an introduction to multivariable calculus including partial derivatives and their applications as well as, multiple integrals and their applications.

1. The ability to apply the techniques of multivariable Calculus when it is encountered in other subject matter.
2. The ability to utilize the appropriate technology when solving problems involving multivariable calculus.

III.             Statement of Objectives

A. The student who successfully completes this course will, in the area of

     attitudes,

     1. appreciate the role Multivariable calculus  plays in

the deeper studies of science and engineering.

     2. appreciate the usefulness and power of current technologies in the study of

mathematics.

3. gain confidence in their ability to correctly execute mathematical processes

in various situations.

             B. The student who successfully completes this course will, in the area of

     knowledge,

1. Understand three dimensional coordinate systems including Cartesian, Cylindrical and Spherical
2. Understand partial derivatives and directional derivatives in the geometric and non-geometric contexts
3. Understands second partial derivatives and their applications including optimization.
4. Understand the benefits to changing variables and coordinate systems.
5. Understand multiple integration and its uses.
6. understand the chain rule for functions with several variables

C. The student who successfully completes this course will, in the area of

     Skills will be able to

1. graph functions of more than one variable both by hand and using technology

2. perform vector arithmetic including the cross product and dot product.

3. graph equations using cylindrical and spherical coordinate systems and apply this to problem solving.

4.  be able to calculate partial derivatives and apply them to problem solving.

5.  apply the chain rule when differentiating a function of more than one variable.

6. Find the extrema of multivariable functions subject to constraints.

7.   compute double integrals and apply them to problem solving.

8.   compute triple integrals and apply them to problem solving.

IV.            Course Outline

A.            Unit 1:            Vectors and the Geometry of Space                        15 Sessions

i.  Space Coordinates and Vectors in Space

ii. Dot product of two vectors

iii. Cross product of two vectors in space

iv. Lines and planes in Space

v. Surfaces in Space

vi. Cylindrical and spherical coordinates

B.            Unit 2:            Functions of several variables                        22 Sessions

i.  Limits and Continuity of functions with multiple variables

ii. Partial Derivatives

iii. Differentials of functions with multiple variables

iv Chain rules for functions of several variables

v. Directional derivatives and gradients

vi. Tangent planes and normal lines

vii Extrema of Functions of two variables with applications

vi Lagrange multipliers

C.            Unit 3:            Multiple Integration                                                       15 Sessions

i. Iterated Integrals

ii. Double Integrals and applications

iii. Change of Variables: Jacobians

v. Surface Area

vi Triple Integrals and Applications

V.            Learning Materials

A.            Required resources to be purchased by the student.

• Calculus by Larson, Hostetler and Edwards 9^th Edition. Published by Houghton Mifflin. ISBN 0-618-14918-X

• Graphing Calculator

B.        Optional resources to be purchased by the student

• None required

C.            Outline or syllabus material to be provided to the student.

• Will be provided at class time

D.            Other learning materials required or optional.

• None required

E.            Proposed use of community resources and resource persons, field trips or

other unique instructional components.

• None required

VI.                 Evaluation Procedures

Homework (including quizzes):                                    20%

First midterm                                                                        25%

2^nd Midterm Exam:                                                            25%

Final Exam:                                                                        30%

VII.      Course Management

1. Recommended number of meetings per week.

This course will meet 4 times per week in 60 minute periods.

B.   Credit hours assigned (Semester Credits):

The completion of this course will result in 3 credit hours of credit.

C.   Outside student preparation expected in clock hours per week.

The student is expected to spend 6-8 hours per week outside of class.

D.    Place of course in the curriculum. MTH 213, Multi Variable Calculus is an elective course to be taken after Math 212 Calculus II

I.            Course Identification:

1. Course Number and Title: Mth 342: Linear Algebra with Differential Equations
2. Department: Math-Science

II            Goals:  Students successfully completing MTH 342 Linear Algebra will have:

A.     a thorough understanding of the principals of matrix operations and linear

transformations with applications to various fields of study.

            B.      the ability to utilize matrix processes when they are encountered in the

context of different courses.

C.      the ability to utilize the appropriate technology when analyzing matrix

problems and linear transformations in various situations.

D        a thorough understanding of eigenvalues, eigenvectors and their applications in difference equations as well as               differential equations.

III.             Statement of Objectives

A. The student who successfully completes this course will, in the area of

attitudes,

     1. appreciate the role that matrix operations and linear transformations play in

the deeper studies of science, social science and business 

     2. appreciate the usefulness and power of current technologies in the study of

mathematics.

3. gain confidence in their ability to correctly execute mathematical processes

in various situations.

B. The student who successfully completes this course will, in the area of

knowledge,

1.  understand matrices as linear transformations.

2.  understand linear transformations in terms of image and kernel.

3.  understand independence of basis elements.

4. understand orthogonality of basis elements.

5.  understand invertibility of  a matrix in terms of a null kernel

6.  understand the relationship between Gauss-Jordan elimination and the

solutions to linear systems.

7.  understand orthogonal projections in matrix and geometric contexts.

8.  understand least squares curve fitting procedures.

9.  understand the non-commutative nature of matrix multiplication.

10. understand the notion of zero divisors in the context of matrices.

11. understand algebraic and geometric interpretations of eigenvalues and eigenvectors

12. understand the role that eigenvalues and eigenvectors play in solving difference and differential equations

13 understand the role of complex numbers in solving differential equations.

C. The student who successfully completes this course will, in the area of

skills,

1.  be able to multiply matrices

2.  be able to find inverses of matrices or determine they don’t exist.

3.  be able to perform Gauss-Jordan elimination.

4.  be able to find solutions to linear systems.

5.  be able to find orthogonal projections.

6.  be able to find the image and kernel of a linear transformation.

7.  be able to transform one basis to another

8.  be able to create an othonormal basis from the Gram-Schmidt

orthoganilazation process.

9. be able to fit data to a regression line

10   be able to calculate eigenvalues and eigenvectors for a matrix

11   be able to solve first order homogeneous linear systems of differential equations

12   be able to create and interpret phase diagrams for dynamic systems

13   be able to use complex numbers to solve differential equations

14   be able to use Markov matrices to model systems

IV.            Course Outline

A.            Unit 1:            Linear Equations                                    5 Sessions

i.  Introduction to Linear Systems

ii. Gauss-Jordan Elimination

iii. Solutions to Linear Systems

B.            Unit 2:            Linear Transformations                        5 Sessions

i.  Linear Transformations and Their Inverses

ii. Geometric Interpretations

iii. Matrix Products

C.            Unit 3:            Subspaces of Rⁿ                                                       5 Session

i. Image and Kernel of a Linear Transformation

ii. Bases and Linear Independence

iii. Coordinates

D.            Unit 4:            Linear Spaces                                                5 Sessions

i. Introduction

ii. Isomorphism

iii. Coordinates in a Linear Space

E.            Unit 5: Orthogonality and Least Squares            6 Sessions

i.  Orthonormal Bases

ii. Orthogonal Projections

iii. Orthogonal Transformations

iv.  Data Fitting

Unit 6 Determinants                                                            4 Sessions

1. Introduction to determinant
2.  Properties of the Determinant
3. Geometrical Interpretations of the Determinant – Cramer’s Rule

Unit 7 Eigenvalues and Eigengvectors                        15 sessions

1. Finding eigenvalues of a matrix
2. Finding eigenvectors of a matrix
3. Digitalization of  a matrix
4. Complex eigenvalues
5. Markov Matrices
6. Linear Differential equations
7. Phase diagrams
8. An Introduction to Continuous Dynamical Systems
9. Stability

V.            Learning Materials

A.            Required resources to be purchased by the student.

• Bretscher, Otto (2001) Linear Algebra with Applications 4th Ed.   Prentice Hall Publishers, ISB

• Graphing Calculator

VI.                 Evaluation Procedures

Homework:                                    20%

First Midterm Exam:                    20%

Second Midterm Exam                20%

Term Paper                                    10%

Final Exam:                                    30%

VII.      Course Management

1. Recommended number of meetings per week.

This course will meet four times per week in 60 minute periods.

B.   Credit hours assigned  (Semester Credits):

The completion of this course will result in 3 credit hours of credit.

C.   Outside student preparation expected in clock hours per week.

The student is expected to spend 8-10 hours per week outside of class

D.    Place of course in the curriculum.

MTH 342, Linear Algebra is and elective course for secondary math students that could be taken in place of MTH 341

This course should be the first 300 level math course a student takes. It

should follow the completion on MTH 212.