SANTA BARBARA CITY COLLEGE

ASSOCIATE DEGREE CREDIT COURSE OUTLINE

 

Department:  Mathematics

Subject Area and Course Number:  Mathematics 200

Course Title:   Multivariable Calculus

Discipline:  Mathematics

Units:  4

Repeatability:  None

Catalog Course Description:  Functions of several variables, multiple integrals and applications, partial differentiation and applications, calculus of vector functions, Green's Theorem, Stokes' Theorem, and Divergence Theorem.

Description for Schedule of Classes:  Functions of several variables, multiple integrals, partial differentiation, calculus of vector functions, Green's Theorem, Stokes' Theorem, and Divergence Theorem.

Lecture Hours per Week:  4.3             (64-72 Total Semester Hours)

Laboratory Hours per Week:  None

Plus Hours:      None

Prerequisites:  Math 160, with a grade of "C" or better.

Co-requisites: None

Skills Advisories:  None

Course Advisories:  None

Limitation on Enrollment:  None

Course Objectives:  By the end of the course, the student will be able to:

1.             Describe the geometry of lines and planes in three-dimensional space.

2.             Use vectors in geometric applications.

3.             Use vector-values functions to describe curvilinear motion in two and three dimensions.

4.             Analyze regions in three dimensions.

5.             Compute extrema using the second partials test and Lagrange multipliers.

6.             Compute double and triple integrals in different coordinate systems.

7.             Compute line and surface integrals.

8.             Apply Green's Stokes', the Divergence Theorem, and the Fundamental Theorem of Calculus for Line Integrals.

9.             Compute and analyze the divergence and curl of a vector field.

 

Course Content and Scope 

1.            Vectors, Lines and Planes

                a.            Vectors in the plane and in space

                b.            The dot product

                c.            The cross product

                d.            Lines in space

                e.            Planes in space

2.            Vector valued functions

                a.            Limits and continuity for vector valued functions

                b.            Derivatives and integrals for vector valued functions

                c.            Arc length for space curves

                d.            Tangent and normal vectors

                e.            Curvature

3.            Partial Derivatives

                a.            Limits and continuity for functions of several variables

                b .           Partial derivatives

                c.            Chain rule for functions of several variables

                d.            Directional derivatives

                e.            Gradient

                f.             Tangent plane and differentials

                g.            Extreme values and Lagrange multipliers

4.            Multiple Integration

                a.            Double integrals

                b.            Surface area

                c.            Triple integrals

                d.            Integrals in polar, cylindrical and spherical co-ordinates

5.            Calculus of Vector Fields

                a.            Line integrals and work; surface integrals and flux

                b.            Green's theorem

                c.            Curl and divergence

                d.            Stokes's theorem

                e.            Divergence theorem

Methods of Instruction:  Lecture, problem solving and the use of mathematical software are the central instructional techniques. Students are expected to work outside of class on reading the text, on assigned exercises, and on computer assignments using mathematical software.

Required Assignments

1.         Appropriate Readings:  Students are required to read assigned chapters in texts. 

2.         Writing Assignments:  Students must work assigned mathematical problems requiring the understanding of abstract ideas.

3.         Appropriate Outside Assignments:  Students will be expected to spend a sufficient amount of time outside of class to practice techniques taught during class time, read assigned materials, and complete frequent homework and computer assignments.

4.         Appropriate Assignments that Demonstrate Critical Thinking:  Students must demonstrate mathematical skills which involve analyzing information, recognizing concepts in new contexts, and analysis of logical arguments. Students will learn to apply their abstract knowledge to solve problems in the sciences and will learn to use computer software in solving problems from multivariable calculus.

Methods of Evaluation:  A student's grade will be based on multiple measures of performance in the solving of problems, designing mathematical models, preparations and analysis of graphs, and analysis of logical arguments. Such measures will typically include three exams and a comprehensive final examination requiring demonstrations of problem solving skills.  In addition, instructors may make use of quizzes, written homework assignments, computer assignments, or other appropriate means to judge a student's dexterity with arithmetic skills and familiarity with mathematical vocabulary and methods of proof.

 

Appropriate Texts and Supplies:

Stewart, Calculus with Early Transcendentals, 6th Ed., Cengage Publishing, 2008

Rogawski, Calculus with Early Transcendentals, 1st Ed., Freeman Publishing, 2008

TI-84 Graphing Calculator, Maple, or equivalent computer algebra system

 

Student Learning Outcomes:

1.                  Parameterize curves and surfaces in space.

2.                  Determine extreme values of functions of several variables.

3.                  Set up and evaluate double and triple integrals that represent areas and volumes.

4.                  Set up and evaluate line and surface integrals that represent work and flux.

5.                  Apply an appropriate Fundamental Theorem of Calculus to evaluate line and surface integrals.

6.                  Solve problems from the sciences using vector calculus.

 

 

 

 

 

 

JK/mej

Approved December 4, 2006

Revised 5/13/09; 8/24/09

FRC (WPC)