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Math 254

COURSE DESCRIPTION: (for catalog)

This course is a study of multivariable and vector-valued functions including parametric curves in space, motion, surfaces, lines, planes, gradients, directional derivatives, and multiple integrals. The TI-89 calculator is required. A required computer laboratory component is included.

PREREQUISITE:

MTH 253 with a C or better

INSTRUCTIONAL MATERIALS REQUIRED OF STUDENT: (text, supplies, etc.)

LIST THE PERFORMANCE OBJECTIVES OR STUDENT LEARNING COMPETENCIES AS DESCRIBED IN THE INSTRUCTIONAL LEARNING SYSTEM FOR THIS COURSE:

1. COURSE CONTENT
1. Linear Algebra
1. Vector Space
2. Matrices
3. Determinants
4. Vector Dot Products
5. Eigenvalues
6. Vector Cross Products
7. Vector Function Differentiation and Integration
2. Gradients and Directional Derivatives
1. Define gradient of multivariable function, pointwise evaluation
2. Define directional derivative, pointwise evaluation
3. Extreme values of directional derivative at a point
3. Maximum and Minimum of Multivariable Functions
1. Method of Lagrange multipliers
4. Space Geometry
1. Space curves represented parametrically and in vector form
2. Smooth and piecewise smooth space curves
3. Parametric form of the derivative, motion along a space curve
4. Definite integrals in parametric form
5. Application of parametric integrals to arc length and surface area
6. Quadric surfaces and surfaces of revolution
7. Normal vectors to smooth surfaces
8. Tangent planes to smooth surfaces and their angle of inclination
2. PERFORMANCE OBJECTIVES
Upon completion of the course, the student should be able to:
1. Linear Algebra
1. Determine if a mathematical system is a vector space.
2. Use the algebra of vectors to solve mathematical problems.
3. Determine a basis for a vector space.
4. Apply the algebra of matrices to solve matrix equations.
5. Represent a mapping from one vector space to another vector space in matrix form. (optional)
6. Determine if a mapping is a linear map. (optional)
7. Evaluate an n x n determinant.
8. Solve a system of linear equations using the method of Gaussian elimination.
9. Find the rank of a matrix.
10. Determine the existence and uniqueness of a system of linear equations.
11. Find the eigen values, eigen vectors and eigen space for certain matrices.
12. Use a similarity transformation to diagonalize a square matrix. (optional)
13. Find the dot product of vectors in a plane or in space.
14. Find the cross product of vectors in a plane or in space.
2. Gradients and Directional Derivatives
Upon completion of the course, the student should be able to:
1. Define the gradient of a multivariable function using the "del" notation.
2. Evaluate the gradient of certain multivariable functions at specified points.
3. Define the directional derivative of a function of two or three independent variables and evaluate it point wise.
4. Find the maximum and minimum values of certain directional derivatives at specified points.
5. Interpret directional derivative values as slopes of tangent lines.
6. Interpret certain gradients as direction vectors normal to tangent planes.
3. Maximum and Minimum Values of Multivariable Functions
Upon completion of the course, the student should be able to:
1. Define critical point for a multivariable function.
2. Find the relative extreme points of certain multivariable functions.
3. Identify relative maxima, relative minima and saddle points, where possible, for a function of two independent variables using the second partials test.
4. Find the relative extreme points for certain functions of two or three independent variables using the method of Lagrange multipliers.
4. Space Geometry
   Upon completion of the course, the student should be able to:
   1. Represent space curves parametrically or in vector form.
   2. Define a smooth or piecewise smooth space curve.
   3. Determine if certain space curves are smooth or piecewise smooth.
   4. Construct and evaluate the parametric form of a derivative to describe motion along certain space curves.
   5. Construct and evaluate certain definite integrals in parametric form.
   6. Use definite integrals in parametric form to measure arc length of piecewise smooth space curves and the area of certain surfaces.
   7. Identify certain quadric surfaces by their equations or graphs.
   8. Identify certain surfaces of revolution by their equations or graphs.
   9. Sketch certain quadric surfaces given their equations.
   10. Sketch certain surfaces of revolution given their equations.
   11. Find normal vectors to certain smooth surfaces at specified points.
   12. Find equations for and angles of inclination of planes tangent to certain smooth surfaces at specified points.

GENERAL INSTRUCTIONAL METHODS:

Because of the prerequisite of a year long sequence in single variable calculus, the student population in MTH 254 is usually mathematically skilled and highly motivated. The students should be expected to commit themselves to a higher than average level of performance.

Lectures and class discussions should include in-depth discussions of the theory (including formal proofs) as well as both drill and application type problems. Student writing skills should be developed enough to expect a more formal style of write-up for problem assignments. This requirement anticipates the students' continued study in university courses and eventual professional employment.

Problems in MTH 254 tend to be more involved but students seem to learn concepts without extensive repetition. Thus written assignments will involve fewer problems. These assignments should be commented upon regarding proper format and writing style as well as correctness of solution.

Students should be encouraged to relate the content of this course to that in the fields of science and engineering whenever possible. Discuss varieties of notation possible and use them consistently. Students should be challenged with difficult problems requiring creativity in the solution process.

Discussions of eigenvalues and eigenvectors will require use of supplementary materials since most calculus texts do not include such topics. A good source for these topics would be a compatible linear algebra text written by the same author as the calculus text being used.

Computer software such as DERIVE can be used to review some topics concerning multivariable functions, for example, graphs of two variable functions, partial derivatives and multiple integrals. This prepares the student to use the computer software to determine extreme values of a two variable function. Students should be given about one week to complete each of about five assignments that require calculus level computer software.

EVALUATION PROCESS:

Students should be expected to perform mostly at the A and B grade level. C grades denote weakness and performance below this level should constitute failure.

Problems, both theoretical and applied in nature, should be included on exams. These problems tend to be rather involved suggesting the use of exams written outside of class. Encouraging students to work together in small groups suggests a team grade for every student in a particular group. Some evaluation on an individual basis is important, especially to identify weaker students.

Consult a sample course syllabus for more specific guidelines on course content, pace, and grading policy.

Exams should be constructed for completion both inside and outside the classroom. For exams completed outside of class use of computers should be allowed if not required.