1. Course Description:

Prerequisite: MAC 2312 or satisfactory score on the mathematics placement test. This course is designed to follow MAC 2312. Topics include, vectors in the plane and space, three-dimensional surfaces, various coordinate systems, vector-valued functions, differential calculus of functions of several variables, gradients, directional derivatives, applications of partial derivatives, multiple integration, vector analysis, line integrals, surface integrals and applications. (Credit is not also given for MAC 2254) Four hours weekly.

2. Major Learning Outcomes:

1. The student will be able to apply concepts of the geometric properties and calculus concepts involving surfaces, two- and three-dimensional vectors, vector-valued functions, planes, lines and the cylindrical and spherical coordinate systems.
2. The student will be able to apply the concepts of limits, continuity, differentiability and the chain rule to functions of several variables.
3. The student will be able to apply the theory of the calculus of functions of several variables to applied problems.
4. The student will be able to apply the extension of the concept of the "definite integral" to a two- and three-dimensional setting and acquire an understanding of the theoretical development with respect to Riemann sums.
5. The student will be able to apply concepts of multiple integrals to applied problems.
6. The student will be able to apply the concepts of vector analysis to applied problems.

3. Course Objectives Stated in Performance Terms:

1. The student will be able to apply concepts of the geometric properties and calculus concepts involving surfaces, two- and three-dimensional vectors, vector-valued functions, planes, lines and the cylindrical and spherical coordinate systems.

1. The student will be able to compute the following when given two- or three-dimensional vectors: sum, difference, scalar product, magnitude, dot product, cross product (3-dim. only), vector projection, scalar projection.
2. The student will be able to find the equations of planes and the parametric and symmetric equations of a line when given sufficient information.
3. The student will be able to find the derivatives and integrals of vector-valued functions and apply these to applied problems concerning both the motion of a particle, and tangent and normal vectors.
4. The student will be able to use vector dot products and cross products in order to compute distances between points, skew lines and planes.
5. The student will be able to use cylindrical or spherical coordinates to solve problems dealing with three-dimensional geometry.

2. The student will be able to apply the concepts of limits, continuity, differentiability and the chain rule to functions of several variables.

1. The student will be able to find limits of functions of two or three variables if the limits exist; show that a limit does not exist by using different paths.
2. The student will be able to determine if a given function of two or three variables is continuous or differentiable at a point.
3. The student will be able to determine the partial derivatives of functions of two or more independent variables.
4. The student will be able to determine the partial derivatives of composite functions of two or more variables by using the chain rule.
5. The student will be able to find the first or second partial derivatives of functions of two variables at a given point by using the definition of partial derivative.

3. The student will be able to apply the theory of the calculus of functions of several variables to applied problems.

1. The student will be able to use the differential to demonstrate the differentiability of a function of several variables.
2. The student will be able to compute the directional derivative for a function of two or three variables and find the equations of the tangent plane and normal line to a point on a given surface.
3. The student will be able to compute the gradient and use it in applied problems.
4. The student will be able to compute the possible extreme for a function of two variables.

4. The student will be able to apply the extension of the concept of the "definite integral" to a two- and three-dimensional setting and acquire an understanding of the theoretical development with respect to Riemann sums.

1. The student will be able to approximate the value of an integrable function of two variables by using Riemann sums.
2. The student will be able to compute the values of double and triple integrals by using iterated integrals in rectangular coordinates, polar coordinates, cylindrical coordinates and spherical coordinates.

5. The student will be able to apply concepts of multiple integrals to applied problems.

1. The student will be able to compute areas, volumes, centers of mass and surface areas by using double or triple integrals in rectangular, polar, cylindrical and spherical coordinates.

6. The student will be able to apply the concepts of vector analysis to applied problems.

1. The student will be able to compute the line integral of a given 2 - or - 3 dimensional vector function by:

1. paremetric equations
2. determining if conservative and using the Fundamental Theorem of Line Integrals
3. Green's Theorem

2. The student will be able to compute the curl and divergence of a vector field.
3. The student will be able to evaluate surface integrals including flux integrals both with and without the Divergence Theorem.
4. The student will be able to apply line integrals and surface integrals to applied problems.

4. Criteria Performance Standard:

In order to earn a grade of C or better the student will achieve at the 70 percent level or higher on classroom measures.

Revised 8/84
DBT 11/20/90
Effective Session 19911
3 YR C&I Review 8/94
C&I 3/17/98; DBT 4/20/98
Effective Session 19981
C&I 4/28/98; DBT 5/29/98
Effective Session 19981