Course Details

1. Explain the concepts of rectangular coordinates, 3-D vectors, and vector-valued functions, calculate 3-D vectors, the vector (cross) and dot products of two vectors and explain their geometric meaning, and find the equation of a plane containing three points.
2. Recall and employ the standard graphs of straight lines, the circle, the parabola, the ellipse, and the hyperbola, in order to solve problems in space involving general cylinders, quadric, and more general surfaces.
3. Explain the concept of real-valued functions of several variables, employ partial differentiation including implicit and the multivariable chain rule to find the gradient vector and the equation of tangent planes.
4. Find and classify the critical points of functions of several variables, and solve constrained maximum-minimum problems by using the Method of Lagrange Multipliers
5. Evaluate double and triple integrals over general regions, including surface integrals, by changing the order of integration or converting to polar, cylindrical, and spherical coordinates.
6. Explain the notion of vector fields, calculate their divergence and curl, determine conservative vector fields, and state the Fundamental theorem of independence of path for conservative vector fields.
7. Explain the concept of line integrals, and evaluate them by employing several methods, including the Fundamental theorem for conservative vector fields.
8. State Green’s, Divergence, and Stoke’s theorems, and choose the most appropriate technique, according to the specific problem, to solve the integrals involved.

Three Dimensional Space, Vectors: Rectangular coordinates & 3-D vectors, the vector (cross) and dot products of two vectors, lines and planes in space, quadric and more general surfaces.

Vector Valued Functions: Vector valued functions, curves and motion in space.

Functions of several variables and optimization: Functions of several variables and the chain rule, directional derivatives and the gradient vector, tangent planes, maximum and minimum values of functions of several variables, the 2^nd derivative test for functions of two variables, Lagrange Multipliers and constrained max-min problems.

Double Integrals: Double integrals over general regions, area and volume by double integration, change of variables in double integrals, double integrals in polar coordinates.

Vector Fields and Line Integrals: The del operator (div, grad, and curl in rectangular coordinates), vector fields, line integrals, the fundamental theorem and independence of path, Green’s theorem.

Triple Integrals: Triple integrals, volume by triple integration, change of variables in triple integrals, triple integrals in cylindrical and spherical coordinates.

Surface area and Surface integrals.

Divergence and Stoke’s theorems.  

• Anton H., Bivens I., and Davis S., Calculus, 7th edition, John Wiley & Sons, 2002.

• Schey H. M., Div, Grad, Curl, And All That: An Informal Text On
• Jerrold E. Marsden, Anthony J. Tromba, Vector Calculus, 4th edition,

The theory is taught, and several examples are solved, on the white board, with the participation of students. Also, students are asked to work on practice problems on their own during class hours, and they are encouraged to ask questions. Many additional exercises as well as their analytic solutions are given to students through the internet to work at home. Students are encouraged to attend office hours for extra help. For the assessment, the students are given two mid-term tests and a final exam.