Math 2720      Vector Calculus                      


Text: Vector Calculus, 2nd edition by T. H. Barr
General: The selected text must be used carefully, since it is easy to over-emphasize some topics at the expense of others.  The syllabus below provides a path giving full coverage to the course topics - as described in the undergraduate catalog:

Catalog Description:Vectors and geometry in two and three dimensions, matrix algebra, determinants, vector differentiation, functions of several variables, partial differentiation, linear transformations, multiple integration, and change of variables.  The computer algebra system Maple will be used to explore some of these topics.

Prerequisites: A passing grade (C or better) in Math 1230 or Math 1710.

Objectives:

1. Understand  cartesian, polar and spherical coordinates. Understand vectors and vector operations, dot and cross products, their geometric and physical meaning, use of vector algebra for solving geometric and physical  problems.
2. Understand  equations of lines and planes in three-dimensional space, use of vector algebra in deriving these equations. Understand vector representation of curves(paths) in two- and three-dimensional spaces, derivatives and integrals of vector-functions of one variable, their geometric and physical meaning.
3. Understand  vectors and vector operations in n-dimensional space, understand matrices and matrix operations,determinants, inverse and transpose matrices. Understand quadratic forms, their matrices and criteria for their strict positiveness. Understand linear transformations and  formulas for areas and volumes of images of parallelograms and parallelopipeds under linear transformation.
4. Understand a concept of vector-function of several variables, limit and continuity. Student should be able to check continuity of functions at a given point. Understand partial derivatives including higher-order ones,  a concept of differentiable function and total derivartives, chain rule. Vector analysis operations: gradients, divergence, curl, their physical meaning. Understand approximation of functions by Taylor polynomials, use them for deriving first- and second-order conditions for minma and maxima. Understand Lagrange multiplier rule for optimization problems with constraints.
5. Understand  line integrals, geometric and physical applications. Understand multiple integrals, their geometric and physical applications, change of variables for multiple integrals. Understand surface integrals. Understand Green, Gauss and Stokes theorems.
6. Understand the possibilities of modern computer algebra systems Maple  for visualization of curves and surfaces, for computation of line, surface and multiple integrals.
7. Improve problem-solving skills.

Maple: Suggested sequence of Math 2720 assignments in Maple

  1. Introduction:  Cover basic Maple syntax and some calculus operations.
  1. Plotting:  Done with curves in 3 dimensions.  Some examples of graphing functions of 2 variables and curves in space.  Include some vector notation in Maple.
  1. Animating linear transformations for geometry.  More graphics work and understanding of linear transformations.
  1. Do unconstrained max and min in two and three dimensions through level sets and Sylvester's Theorem.

 More Maple projects can be found in A Maple Approach to Calculus, by John Gresser, Prentice Hall. 

Schedule: The schedule below allows for covering 33 sections  in 44 units (about 11 weeks), leaving about 2.5 weeks for expanding, testing and extra materials.  You might have time to do some of the following additional topics: Lagrange multipliers (sec 4.5), Surface integrals (sec 5.5-5.6).
Math 2720 Suggested Schedule
Section Unit Contents and Remarks
1.1
 1 Coordinates and distance
1.2
 1 Graphs of functions of two variables
1.3
 2 Quadraic surface
1.4
 2 Cylindrical and spherical coordinates. Maple Lab, Introduction and ploting. Maple assignment 1 and 2.
1.5
 1 Vectors in R3
1.6
 1 Dot product, projection, and work. Don't omit the concept of work, which will be revisited in section 5.2.
1.7
 2 The cross product and determinants
1.8
 1 Planes and lines in R3
1.9
 1 Vector valued functions
1.10
1 Derivatives and motion

Chapter 1 lays foundation for what follows. Don't rush.
2.1
 1 Vectors in Rn
2.2
 2 Matrices. Save time by using a calculator (not Theorem 2.2.2) to find matrix inverses beyond 2 x 2.
2.3
 1 Linear Transform
2.4
1 Geometry of linear transform. Transformations of areas and volumes are important for multiple integration (sections 5.5 - 5.8). Maple assignment 3 will be helpful.

2.5 Quadratic forms in this chapter  may be postponed until just before section 4.3
3.1
 1 Graphs, level sets and vector fields: geometry
3.2
 1 Limits and continuity. This is hard stuff so don't get too deep.  A few good examples will illustrate how continuity in several variables differs from continuity in one variable.
3.3
 1 Open, closed sets and continuity. Go light on the topology, emphasizing closed and bounded sets.  Focus on the extremal problems (based on Theorem 3.3.2).
3.4
1 Partial derivatives
3.5
2 Total derivative. Viewing the derivative as a linear transformation is a change in point of view from Calculus 1.
3.6
 2 The chain rule. Chain rule and implicit differentiation are important but hard for students. It is worth to spend a little more time.

 This chapter is difficult for students. May consider to spend some extra time if needed.
4.1
 1 The gradient and directional derivative
2.5
 2 Quadratic forms
4.3
1 Emphasize 1st and 2nd order Taylor polynomials.
4.4
2 Local extremal. Don't ignore the general case, but focus on n = 2.   Ask students to do maple assignment 4.
4.5
 1 Constrained optimization

Omit 4.2. Section 4.5 is optional.
5.1
 1 Path and arc length
5.2
1 Line integeral
5.3
2 Double integral
5.4
1 Triple integral
5.7
 1.5 Change of variable in double integral. Discuss the general case using the ideas of section 2.4, then focus on polar coordinates.
5.8
 1.5 Change of variable in triple integral. Similar to section 5.7.  Focus on cylindrical and spherical coordinates.

If time allows may consider conver sections 5.5 and 5.6 on surface integral.
6.1
 1 Line intergration. (Optional)
6.2
2 Green's theorem. (Optional)
   
 
    Sections 6.1 and 6.2 let students taste a generalization of fundamental theorem of calc in higher dimension. Recommended if you have time.

Approved by the Department Curriculum Committee 4/08