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
threedimensional space, use of vector algebra in deriving these
equations. Understand vector representation of curves(paths) in two
and threedimensional spaces, derivatives and integrals of
vectorfunctions of one variable, their geometric and physical meaning.
3. Understand vectors and vector operations in ndimensional
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 http://www.wmich.edu/math/Faculty Photos/
of parallelograms and parallelopipeds under linear transformation.
4. Understand a concept of vectorfunction of several variables, limit
and continuity. Student should be able to check continuity of functions
at a given point. Understand partial derivatives including higherorder
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
secondorder 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 problemsolving skills.
Maple: Suggested sequence of Math 2720 assignments in Maple
More
Maple projects can be found in A
Maple Approach to Calculus, by John Gresser, Prentice Hall.


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 

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. 