List of Topics


The following is a list of topics that will be covered in this course along with the chapter and section numbers. The chapter and section numbers are taken from the textbook "Vector Calculus" (4th Edition), by J. Marsden and A. Tromba. There will also be several concepts presented that are not in the book.

CHAPTER 1. THE GEOMETRY OF EUCLIDEAN SPACE

1.1 Vectors in two and three dimensional space
1.2 The inner product, length, and distance
1.3 Matrices, determinants, and cross product
1.4 Cylindrical and spherical coordinates
1.5 n-dimensional Euclidean space

CHAPTER 2. DIFFERENTIATION

2.1 The geometry of real-valued functions
2.2 Limits and continuity
2.3 Differentiation
2.4 Introduction to paths
2.5 Properties of the derivative
2.6 Gradients and directional derivatives

CHAPTER 3. HIGHER-ORDER DERIVATIVES; MAXIMA AND MINIMA

3.1 Iterated partial derivatives
3.2 Taylor's theorem
3.3 Extrema of real-valued functions
3.4 Constrained extrema and Lagrange multipliers
3.5 The implicit function theorem
3.6 Some applications

CHAPTER 4. VECTOR-VALUED FUNCTIONS

4.1 Acceleration and Newton's second Law
4.2 Arc Length
4.3 Vector fields
4.4 Divergence and curl

CHAPTER 5. DOUBLE AND TRIPLE INTEGRALS

5.2 The double integral over a rectangle
5.3 The double integral over more general regions
5.4 Changing the order of integration
5.5 Some technical integration theorems
5.6 The triple integral