## M427L Syllabus

ADVANCED CALCULUS FOR APPLICATIONS II

**Prerequisite and degree relevance:** The prerequisite is 408D, 408L, 408S with a grade of at least C-.

**Course description:** Topics include matrices, elements of vector analysis and calculus functions of several variables, including gradient, divergence, and curl of a vector field, multiple integrals and chain rules, length and area, line and surface integrals, Greens theorem in the plane and space. If time permits, topics in complex analysis may be included. This course has three lectures and two problem sessions each week. It is anticipated that most students will be engineering majors. Five sessions a week for one semester.

**Text: Marsden & Tromba, Vector Calculus 5th edition**

- I THE GEOMETRY OF EUCLIDEAN SPACE (6 days)
- 1. I Vectors in two- and three-dimensional space
- 1.2 The inner product, length, and distance
- 1.3 Matrices, determinants, and the cross product
- 1.4 Cylindrical and spherical coordinates
- 1.5 n-dimensional Euclidean space

- 2 DIFFERENTIATION (5-6 days)
- (add discussion of linear maps, matrices) 2.1 The geometry of real-valued functions
- 2.2 Limits and continuity (assign to read)
- 2.3 Differentiation
- 2.4 Introduction to paths
- 2.5 Properties of the derivative
- 2.6 Gradients and directional derivatives

- 3 HIGHER-ORDER DERIVATIVES (3 days)
- 3.1 Iterated partial derivatives (briefly)
- 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(if time permits) 3,5

- 4 VECTOR-VALUED FUNCTIONS (5 days)
- 4.1 Accelerationand Newton's Second Law
- 4.2 Arc length
- 4.3 Vector fields
- 4.4 Divergence and curl

- 5 DOUBLE AND TRIPLE INTEGRALS (3 days)(cover first three sections in one lecture)
- 5.1 Introduction
- 5.2 The double integral over a rectangle
- 5.3 The double integral over more general regions
- 5.4 Changing the order of integration
- 5.6 The triple integral

- 6 THE CHANGE OF VARIABLES FORMULA (3 days)
- 6.1 The geometry of maps (not crucial)
- 6.2 The change of variables theorem (lightly)
- 6.3 Applications of double, triple integrals(if time permits)

- 7 INTEGRALS OVER PATHS AND SURFACES (7 days) (next chapter
- depends heavily on this) 7.1 The path integral
- 7.2 Line integrals
- 7.3 Parametrized surfaces
- 7.4 Area of a surface
- 7.5 Integrals of scalar functions over surfaces
- 7.6 Surface integrals of vector functions

- 8 THEOREMS OF VECTOR ANALYSIS (5-6 days) (may reorder as (8.1, 8.4, 8.2, 8.3)
- 8.1 Green's theorem
- 8.2 Stokes' theorem
- 8.3 Conservative fields
- 8.4 Gauss' theorem

**Responsible party:** Kathy Davis 1998