Applied Calculus III
Functions of Multiple Variables

Volume: 15 weeks × 4 lecture periods/week

Textbook: The first part of this course gives some necessary background in linear algebra. There are several books that might be helpful:

• J. Hefferon, Linear Algebra can be obtained here.

• Jänich, K., Linear Algebra, Springer, Link to publisher's description

For the remainder of the course, focusing on multidimensional calculus, we use the second half of

• James Stewart, Calculus, 5th Edition, Brooks-Cole Publishers (International Edition),
  Link to publisher's description

Prerequisites: Vv156.

Last Taught: Summer Term 2010

Background and Goals: The sequence Applied Calculus Vv156-255-256 is an introduction to basic calculus. It differs from the Honors Math sequence in that new concepts are often introduced and extended from concrete examples, remaining closely aligned to applications. Most theorems are stated rigorously and motivated from examples, but complicated proofs and abstract generalizations are often omitted. The emphasis is on applying mathematical results to concrete problems.

The present course includes an overview of linear algebra followed by the calculus of functions in multidimensional euclidean space.

Key Words: Linear systems of equations and the Gauss-Jordan algorithm; finite-dimensional vector spaces (with an emphasis on euclidean space), linear independence and bases; scalar products and Gram-Schmidt orthonormalization; linear maps and matrices; determinants; analytic geometry of lines and planes; parametric representation of curves and surfaces; partial derivatives and applications; line, surface and volume integrals; vector fields the classical theorems of vector analysis in three dimensions (Green, Gauss and Stokes) and applications.

Detailed Content: The first part functions as a general introduction to linear algebra. Many of the components of a standard linear algebra course are touched upon, such as the Gauß-Jordan algorithm for solving systems of equations, the theory of finite dimensional vector spaces, linear maps, matrices and determinants. These elements will serve as an essential toolbox for the following calculus in ${ℝ}^n$.

The second part of the course deals with scalar (potential) functions of multiple variables, also including elements of analytic geometry. In particular, partial and directional derivatives, extrema and integration of scalar functions will be discussed, as well as curves and surfaces in ${ℝ}^2$ and ${ℝ}^3$.

In the third part, we introduce vector-vaued functionfs of multiple variables, focussing on vector fields. Topics include potential and conservative fields, line (work) and surface (flux) integrals of vector fields, classical vector differential operators (divergence, rotation, gradient) and the fundamental laws of integration (Green, Gauß, Stokes).

Alternatives: Vv285 (Honors Mathematics III) is a more theoretical course, which covers much of the same material.

Subsequent Courses: Vv256 (Applied Calculus IV) is the natural sequel.

Overview:

Week	Lecture Subject	Date
1	Systems of Linear Equations Finite-Dimensional Vector Spaces	11-5-2010 13-5-2010
2	Euclidean and Unitary Vector Spaces Matrices and Linear Maps	18-5-2010 20-5-2010
3	Matrices and Linear Maps	25-5-2010 27-5-2010
4	Determinants	1-6-2010 3-6-2010
5	Elements of Analytic Geometry Curves in ${ℝ}^n$	8-6-2010 10-6-2010
6	Curves in ${ℝ}^n$ First Midterm Exam	13-6-2010 17-6-2010
7	Scalar Line Integrals and Functions of Several Variables Partial Derivatives and Applications	22-6-2010 24-6-2010
8	Integration over Flat Domains Scalar Line and Surface Integrals	29-6-2010 1-7-2010
9	Extrema of Real Functions Extrema with Constraints	6-7-2010 8-7-2010
10	Vector Fields and Potential Functions Second Midterm Exam	13-7-2010 15-7-2010
11	Surfaces and Surface Integrals of Potential Functions	20-7-2010 22-7-2010
12	Extrema of Potential Functions with and without Constraints	27-7-2010 29-7-2010
13	Vector Fields	3-8-2010 5-8-2010
14	Theorems of Green, Stokes and Gauß	10-8-2010 12-8-2010
15	Final Exam	17-8-2010 19-8-2010

Assignments (2010 version; PDF files):

• Assignment 1

• Assignment 2

• Assignment 3

• Assignment 4

• Assignment 5

• Assignment 6

• Assignment 7

• Assignment 8

• Assignment 9

• Assignment 10

• Assignment 11

• Assignment 12

(There will always be minor modifications from one iteration of the course to the next; if you are presently taking the course, your assignments may differ from these.)