## Syllabus: M 408M

MULTIVARlABLE CALCULUS

**Text: Stewart, Calculus, Early Transcendentals, Eighth Edition**

**Responsible Parties:** Ray Heitmann and Jane Arledge, May 2012

**Prerequisite and degree relevance:** M308L, M408L or M308S, M408S, with a grade of C- or better. M 408D and M 408M cannot both be counted toward a degree.

Calculus is offered in two equivalent sequences: a two-semester sequence, M 408C/D, or either of two three-semester sequences, M 408N/S/M (for College of Natural Science Students) or M 408K/L/M. Completion of one of these sequences is required for mathematics majors, with a C- or better in each course.

For some degrees, M 408N/S or M 408K/L satisfies the calculus requirement . This sequence is also a valid prerequisite for some upper-division mathematics courses, including M 325K, M 427K, M 340L, and M 362K.

**Course description:** M 408M is directed at students in the natural and social sciences and at engineering students. In comparison with M408D, it covers fewer chapters of the text. However, some material is covered in greater depth. This is not a course in the theory of calculus.

The content includes an introduction to the theory and applications of differential and integral calculus of functions of several variables, including parametric equations, polar coordinates, vectors, vector calculus, functions of several variables, partial derivatives, gradients, and multiple integrals.

### Forty Class Days As:

- 10 Parametric Equations and Polar Coordinates (seven days)
- 10.1 Curves Defined by Parametric Equations
- 10.2 Calculus with Parametric Curves
- 10.3 Polar Coordinates
- 10.4 Areas and Lengths in Polar Coordinates
- 10.5 Conic Sections
- 10.6 Conic Sections in Polar Coordinates

- 12 Vectors and the Geometry of Space (eight days)
- 12.1 Three-Dimensional Coordinate Systems
- 12.2 Vectors
- 12.3 The Dot Product
- 12.4 The Cross Product
- 12.5 Equations of Lines and Planes
- 12.6 Cylinders and Quadric Surfaces

- 13 Vector Functions (five days)
- 13.1 Vector Functions and Space Curves
- 13.2 Derivatives and Integrals of Vector Functions
- 13.3 Arc Length and Curvature
- 13.4 Motion in Space: Velocity and Acceleration

- 14 Partial Derivatives (ten days)
- 14.1 Functions of Several Variables
- 14.2 Limits and Continuity
- 14.3 Partial Derivatives
- 14.4 Tangent Planes and Linear Approximations
- 14.5 The Chain Rule
- 14.6 Directional Derivatives and the Gradient Vector
- 14.7 Maximum and Minimum Values
- 14.8 Lagrange Multipliers

- 15 Multiple Integrals (ten days)(first three sections are review)

- 15.1 Double Integrals over Rectangles
- 15.2 Double Integrals over General Regions
- 15.3 Double Integrals in Polar Coordinates
- 15.4 Applications of Double Integrals (optional)
- 15.9 Change of Variables in Multiple Integrals (if time permits)