Summary
These lecture notes for Calculus III (Math 251) at Simon Fraser University cover multivariate calculus topics. The booklet is designed for students to use during lectures, filling in blanks, and includes interactive applets. Key subjects discussed are integral theorems, the geometry of space including cylinders and quadric surfaces, and detailed properties of the dot product. Definitions and theorems are clearly distinguished using colored boxes.
* **Course & Authorship:** Calculus III (Multivariate Calculus) lecture notes for Math 251 at Simon Fraser University, authored by Veselin Jungic & Jamie Mulholland [S1, S3].
* **Intended Use:** Students are expected to use the booklet during lectures to follow along and fill in details [S1].
* **Document Features:** Definitions are in orange boxes, theorems in blue boxes [S1].
* **Interactive Applets:** Links to online interactive applets are provided, accessible via http://www.sfu.ca/~jtmulh [S1].
* **Integral Theorems:** Covers the Fundamental Theorem for Line Integrals, Green’s Theorem, Curl and Divergence, and a summary of integral theorems [S1].
* **Cylinders:** Defined as a surface with rulings parallel to a given line and passing through a given plane curve [S2].
* **Quadric Surfaces:** Graphs of second-degree equations in three variables (x, y, z), often simplified to standard forms [S2].
* **Dot Product Properties:** Includes ⃗a ·⃗a = |⃗a|^2, commutative (⃗a ·⃗b = ⃗b ·⃗a), and distributive properties [S4].
* **Geometric Interpretation of Dot Product:** ⃗a ·⃗b = |⃗a||⃗b|cos θ, where θ is the angle between vectors [S4].
* **Perpendicular Vectors:** Two nonzero vectors ⃗a and ⃗b are perpendicular if and only if their dot product ⃗a ·⃗b = 0 [S4].
* **Direction Angles:** Angles a nonzero vector ⃗a makes with the positive x-, y-, and z-axes [S4].
* **Vector Functions:** The content also introduces vector functions, space curves, and their derivatives and integrals [S3].
* **License:** Granted to print this document for personal/educational use [S3].