# 9: Multivariable and Vector Functions

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http://scholarworks.gvsu.edu/books/14/

• 9.1: Functions of Several Variables and Three Dimensional Space
We will see that many of the ideas from single variable calculus translate well to functions of several variables, but we will have to make some adjustments as well. In this chapter we introduce functions of several variables and then discuss some of the tools (vectors and vector-valued functions) that will help us understand and analyze functions of several variables.
• 9.2: Vectors
• 9.3: Dot Product
In this section, we will introduce a means of multiplying vectors.
• 9.4: The Cross Product
In this section, we will meet a final algebraic operation, the cross product, which again conveys important geometric information.
• 9.5: Lines and Planes in Space
• 9.6: Vector-Valued Functions
• 9.7: Derivatives and Integrals of Vector-Valued Functions
A vector-valued function determines a curve in space as the collection of terminal points of the vectors r(t). If the curve is smooth, it is natural to ask whether r(t) has a derivative. In the same way, our experiences with integrals in single-variable calculus prompt us to wonder what the integral of a vector-valued function might be and what it might tell us. We explore both of these questions in detail in this section.
• 9.8: Arc Length and Curvature
Given a space curve, there are two natural geometric questions one might ask: how long is the curve and how much does it bend? In this section, we answer both questions by developing techniques for measuring the length of a space curve as well as its curvature.

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