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Unformatted text preview: Section 17.1 Vector Fields Vector Fields, or Vector Valued Functions In this chapter we consider a new type of integral where instead of integrating a scalar valued function, we consider integrating vector val- ued functions. Though the definitions are very different, we shall see that the new integrals we shall define can be interpreted as regular double and triple integrals and evaluated using generalizations of the Fundamental Theorem of Calculus. 1. Vector Fields We start with a formal definition of a vector field. Definition 1.1. A vector field is a function vector F on R 2 (or R 3 ) which assigns to each point in R 2 (or R 3 ) a two dimensional vector vector F ( x, y ) in 2-space (or a vector vector F ( x, y, z ) in 3-space). Since vectors can be decomposed, we can also decompose vector fields. Specifically, we can write vector F ( x, y ) = P ( x, y ) vector i + Q ( x, y ) vector j for a 2 dimen- sional vector field where P and Q are functions of two variables, and we can write vector F ( x, y, z ) = P ( x, y, z ) vector i + Q ( x, y, z ) vector j + R ( x, y, z ) vector k for a 3 dimensional vector field where P , Q and R are functions of three variables. We call P , Q and R the component functions of vector F ....
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