Chapter 17

Chapter 17 - 5E-17(pp 1090-1099 2:27 PM Page 1090 CHAPTER...

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Vector Calculus C H A P T E R 1 7 The collection of wind velocity vectors at any given time during a tornado is an example of a vector field. 5E-17(pp 1090-1099) 1/19/06 2:27 PM Page 1090

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In this chapter we study the calculus of vector fields. (These are functions that assign vectors to points in space.) In par- ticular we define line integrals (which can be used to find the work done by a force field in moving an object along a curve). Then we define surface integrals (which can be used to find the rate of fluid flow across a surface). The connections between these new types of integrals and the single, double, and triple integrals that we have already met are given by the higher-dimensional versions of the Fundamental The- orem of Calculus: Green’s Theorem, Stokes’ Theorem, and the Divergence Theorem. |||| 17.1 Vector Fields The vectors in Figure 1 are air velocity vectors that indicate the wind speed and direction at points 10 m above the surface elevation in the San Francisco Bay area. We see at a glance from the largest arrows in part (a) that the greatest wind speeds at that time occurred as the winds entered the bay across the Golden Gate Bridge. Part (b) shows the very dif- ferent wind pattern at a later date. Associated with every point in the air we can imagine a wind velocity vector. This is an example of a velocity vector field. (a) 12:00 P . M ., June 11, 2002 FIGURE 1 Velocity vector fields showing San Francisco Bay wind patterns (b) 4:00 P . M ., June 30, 2002 1091 5E-17(pp 1090-1099) 1/19/06 2:27 PM Page 1091
Other examples of velocity vector fields are illustrated in Figure 2: ocean currents and flow past an airfoil. Another type of vector field, called a force field, associates a force vector with each point in a region. An example is the gravitational force field that we will look at in Example 4. In general, a vector field is a function whose domain is a set of points in (or ) and whose range is a set of vectors in (or ). Definition Let be a set in (a plane region). A vector field on is a func- tion that assigns to each point in a two-dimensional vector . The best way to picture a vector field is to draw the arrow representing the vector starting at the point . Of course, it’s impossible to do this for all points , but we can gain a reasonable impression of by doing it for a few representative points in as in Figure 3. Since is a two-dimensional vector, we can write it in terms of its com- ponent functions and as follows: or, for short, Notice that and are scalar functions of two variables and are sometimes called scalar fields to distinguish them from vector fields. Definition Let be a subset of . A vector field on is a function that assigns to each point in a three-dimensional vector . A vector field on is pictured in Figure 4. We can express it in terms of its compo- nent functions , , and as As with the vector functions in Section 14.1, we can define continuity of vector fields and show that is continuous if and only if its component functions , , and are continuous.

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