Exercise 1. Going back to Section 16.2, read the subsection "Gradient Fields" (p. 977 - 978). According to this reading, what physical quantity does the potential function f represent in many applications? Do not refer to the general direction in your answer. Note that we're referring to the lowercase f for this exercise.

16.2 Vector Fields and Line Integrals: Work, Circulation, and Flux 977 defined on a region in space. These and other fields are illustrated in Figures 16.7-16.16. To sketch the fields, we picked a representative selection of domain points and drew the vectors attached to them. The arrows are drawn with their tails, not their heads, attached to the points where the vector functions are evaluated. FIGURE 16.9 Vectors in a gravitational field point toward the center of mass that gives the source of the field. FIGURE 16.10 A surface might represent a filter, or a net, or a parachute, in a vector field representing water or wind flow veloc- ity vectors. The arrows show the direction of fluid flow, and their lengths indicate speed. FIGURE 16.11 The field of gradient vectors 7f on a level surface fix. > 2) = c. The function f is constant on the surface, and each vector points in the direction where f is increasing fastest.FIGURE 16.13 A "spin" field of rotat- FIGURE 16.12 The radial field ing unit vectors F - al + yj formed by the position vectors of points in the plane. Notice the convention F = (-yi + xj/(x + y2)1/2 that an arrow is drawn with its tail, not its in the plane. The field is not defined at the head, at the point where I' is evaluated. origin. Gradient Fields The gradient vector of a differentiable scalar-valued function at a point gives the direction of greatest increase of the function. An important type of vector field is formed by all the gradient vectors of the function (see Section 14.5). We define the gradient field of a dif- ferentiable function f(x. y. z) to be the field of gradient vectors FIGURE 16.14 The flow of fluid Of = ax in a long cylindrical pipe. The vectors v = (al - " )k inside the cylinder that At each point (x, y, 2), the gradient field gives a vector pointing in the direction of greatest have their bases in the xy-plane have their increase of f, with magnitude being the value of the directional derivative in that direction. tips on the paraboloid z = a' - p'. The gradient field might represent a force field, or a velocity field that gives the motion of978 Chapter 16 Integrals and Vector Fields 40 AD 50 40 wind speed (miles per hour] Latitude(degree:) 20 28 24 -BU -84 -12 -76 -74 Longitude[cegross FIGURE 16.15 The velocity vectors v(/) FIGURE 16.16 Data from NASA's QuikSCAT satellite were used to create this representation of of a projectile's motion make a vector field windspeed and wind direction in Hurricane Irene approximately six hours before it made landfall in along the trajectory. North Carolina on August 27, 2011. The arrows show wind direction, while speed is indicated by color (rather than length). The maximum wind speeds (over 130 km/hour) occurred over a region too small to be resolved in this illustration. a fluid, or the flow of heat through a medium, depending on the application being consid ered. In many physical applications, f represents a potential energy, and the gradient vec- tor field indicates the corresponding force. In such situations, / is often taken to be nega- tive. so that the force gives the direction of decreasing potential energy.EXAMPLE 1 Suppose that a material is heated, that the resulting temperature 7 at each point (x, y, z) in a region of space is given by 7 = 100 - x - y? - :, and that F(r. y. z) is defined to be the gradient of 7. Find the vector field F. Solution The gradient field F is the field F = VT = -2ri - 2yj - 2zk. At each point in the region, the vector field F gives the direction for which the increase in temperature is greatest. The vectors point toward the origin. where the temperature is greatest. See Figure 16.17. FIGURE 16.17 The vectors in a temper- Line Integrals of Vector Fields ature gradient field point in the direction In Section 16.1 we defined the line integral of a scalar function f(x, y, z) over a path C. We of greatest increase in temperature. In this turn our attention now to the idea of a line integral of a vector field F along the curve C. case they are pointing toward the origin. Such line integrals have important applications in studying fluid flows, work and energy, and electrical or gravitational fields. Assume that the vector field F = M(x. y. z)i + N(x. >. =)j + P(x, y, z)k has continu- ous components, and that the curve C has a smooth parametrization r() = g(0)i + Moj + kok, a = = b. As discussed in Section 16.1, the parametrization r() defines a direction (or orientation) along C which we call the forward direction. At each point along the path C, the tangent vector T = dr/ds = v/ vis a unit vector tangent to the path and pointing in this forward direction. (The vector v = dr/dr is the velocity vector tangent to C at the point, as discussed in Sections 13.1 and 13.3.) The line integral of the vector field is the line integral of the scalar tangential component of F along C. This tan- gential component is given by the dot product F . T = p.dr ds so we are led to the following definition