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LineIntegral - 820 CHAPTER 15 Vector Fields 19 Show that...

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820 CHAPTER 15 Vector Fields 19. Show that the gradient of a function expressed in terms of polar coordinates in the plane is φ( r ,θ) = ∂φ r ˆ r + 1 r ∂θ ˆ θ . (This is a repeat of Exercise 16 in Section 12.7.) 20. Use the result of Exercise 19 to show that a necessary condition for the vector ±eld F ( r = F r ( r ˆ r + F θ ( r ˆ θ (expressed in terms of polar coordinates) to be conservative is that F r r F θ r = F θ . 21. Show that F = r sin 2 θ ˆ r + r cos 2 θ ˆ θ is conservative, and ±nd a potential for it. 22. For what values of the constants α and β is the vector ±eld F = r 2 cos θ ˆ r + α r β sin θ ˆ θ conservative? Find a potential for F if α and β have these values. 15.3 Line Integrals The de±nite integral, R b a f ( x ) dx , represents the total amount of a quantity distributed along the x -axis between a and b in terms of the line density , f ( x ) , of that quantity at point x . The amount of the quantity in an in±nitesimal interval of length at x is f ( x ) , and the integral adds up these in±nitesimal contributions (or elements ) to give the total amount of the quantity. Similarly, the integrals RR D f ( x , y ) dA and RRR R f ( x , y , z ) dV represent the total amounts of quantities distributed over regions D in the plane and R in 3-space in terms of the areal or volume densities of these quantities. It may happen that a quantity is distributed with speci±ed line density along a curve in the plane or in 3-space, or with speci±ed areal density over a surface in 3-space. In such cases we require line integrals or surface integrals to add up the contributing elements and calculate the total quantity. We examine line integrals in this section and the next and surface integrals in Sections 15.5 and 15.6. Let C be a bounded, continuousparametric curve in 3 . Recall (from Section 11.1) that C is a smooth curve if it has a parametrization of the form r = r ( t ) = x ( t ) i + y ( t ) j + z ( t ) k , t in interval I , with “velocity” vector v = d r / dt continuous and nonzero. We will call C a smooth arc if it is a smooth curve with ±nite parameter interval I = [ a , b ]. In Section 11.3 we saw how to calculate the length of C by subdividing it into short arcs using points corresponding to parameter values a = t 0 < t 1 < t 2 < ··· < t n 1 < t n = b , adding up the lengths | 1 r i |=| r i r i 1 | of line segments joining these points, and taking the limit as the maximum distance between adjacent points approached zero. The length was denoted Z C ds and is a special example of a line integral along C having integrand 1. The line integral of a general function f ( x , y , z ) can be de±ned similarly. We choose a point ( x i , y i , z i ) on the i th subarc and form the Riemann sum S n = n ± i = 1 f ( x i , y i , z i ) | 1 r i | .
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SECTION 15.3: Line Integrals 821 If this sum has a limit as max | ± r i |→ 0, independent of the particular choices of the points ( x i , y i , z i ) , then we call this limit the line integral of f along C and denote it ± C f ( x , y , z ) ds .
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