17.3 Prel 1-3, Ex 1-4

17.3 Prel 1-3, Ex 1-4 - 1094 C H A P T E R 17 L I N E A N D...

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1094 CHAPTER 17 LINE AND SURFACE INTEGRALS (ET CHAPTER 16) = Z t 1 t 0 ( y ( t ) x 0 ( t ) + x ( t ) y 0 ( t ) ) dt = Z t 1 t 0 d ( x ( t ) y ( t )) The last equality follows from the Product Rule for differentiation. We now use the Fundamental Theorem of Calculus to obtain: Z c F · d s = x ( t ) y ( t ) ¯ ¯ ¯ ¯ t 1 t = t 0 = x ( t 1 ) y ( t 1 ) x ( t 0 ) y ( t 0 ) = cd ab 17.3 Conservative Vector Fields (ET Section 16.3) Preliminary Questions 1. The following statement is false. If F is a gradient vector Feld, then the line integral of F along every curve is zero. Which single word must be added to make it true? SOLUTION The missing word is “closed” (curve). The line integral of a gradient vector ±eld along every closed curve is zero. 2. Which of the following statements are true for all vector ±elds, which are true only for conservative vector ±elds? (a) The line integral along a path from P to Q does not depend on which path is chosen. (b) The line integral over an oriented curve C does not depend on how the C is parametrized. (c) The line integral around a closed curve is zero. (d) The line integral changes sign if the orientation is reversed. (e) The line integral is equal to the difference of a potential function at the two endpoints.
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17.3 Prel 1-3, Ex 1-4 - 1094 C H A P T E R 17 L I N E A N D...

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