ch14_09 - P1: OSO/OVY GTBL001-14-nal P2: OSO/OVY QC:...

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1208 CHAPTER 14 . . Vector Calculus 14-94 25. Show that ± C ( f f ) · d r = 0 for any simple closed curve C and differentiable function f . 26. Show that ± C ( f g + g f ) · d r = 0 for any simple closed curve C and differentiable functions f and g . 27. Let F ( x , y ) M ( x , y ) , N ( x , y ) ² be a vector field whose com- ponents M and N have continuous first partial derivatives in all of R 2 . Show that ∇· F = 0i f and only if ² C F · n ds = 0 for all simple closed curves C . (Hint: Use a vector form of Green’s Theorem.) 28. Under the assumptions of exercise 27, show that ² C F · n = 0 for all simple closed curves C if and only if ² C F · n is path-independent. 29. Under the assumptions of exercise 27, show that F = 0 if and only if F has a stream function g ( x , y ) such that M ( x , y ) = g y ( x , y ) and N ( x , y ) =− g x ( x , y ). 30. Combine the results of exercises 27–29 to state a two-variable theorem analogous to Theorem 8.3. 31. If S 1 and S 2 are surfaces that satisfy the hypotheses of Stokes’ Theorem and that share the same boundary curve, under what circumstances can you conclude that ³³ S 1 ( ∇× F ) · n dS = S 2 ( F ) · n dS ? 32. Give an example where the two surface integrals of exercise 31 are not equal. 33. Use Stokes’ Theorem to verify that ´ C ( f g ) · d r = S ( f ×∇ g ) · n dS , where C is the positively oriented boundary of the surface S . 34. Use Stokes’ Theorem to verify that ´ C ( f g + g f ) · d r = 0, where C is the positively oriented boundary of some surface S . EXPLORATORY EXERCISES 1. The circulation of a vector field F around the curve C is de- fined by ² C F · d r . Show that the curl F (0 , 0 , 0) is in the same direction as the normal to the plane in which the circula- tion per unit area around the origin is a maximum as the area around the origin goes to 0. Relate this to the interpretation of the curl given in section 14.5. 2. The Fundamental Theorem of Calculus can be viewed as re- lating the values of the function on the boundary of a region (interval) to the sum of the derivative values of the function within the region. Explain what this statement means and then explain why the same statement can be applied to The- orem 3.2, Green’s Theorem, the Divergence Theorem and Stokes’ Theorem. In each case, carefully state what the “re- gion” is, what its boundary is and what type derivative is involved. 14.9 APPLICATIONS OF VECTOR CALCULUS Through the past eight sections, we have developed a powerful set of tools for analyzing vector quantities. You can now compute flux integrals and line integrals for work and circulation, and you have the Divergence Theorem and Stokes’ Theorem to relate these quantities to one another. To this point in the text, we have emphasized the conceptual and computational aspects of vector analysis. In this section, we present a small selection of applications from fluid mechanics and electricity and magnetism. As you work through the examples in this section, notice that we are using vector calculus to derive general results that can be applied to any specific vector field that you may run across in an application.
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ch14_09 - P1: OSO/OVY GTBL001-14-nal P2: OSO/OVY QC:...

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