Therefore with the same reason we cannot apply the

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Unformatted text preview: nto a double integral or surface integral. As our experience, to solve a double integral, we need to know the expression of the function, and substitute the parametric equations of the curve C into the function f . Therefore, with the same reason, we cannot apply the two theorem to this problem. Later, we will compare the methods with more details. Therefore, based on the above analysis, we have to use Fundamental Theorem of Line Integral for this question. Use the Fundamental Theorem of Line Integral, we know that: C ∇f • dr = f (terminal point) − f (initial point). 182 The function f is given in the table: ........................................................................................... . . . .. . . .......................................................................................... . . ...... . . . . .... . . . . . .... . . . . . . .... . . . . . .... . . . . .... . . . . . . .... . . . . . . . . .... . . . ................... . ............................................................ . ................ . . . . ................. ....................................... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ........................................................................... . ........................................................ . . . . . . .................. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ...................................... . .......................................................................... . . . . ..................................... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ......................................................................... . . . . ........................................................................... . x y 0 1 2 0 1 3 8 1 6 5 2 2 4 7 9 (a) C has parametric equation r(t) = (t2 + 1)i + (t3 + t)j, 0 ≤ t ≤ 1. So, • t = 0 implies the initial point, which is r(0) = (1, 0). • t = 1 implies the terminal point, which is r(1) = (2, 2). Therefore, by Fundamental Theorem of Line Integral, C ∇f • dr = f (2, 2) − f (1, 0). Check the function values in the table, we know that f (2, 2) = 9, f (1, 0) = 3 =⇒ C ∇f • dr = f (2, 2) − f (1, 0) = 9 − 3 = 6. (b) C is the unit circle x2 + y 2 = 1. Since C is a a closed curve and so the initial point is the same as the terminal point, saying A. Thus, apply the Fundamental Theorem of Line Integral to the conservative field ∇f : C 31 ∇f • dr = f (A) − f (A) = 0. 2 31 Note that: by definition of conservative field, for any function f , the gradient ∇f is always conservative. 183 2. The base of...
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This note was uploaded on 05/05/2009 for the course MA MA1505 taught by Professor Ma1505 during the Spring '09 term at National University of Juridical Sciences.

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