Tutorial 9 solution

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Unformatted text preview: NATIONAL UNIVERSITY OF SINGAPORE Department of Mathematics MA 1505 Mathematics I Tutorial 9 (Solution Notes) 1. A table of values of a function f is given below. . . . . .......................................................................................... .. .. . . . . .......................................................................................... . . ...... . . . . ... . .... . . . . . .... . . . . . . . . .... . .... . . . . . . . . . . .. . . . . . ... . . .......................................................................... . . . ........................................................................... . .. . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ........................................................................... ........................................................ . . . . . .................. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ......................................................................... . . . . ........................................................................... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ......................................................................... . . . . . ........................................................................... . x y 0 1 2 0 1 3 8 1 6 5 2 2 4 7 9 For example, f (1, 1) = 5. Find C ∇f • dr, where (a) C has parametric equations x = t2 + 1, y = t3 + t, 0 ≤ t ≤ 1. (b) C is the unit circle x2 + y 2 = 1. Preliminaries Recall: Fundamental Theorem of Calculus If f is continuous at every point of [A, B ], then B A f (x) dx = f (B ) − f (A). Remark: 1. This is the second type of Fundamental Theorem of Calculus, and the other type is equivalent to the above one. 2. For multi-variable functions, the gradient ∇f is composed of partial derivatives: ⎞ ⎛ fx fx ⎟ ⎜ , ∇f (x, y, z ) = ⎝ fy ⎠ , · · · ∇f (x, y ) = fy fz Therefore, if we replace f with ∇f in the Fundamental Theorem for multi-variable function f , we can get the general form of Fundamental Theorem: (Of course, here the integral is along lines in vector spaces, so line integrals!) Fundamental Theorem of Line Integral If the integral is independent of the path from A to B , its value is B A ∇f • dr = f (B ) − f (A). 177 Remark: 1. Here the path can be any curve: straight line, curve in a plane, curve in vector space. 2. The single-variable integration is a special case of line integration, that is do the integration along x-axis. 3. The Fundamental theorem says that the value of a line integral is just depending on the limit points of the integral. Such functions (equal to so...
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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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