Tutorial 9 solution

# That means the fee of mrt transportation is path

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Unformatted text preview: wo choices: • Take East West Line directly go there; • Take East West Line to Ourtram Park ﬁrst, and take North East Line, and then change to North South Line to go to Jurong East. That means the fee of MRT transportation is path independent (Conservative). 181 Solution Before doing this question, let us recall the methods to deal with line integrals. (a) Direct Computation We can not use this method for this question. This is because we cannot ﬁnd the expression of f . Actually, we only know the values on 9 discrete points: ........................................................................................... . . . .. . . . .......................................................................................... . . ...... . . . . .... y . . . . ... . . . . .... . . .... . . . .... . . 0 ....... 1 ........ 2 ........ . x ....... . . . . ............................................................ ........................................................................... . . ............... . . . . . . . . . . . . . . . . . . . . . . . 0.........................1........................6.........................4................ . . ........................................................................... .. .. .. . ......... . . . . . . . . . . . . . . . . . . . . . . . . . 1.........................3.......................5........................7............... . . ........................................................................... . .. .. .. . ......... . . . . . . . . . . . . . . . . . . . . . . . . 2........................8.......................2........................9............... . . ........................................................................... . ......... . .. .. .. We don’t know any other property of f , such as linearity, parity; and we don’t know whether f is a polynomial or a trigonometric function or an even more complicated function. So we cannot ﬁnd the expression. Remark: If the question tells that the function is a cubic function for x and y , it is still need much time to solve out the coeﬃcients of each term containing x and y . Use direct computation is not an ideal way for such functions. (b) Fundamental Theorem of Line Integral We can use fundamental theorem for this question. Since the theorem only needs two values of the function, and we do not need to suﬀer ourselves to get the expression of the function on the whole region. But one requirement is that the initial point and the terminal point must be contained in the table. Actually, the following solution is using the Fundamental Theorem of Line Integral. (c) Green’s Theorem & Stokes’ Theorem We have not touch this part in the tutorial, but we know that the formulas in the two theorems are quite complicated. The basic idea of the two theorem is by changing a line integral i...
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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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