Engineering Calculus Notes 219

Engineering - integrals Remark 2.5.4 The path integral of a function over a parametrized curve is unchanged by reparametrization when the

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2.5. INTEGRATION ALONG CURVES 207 we have, for any two partitions P i , i = 1 , 2, L ( P 1 ,f ) ≤ U ( P 2 ,f ) We will say the function f ( −→ p ) is integrable over the arc C if sup P L ( P ,f ) = inf P U ( P ,f ) and in this case the common value is called the path integral or integral with respect to arclength of f along the arc C , denoted i C f d s . As in the case of the usual Riemann integral, we can show that if f is integrable over C then for any sequence P k of partitions of [ a,b ] with mesh( P k ) 0, the Riemann sums using any sample points t j I j converge to the integral: R ( P k ,f, { t j } ) = n s j =1 f ( t j ) s j i C f d s . It is easy to see that the following analogue of Remark 2.5.3 holds for path
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Unformatted text preview: integrals: Remark 2.5.4. The path integral of a function over a parametrized curve is unchanged by reparametrization; when the parametrization −→ p : R → R 3 is regular, we have i C f d s = i b a f ( −→ p ( t )) v v v ˙ −→ p ( t ) v v v dt. As two examples, let us take C to be the parabola y = x 2 between (0 , 0) and (1 , 1), and compute the two integrals i C xd s i C y d s . To compute the Frst integral, we use the standard parametrization in terms of x −→ p ( x ) = ( x,x 2 ) , ≤ x ≤ 1;...
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This note was uploaded on 10/20/2011 for the course MAC 2311 taught by Professor All during the Fall '08 term at University of Florida.

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