Unformatted text preview: he curve C be given
≤ ≤ . Then
2 (x) = ( ) 1 C =1 and we need to check that this expression doesn’t change if we substitute = ( ).
Assume that = ( ) is a bijective continuously diﬀerentiable function from [α β] to
[. Then
= · = and we need to consider two diﬀerent situations — 3 > 0 and < 0. > 0, then First, if 2 ( ) 1 2 β = ( ) 1 α =1 =1
=1 Similarly, if < 0, then
2 ( 1 ) 2 α ( = 1 β =1 − )
=1 =−1 Both formulae in are the same as the original one in . :) E
Prove that the value of a line integral of a vector ﬁeld does not depend on
the parametrization.
S
Consider a vector ﬁeld in R with components P1
be given by 1 ( )
( ) for ≤ ≤ . Then
P P and let the curve C P = C =1 =1 and we need to check that this expression doesn’...
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This document was uploaded on 02/10/2014.
 Spring '13

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