Line Integrals and Potentials
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The line integral does not depend on the parameter
So long as the path C along which the vector function F is integrated remains the same, and the orientation is not altered, then a choice of different parameter does not affect the value of the line integral. Proof Let [Equation goes here - download the original pdf to see it.] be two different parametizations of the same curve C, such that both have the same orientation and their end-points correspond [Equation goes here - download the original pdf to see it.] Then [Equation goes here - download the original pdf to see it.] is a function that maps the parameter s to the parameter t such that [Equation goes here - download the original pdf to see it.] By the chain rule [Equation goes here - download the original pdf to see it.] and [Equation goes here - download the original pdf to see it.] So whilst the line integral does depend on the end-points and the path of integration, it does not depend on the parameter. This is to be expected, if the line integral is to define something of physical significance, the value of that physical property should not depend on the particular choice of parameter. Whilst different parameters may be chosen, two with physical relevance are time (usually, t ) and displacement (usually, s). This result shows that we can "switch" between the two without affecting the physical interpretation of the result. This is important in what follows.
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Contents of
Line Integrals and Potentials
1 Line integrals that are independent of path
2 Curve of integration
3 Potentials
4 Line integrals
5 Proof of the theorem
6 The line integral depends on path
7 Potentials and integration around a closed curve
8 The line integral does not depend on the parameter
9 Conservative scalar fields
10 Work integral
11 Exact differential forms
12 Work done is equal to gain in kinetic energy
13 Simply connected domains
14 Exactness and independence of path
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