Line Integrals and Potentials
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Line integrals that are independent of path
Equations are omitted for technical reasons - download the original pdf
In general, the line integral of the vector function [Equation goes here - download the original pdf to see it.] over a curve, C, which is defined to be [Equation goes here - download the original pdf to see it.] depends not only on the end points [Equation goes here - download the original pdf to see it.] but also on the path of integration, C. This means that if [Equation] are two different paths connecting A to B, then we cannot infer that [Equation goes here - download the original pdf to see it.] On the other hand, there are many physical situations where the value of an integral depends only on the endpoints. For example, when climbing a hill, the amount of work done against gravity depends only on the starting and the end point - we do the same amount of work whether we go up the steep or the gentle slope! Thus, a line integral will be defined to be independent of path, if, whatever the endpoints A and B, the line integral is the same for all other paths that start at A and end at B. [Equation goes here - download the original pdf to see it.]
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Contents of
Line Integrals and Potentials
1 Line integrals that are independent of path
2 Curve of integration
3 Line integrals
4 Potentials
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 Work done is equal to gain in kinetic energy
12 Exact differential forms
13 Simply connected domains
14 Exactness and independence of path
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