Line Integrals and Potentials
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Work integral
When a constant force moves through a displacement d, the product is defined to be the work done by the force. [Equation goes here - download the original pdf to see it.] The function representing C [Equation goes here - download the original pdf to see it.] is a parametrization of C. If the parameter is the arc length of C, that is [symbol], where s is the arc length of C, and the vector function, F, is a variable force, then this suggests that the line integral along C is equal to the work done by the variable force F along C. We can show that this definition is consistent, though a little extra effort involving limits is required. We must show that the work done by the variable force F along C is approximated by the work done by F along small chords of C, and that when the size of these chords is decreased, the approximation becomes better, and in the limit, it is equal to the line integral. This can be done, hence The line integral of a variable force F along a path C is equal to the work done by F along C.
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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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