Line Integrals and Potentials
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Work done is equal to gain in kinetic energy
Equations are omitted for technical reasons - download the original pdf
If F is a variable force, the line integral is [Equation goes here - download the original pdf to see it.] Where W is the work done by F. If t is time, then [Equation goes here - download the original pdf to see it.] So the line integral becomes [Equation goes here - download the original pdf to see it.] Newton's second law states [Equation goes here - download the original pdf to see it.] Substituting this into (*) gives [Equation goes here - download the original pdf to see it.] Now [Equation goes here - download the original pdf to see it., so [Equation goes here - download the original pdf to see it.] Since [Equation goes here - download the original pdf to see it.] is the definition of kinetic energy, the work done along the path C by the variable force F is equal to the gain in kinetic energy.
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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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