Line Integrals and Potentials
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Potentials and integration around a closed curve
Theorem The integrand v is independent of path in a domain D if, and only if, its value around every closed path in D is zero. Proof The proof is in two parts, a forward derivation and a reverse derivation. Forward derivation Let the integrand [Equation goes here - download the original pdf to see it.] be independent of path in a domain D. Let A and B be arbitrary points in the domain D. The integrand [Equation goes here - download the original pdf to see it.] is a closed path in D - a line integral from A to B and then from B back to A. Since the integrand is path independent [Equation goes here - download the original pdf to see it.] Hence [Equation goes here - download the original pdf to see it.] Reverse derivation Let [Equation goes here - download the original pdf to see it.] for all arbitrary closed paths in D. Let A and B be arbitrary points in D, and chose any closed path C that passes through A and B. The points A and B cut C into two paths [Equation]. Then [Equation goes here - download the original pdf to see it.] whence [Equation goes here - download the original pdf to see it.] In other words the line integral from A to B is the same regardless of what curve we take it. Hence, the integrand [Equation goes here - download the original pdf to see it.] is independent of path in a domain D.
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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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