Line Integrals and Potentials
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Proof of the theorem
We will now prove the theorem that if F is a vector field in a domain D., then the line integral [Equation goes here - download the original pdf to see it.] is independent of path, if, and only if [Equation goes here - download the original pdf to see it.] for some scalar field f. The proof divides into two parts, the forward derivation and the reverse derivation. Forward derivation Let F be a vector field in a domain D, and [Equation] for some scalar field f. Let C be any path in D with parametrization [Equation goes here - download the original pdf to see it.] such that [Equation goes here - download the original pdf to see it.] The relation [Equation goes here - download the original pdf to see it.] implies [Equation goes here - download the original pdf to see it.] Also the differential, df, is given by [Equation goes here - download the original pdf to see it.] In component form [Equation goes here - download the original pdf to see it.] Reverse derivation Let [Equation goes here - download the original pdf to see it.] be independent of path in D. Let A be a fixed point in D [Equation goes here - download the original pdf to see it.] Let B be an arbitrary point such that [Equation goes here - download the original pdf to see it.] Let f be the function [Equation goes here - download the original pdf to see it.] Since the line integral is independent of path, the value of the function f depends only on [Equation goes here - download the original pdf to see it.] We now show that [Equation goes here - download the original pdf to see it.] Let [Equation goes here - download the original pdf to see it.] [Equation goes here - download the original pdf to see it.] Here [Equation goes here - download the original pdf to see it.] is any parametrization of the curve from A to B. The value of the integral does not depend on the parameter, but only on the end point [Equation goes here - download the original pdf to see it.] Take the partial derivative with respect to x on both sides [Equation goes here - download the original pdf to see it.] [Equation goes here - download the original pdf to see it.] That is, by the fundamental theorem of calculus [Equation goes here - download the original pdf to see it.] By a similar argument [Equation goes here - download the original pdf to see it.] whence [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 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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