Line Integrals and Potentials
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Curve of integration
A line integral is an integral of a function along a curve, C, in space. It might be better to call it a curve integral but the term line integral is standard. The curve, C, is called the curve of integration. It is a smooth curve if it can be represented by a vector [Equation goes here - download the original pdf to see it.] such that its derivative [Equation goes here - download the original pdf to see it.] exists, an nowhere is [Equation goes here - download the original pdf to see it.]. This means that at each point the curve, C, has a unique tangent. The line integral will be an integral from an initial point A to a terminal point B such that [Equation goes here - download the original pdf to see it.] The direction if the integration is orientated, meaning that it is the integral from A to B, and the direction of orientation is indicated on a sketch by an arrow pointing from A to B. If the points A and B are identical, then the curve is called a closed curve. A curve may be made up of segments of curves joined together. In this case it is assumed that such a curve is piecewise smooth, meaning that it consists of finitely many smooth curves.
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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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