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Line Integrals

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Part of the Library of Mathematics book series (LIMA)

Abstract

If ϕ(t) and ψ(t) are continuous functions of t, defined in an interval α ⩽ t ⩽ β, the equations
$$x\; = \;\phi \left( t \right),\;y\; = \;\psi \left( t \right)\;\;\;\;\left( {\alpha \; \le t\; \le \beta } \right)$$
determine a path in the (x,y)-plane. We may think of t as the time and interpret (1) as the motion of a point whose coordinates at time t are (ϕ(t), ψ(t)). For brevity, we often refer to this point as the point t of the path. The initial point and the end point of the path are A = (ϕ(α), ψ(α)) and B = (ϕ(β), ψ(β)) respectively. For a closed path, or loop, we have that
$$\phi \left( \alpha \right)\; = \;\phi \left( \beta \right),\;\psi \left( \alpha \right)\; = \;\psi \left( \beta \right).$$

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Copyright information

© Walter Ledermann 1966

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