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line integral


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If a vector function V(x,y,z) is defined between two points A and B on a curve, the curve is approximately given by a series of equal directed chords Δ1l2l,…Δnl. In each segment i of the curve it is possible to define the scalar product Vi·Δil. If the sum of the scalar products∑in= Vi·Δilis considered, the line integral is defined by∫ABV·dl=limn ∑i=1nV·Δil.An example of a line integral is given by the case of a force F acting on a particle in the field of the force. The line integral ∫ABF·dlis the work done on the particle as it moves from A to B because of the force.

If the line integral is taken round a closed path (loop), the line integral is denoted by

CV·dl

or

V·dl.

A line integral for a scalar function φ(x,y,z) is defined in a similar way and is denoted ∫ABφdl. It is also possible to define another type of line integral for a vector function Vby∫ABV×dl..

Subjects: Physics.


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