## Quick Reference

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 Δ_{1}*l*,Δ_{2}*l*,…Δ_{n}*l*. In each segment *i* of the curve it is possible to define the scalar product *V** _{i}*·Δ

_{i}*l*. If the sum of the scalar products∑

*i*

*n*=

**·**

*V**i*

*Δ**i**l*is considered, the line integral is defined by∫AB

*V*·

*d*

*l*=

*l*

*i*

*m*

*n*→

*∞*∑

*i*=1

*n*

*V*·

*Δ*

*i**l*.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 ∫AB

*F*·d

*l*is 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

∫_{C}*V*·d*l*

or

∮*V*·d*l*.

A line integral for a scalar function φ(*x*,*y*,*z*) is defined in a similar way and is denoted ∫AB*φ*d*l*. It is also possible to define another type of line integral for a vector function *V*by∫AB*V*×d*l*..

**From:**
line integral
in
A Dictionary of Physics »

*Subjects:*
Physics.

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