Line
Integrals (continued)
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Both dot products discussed
here F(x,
y) · dr(x, y) and
F(x, y¸z) ·
dr(x,
y,z) appear in engineering as
the incremental work of the “force” F
along the path, C. i.e. Suppose F is a force acting on a particle in the x-y
plane and r
is the position vector from the origin to the
particle. Let the particle move an
amount ds along the curve, C, in the x-y plane under
the influence of the
force F . The change in the
position vector along the curve (tangent to C) is dr . Then the
incremental work, dW, done on the particle by the force F is
the dot product of F and dr . Thus the line
integral along C gives the work, W, done by the
force F acting on the particle
as it moves along C. W = ∫ F . dr = ∫ F . (dr/dt) dt and dr = T ds, T is
the unit tangential vector to the curve, C C Click here for further discussion of line
integrals. |
Copyright © 2017 Richard C. Coddington
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