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We seek to evaluate an integral along a path P of the dot product w(x,
y, z)dl. There are in general three ways to define a path; the one most conducive to
evaluating the integral is the parametric representation. Alternatively the path
can be defined qualitatively, (for example: it is a circle centered about the
point (a, b, c) of radius d) or it can be defined by an equation (in two dimensions)
or by two equations (in three dimensions) When the path is defined qualitatively you must provide a parametric representation
yourself. Normally we confine our attention to relatively simple paths, that consist
of parts that are arcs of circles or straight lines or portions of a conic section
(ellipse parabola or hyperbola) and there are standard parametric representations
of these. We will assume here that we start with a parametric representation. This means
that we have a parameter s (which often represents time t or can often be chosen
to be one of the variables x, y or z) and formulae which give x, y and z as functions
of s on P. These can be written as (s),
or as (x(s), y(s), z(s)), as x(s)i + y(s)j + z(s)k, or we
can be given expressions for each of the three components here. An example is
the helix: The curve P will also have a beginning parameter value s0 and an
end one , s1. Now w(x, y, z)dl can be written out as , and we can replace dx by with similar replacements for y and z. The middle expression here is an ordinary integral and is the reduction sought here. So how do we construct it? We can write our integral according to the following procedure: Step 1: Express x y and z in terms of s. That is, write
down the parametric representation of P. |