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While the reduction process here is straightforward, there are many details
to handle and it is easy to go astray. Standard errors are the usual loss of a
factor between steps, and also failure to differentiate the parametric expression
for What happens when your path P is presented to you as the solution to two equations,
or by a qualitative description? In order to reduce the path integral to a single integral over one variable,
you almost always have to be able to express position on the path (the values
of x, y and z or some other set of coordinates on the path) as functions of that
variable. This usually requires your being able to produce a parametric representation
of the path with your variable as parameter from the given information. When the path consists of pieces that are straight lines or arcs of circles
you should be prepared to represent them parametrically from knowing the beginning
and end points of the straight line and the center radius and end angles of the
arc. When the path is defined by equations and you are able to solve them to find
all variables in terms of one of them or in terms of some other parameter, you
can handle the problem as if the information were given parametrically from the
start. Otherwise, you can always find lots of points on the curve numerically and
use a numerical approximation to the resulting integral, by means we shall soon
describe. Exercises23.1. Integrate the dot product of w with dl over the helical path above with w given by w = (x,y2,z3). 23.2 Integrate this same integral over the straight line path from (1, 0,
0) to 23.3 Integrate the line integral w·dl with w = (y,x,sin (exp(zy)) around the unit circle in the xy plane centered about the point (5, 0, 0). (hint: use Stokes theorem) |
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before taking the dot product with w as well as misinterpreting the question,
switching addition and multiplication, etc. 