>
Warning, the name changecoords has been redefined
>
Maple for Line Integrals and Vector Fields
Here's how to use Maple to help visualize and also evaluate the line integral of the vector field
(P, Q) = (x-y, x+y)
over a curve---say the upper half of the unit circle.. (This vector field is illustrated in the book in several places. )
First, let's plot the vector field, using the fieldplot command:
> with(plots):
> x := 'x'; y := 'y'; P := x-y; Q := x+y;
> fieldplot( [P,Q], x=-3..3, y=-3..3, grid=[10,10] );
Now let's integrate this vector field along a curve; say the upper half
of the unit circle. The first step is to parametrize the curve:
> x := cos(t); y := sin(t);
Now we find the dx and dy parts:
> dx := diff(x,t); dy := diff(y,t);
Now we put it all together, integrating Pdx + Qdy over the curve:
> int( P*dx + Q*dy, t = 0 .. Pi );
>
Note that the integral is taken from t=0 to t=Pi because that corresponds
to the UPPER HALF of the unit circle.
Note also that the answer is positive --- could you have predicted
this from the field plot picture?
One can now change any part of what's above to calculate other line
integrals. Be sure to start at the top (where P and Q are defined)... otherwise Maple might get confused about conflicting definitions.