7.138 Bug in VectorCalculus:-ArcLength, Maple 8 (23.1.03)

7.138.1 A. van der Meer

The following session shows a serious bug in the procedure ArcLength.

> with(VectorCalculus): 
Warning, the assigned names <,> and <|> now have a global binding 
 
Warning, these protected names have been redefined and unprotected: *, 
+, ., Vector, diff, int, limit, series 
 
> v := VectorField( <vx(x,y),vy(x,y)>, cartesian[x,y] ); 
                                  _             _ 
                    v := vx(x, y) e  + vy(x, y) e 
                                   x             y 
> k := <X(t),Y(t)>; 
 
                        k := X(t) e  + Y(t) e 
                                   x         y 
 
> LineInt( v, Path( k, t=a..b ) ); 
 
        b 
       / 
      |                  /d      \                  /d      \ 
      |   vx(X(t), Y(t)) |-- X(t)| + vy(X(t), Y(t)) |-- Y(t)| dt 
      |                  \dt     /                  \dt     / 
     / 
       a 
 
> ArcLength( k, t=a..b ); 
 
                    b 
                   / 
                  |   //d      \2   /d      \2\1/2 
                  |   ||-- X(t)|  + |-- Y(t)| |    dt 
                  |   \\dt     /    \dt     / / 
                 / 
                   a
 

Perfect.

But when I calculate the lineintegral again I get:

> LineInt( v, Path( k, t=a..b ) ); 
 
     b 
    /                       / 2      \ 
   |      d        d        |d       | 
   |   vx(-- X(t), -- Y(t)) |--- X(t)| 
   |      dt       dt       |  2     | 
  /                         \dt      / 
    a 
 
                                / 2      \ 
              d        d        |d       | 
         + vy(-- X(t), -- Y(t)) |--- Y(t)| dt 
              dt       dt       |  2     | 
                                \dt      /
 

The call of ArcLength seems to have changed the value of k.

> k; 
 
                     /d      \      /d      \ 
                     |-- X(t)| e  + |-- Y(t)| e 
                     \dt     /  x   \dt     /  y
 

It is corrected with Maple 9. (U. Klein)