Line Integrals

****

The line integral in two dimensions is:

    /
   |
   | P(x, y)dx + Q(x, y)dy
   |
  /  C

where x, y are the coordinates along the integration path C. The integration path itself can be specified by the coordinates of its points in terms of a parameter t:

  x = f(t), y = g(t)

For three dimensions, we have:

    /
   |
   | P(x, y, z)dx + Q(x, y, z)dy + R(x, y, z)dz
   |
  /  C

where x, y, z are the coordinates along the integration path C. The integration path itself can be specified by the coordinates of its points in terms of a parameter t:

  x = f(t), y = g(t), z = h(t)

A first Example:

We consider this integrand:

 integrand: x^2*y*diff(x) + y*z*diff(y) + z*x*diff(z);

                                2
     x z del(z) + y z del(y) + x  y del(x)

and this integration path:

 path: [x = cos(t), y = sin(t), z = sin(t)];

   [x = cos(t), y = sin(t), z = sin(t)]

The coordinates of this path are substituted into the integrand:

 sublis(%, integrand);

We obtain:

       2
    sin (t) del(sin(t))
    + cos(t) sin(t) del(sin(t))

         2
    + cos (t) sin(t) del(cos(t))

Now we have to evaluate the derivatives:

 ev(%, diff);

	   2	   2			   2		    2
      - cos (t) sin (t) del(t) + cos(t) sin (t) del(t) + cos (t) sin(t) del(t)

To integrate this expression, we remove the del(t):

  %, del(t) = 1;

		   2	   2		    2	      2
      	      - cos (t) sin (t) + cos(t) sin (t) + cos (t) sin(t)

Now we can integrate

 integrate(%, t, 0, 2*%pi);

and obtain:

      %pi
    - ---
       4

The computation of the line integral requires four steps:

• Substitution of the path equations into the integrand.
• Evaluation of the derivatives
• Removal of the differential of the parameter
• The computation of a definite integral

To simplify this computation and to reduce the risk of typing errors, it is convenient to put these four steps into a function definition:

lineIntegral(fn, path, param, p0, p1) :=
 block ( [substitutedFn, x, xx],
         substitutedFn: sublis(path, fn),
         x : ev (substitutedFn, diff),
         xx: subst(1, diff(param), x),
         integrate(xx, param, p0, p1) 
       )$

You find this definition in file lineInt.mc.

Here is the above example again:

lineIntegral(x^2*y*diff(x) + y*z*diff(y) + z*x*diff(z),
         [x = cos(t), y = sin(t), z = sin(t)],
          t, 0, 2*%pi);

      %pi
    - ---
       4

An example from Spiegels book "Theory and Problems of the Laplace Transforms":

lineIntegral ((x^2 - y)*diff(x) + (y^2 + x)*diff(y),
                    [x = t, y = t^2 + 1],
                    t, 0, 1);

 	       2