|Applications of Exterior Differential Forms|
Phil has investigated Beltrami flows ( a class of 1-forms of Pfaff dimension 3
or more.) Phil also worked out the truth table for the Cartan topology
constructed from the Pfaff Sequence. See
Div Grad and Curl are Dead!
Professor Burke started up a website for differential forms and their applications. Tragically, he recently died. His stuff is still useful reading
In the MAPLE programs that follow below, click on the desired URL to fetch a zip file of the Maple program. Store the file on you hard disk, unzip and run the unzipped program with Waterloo Maple. You can change coefficients and modify the various pieces of the code to suit. The URL above will take you to a list of pdf files that may be of interest.
This Maple program generates the conjugate minimal surface pairs that have degenerate limits as the Catenoid of revolution, or the Right Helicoid. These states act like independent polarization states, with the general minimal surface a linear combination of the two.
The Monkey saddle and its conjugate are computed by assuming a Monge type surface exists as a function of the real part of z^3 or the imaginary part of z^3. Niether of these conjugate surfaces is a minimal surface, but the surface defined by complex function z^3 is holomorphic and therefore generartes a complex Minimal surface in the space of 4 dimensions.
For a pdf file writeup, see
The Monge surface .. by Maple.
Use differential forms to compute E and B fields from potentials. Utilize electomagnetism to gather confindence in forms, then use forms to find some new topological features of electromagnetism.
For implicit surfaces (of multiple components) the 1-form representing the surface need not be integrable. By scaling the normal field by a function which is homogeneous of degree 1 in the components of the original vector, a new vector field can be computed. The Jacobian matrix of this vector field has similarity invariants which may be evaluated. For the special scaling, the product of the eigenvectors (the Jacobian determinant) is always zero. The trace of the Jacobian matrix gives the mean curvature of the resulting surface, while the trace of the adjoint of the Jacobian matrix ( the matrix of cofactors transposed) gives the Gauss curvature.
Sophus Lie proved that a complex holomorphic function defines a two dimensional subspace which is a minimal surface. These surfaces are not represented by a single implicit function in 3D !!! These objects are true artifacts of 4D, and should be recognized as special configurations by physicists and engineers who work in space-time. The engineering employment of such surfaces has been almost zero, yet the concept of harmonic minimal surfaces with zero pressure drops across the surface and admitting shear discontinuities with no dissipation effects is just right for the description of wakes.
These special minimal surfaces are the intersections of two (special) four dimensional implicit functions.
Maple is used to evaluate and plot the Fresnel wave and ray surfaces (Kummer surfaces) that are point sets for which the solution to the Maxwell equations are NOT unique. These point sets can support field discontinuities, and as connected sets will propagate with different phase velocities. Luneberg and Fock define such propagating discontinuities as SIGNALS.
The Maple program allows various constitutive constraints to be placed upon the D, E, B, H fields, and computes as plots the resulting Kummer surfaces.