--- Bill Dubuque <wgd@zurich.ai.mit.edu> wrote:
I'd be grateful if someone could elaborate on Abhyankar's remarks below, esp. precisely how the mathematical principles are realized physically (alas, my physics knowledge has atrophied)
"On the other hand, Riemann's approach appears to have been existential: know that the formulas exist! Many of his existence arguments were based on "Dirichlet's principle," the genesis of which can be expressed by saying something like this: the real and imaginary parts of an analytic function of a complex variable satisfy the same sort of partial differential equation as a gravitational potential or an electric potential (or a fluid flow or heat conduction ...); so the existence of analytic functions or differentials with prescribed boundary behaviour or assigned singularities can be adduced from physical considerations.
The real and imaginary parts of an analytic function are harmonic functions, i.e. satisfy Laplace's equation diff(f,x,x)+diff(f,y,y)=0. The same equation is satisfied by the static electric potential in a homogeneous medium outside of charges, by the static gravitational potential outside of masses, by the steady-state temperature distribution in a homogeneous medium outside of heat sources, and by certain functions describing laminar fluid flow.
To carry the analogy further (or backwards), once could say that the mathematical formula "number of zeros of a function = number of its poles" corresponds to the physical fact that there is no gravitational pull inside a hollow shell,
Not true; a hollow shell is not a gravity shield. However, a hollow spherical shell of uniform density produces no gravitational force within its interior due to its own mass. Consider a cone of small solid angle dw with vertex in the interior. The two ends of the cone intercept the shell in areas A1 = r1^2 dw/(cos t1), A2 = r2^2 dw/(cos t2), where r is the distance from vertex to sphere, and t is angle the sphere normal makes with r. The two areas produce oppositely directed gravitational forces proportional to A/r^2, and by geometry t1=t2. So the forces cancel.
or that there is no electric intensity inside a hollow charged conductor.
An electrical conductor (under static conditions) is at constant electrical potential, as otherwise there would be a electric field and charges would be moving. In the space inside a hollow conductor the potential satisfies Laplace's equation and is constant on the boundary. Therefore it is constant throughout the whole space. But the elctric field is (minus) the gradient of the potential, so there is no field. Gene __________________________________ Do you Yahoo!? Yahoo! SiteBuilder - Free, easy-to-use web site design software http://sitebuilder.yahoo.com