I have what's certainly a totally "newbie" question, but it's something I've been wondering about..(adsbygoogle = window.adsbygoogle || []).push({});

Suppose we have a simple boundary value problem from electrostatics. For instance, suppose we have a conducting sphere held at some potential, [tex]\phi = \phi_0[/tex]. Because the sphere is conducting, and b/c we are dealing w/ the static case, take the entire surface of the sphere to be at the same potential ([tex]\phi_0[/tex] in this case).

We now wish to solve Laplace's Equation, [tex]\nabla^2 \phi = 0[/tex]

Note first that the surface of the sphere divides [tex]R^3[/tex] into two "topologically distinct" spaces: first there is the topologically "closed" region, which is the interior of the sphere, and second there is the toplogically "open" region, which is everything external to the sphere.

Note also the solutions to Laplace's equation (which in both cases are trivial) take two distinct forms in each topological region: in the interior (i.e., closed) region, the solution is to take the potential to be a constant, specifically [tex]\phi = \phi_0[/tex], which in the exterior (i.e., open) region, the solution is to take the potential as a 1/r function, specifically [tex]\phi = Q/r[/tex] for a constant Q.

My point in going through this relatively simple example is simply this: here we have the same differential equation (Laplace's equation), subject to the same boundary condition (potential constant on the surface of a sphere), and yet we derive two distinct solutions to the DE depending on whether we are solving the system in a topologically closed region, versus a topologically open region.

Is this the essence of what differential topology, or differential geometry, concern themselves w/?

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# Differential Topology

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