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Results tagged with differential-topology
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user 766
The study of differentiable manifolds and differentiable maps. One fundamental problem is that of classifying manifolds up to diffeomorphism. Differential topology is what Poincaré understood as topology or “analysis situs”.
141
votes
Accepted
Is the boundary $\partial S$ analogous to a derivative?
The surface area $|\partial S|$ of a (bounded, smooth) body $S$ is the derivative of the volume $|S_r|$ of the $r$-neighbourhoods $S_r$ of $S$ at $r=0$:
$$ |\partial S| = \frac{d}{dr} |S_r| |_{r=0}.$ …
- 113
63
votes
What makes four dimensions special?
The Yang-Mills functional $\int_{{\bf R}^{1+d}} F^{\mu \nu} F_{\mu \nu}\ dx dt$ is dimensionless (scale-invariant) if and only if the spacetime dimension is four. (The integrand is a quadratic functi …
- 114k
72
votes
Accepted
Why is cotangent more canonical than tangent?
If you want to differentiate functions from a manifold to (say) the real line R, then you want to use the cotangent bundle on the manifold.
If instead you want to to differentiate functions to the ma …
- 114k