/sci/ - Science & Math

>>12459677
If [math]f = g \circ h:X\rightarrow Y\rightarrow Z[/math] then [math]df[/math] is the pullback of [math]dh[/math] by [math]g[/math]. If you understand the path-velocity definition of tangents, this means that how the path [math]f^*\gamma(x) = \gamma(f(x))[/math] on [math]Z[/math] moves is dictated by how it moves on [math]Y[/math] along the direction of [math]dg_x[/math] for [math]x\in X[/math]. This is why you dot [math]dg[/math] into [math]dh[/math].
>>12464799
"Force" is by definition [math]\dot{p}[/math]; the mathematics underlying the Hamiltonian dynamics of a system (i.e. "forces") is given precisely by symplectic geometry.
>>12465224
This is incorrect. Symplectic manifolds have a canonical local Darboux frame in which [math]\omega = dpdq[/math] from which the momentum coordinate can be identified. This is a basic structure theorem.
>>12464823
The promotion of Poisson [math]\rightarrow[/math] Dirac bracket can be done in this symplectic formulation via geometric prequantization. The single-particle Hilbert space is the space of [math]L^2[/math]-sections of the prequantum line bundle.
>>12465095
Construct the functional [math]S[\gamma] = \int_{p_1}^{p_2}dt \frac{l(R\gamma(t))}{l(\gamma(t))}[/math] then minimize it.
>>12466584
In essence [math][S^n,\operatorname{Hom}(S^1,X)] \cong [S^n\wedge S^1,X][/math]. Consider the homotopy [math]f:S^n\times I \rightarrow \operatorname{Hom}(S^1,X) [/math] as a map [math](S^n \times S^1)\times I \rightarrow X[/math], then you can push [math]f[/math] off at the base point in [math]S^n\times S^1[/math] such that it descends to the quotient [math](S^n\times S^1)\times I / S^n\vee S^1 = S^n\wedge S^2 \times I[/math]. You can compute easily that [math]S^n \wedge S^1 \simeq S^{n+1}[/math].

>>11065133
https://en.wikipedia.org/wiki/Cavalieri%27s_principle
>>11065835
This suffices when the dimension of the vector space is finite.
>>11065909
Integrate term by term?