/sci/ - Science & Math

>>9681060
Let [math]f:M\rightarrow N[/math] be a diffeomorphism between [math]n[/math]-dimensional manifolds [math]M,N[/math], and let [math]\nu_N \in \Omega^n(N)[/math] be the volume form on [math]N[/math]. On forms, [math]f[/math] pulls back to an isomorphism [math]f^*: \Omega^k(N) \rightarrow \Omega^k(M), 1\leq k \leq n[/math] such that the induced volume form [math]f^*(\nu_N)[/math] satisfies [eqn]\int_M f^*(\nu_N) = \int_N \nu_N.[/eqn] Now if [math]M[/math] admits a trivial cover, [math]\Omega^n(M)[/math] is spanned by [math]dV = dx^1\dots dx^n[/math] as a 1-dimensional [math]C^\infty(M)[/math]-module, and hence there exists a smooth function [math]\phi_f:M \rightarrow N[/math] such that [math]f^*(\nu_N) = \phi_f dV[/math], which leads to [eqn]\int_N\nu_N = \int_M f^*(\nu_N) = \int_M \phi_f dV.[/eqn] Hence if [math]N \hookrightarrow M[/math] is an embedding, then [math]\int_M \phi_f dV[/math] is the volume of [math]N[/math] in [math]M[/math] with respect to the volume form [math]\nu_N[/math].
>>9681171
Consider the case [math]k = 1, q_1 \equiv t [/math], and let [math]\gamma:\mathbb{R} \rightarrow M[/math] be a curve on [math]M[/math]. Your function [math]V[/math] is then the evaluation of an associated function [math]V:M \rightarrow N[/math] along the curve [math]\gamma[/math]. The vector at [math]\gamma(t) = x\in M[/math] tangent to [math]\gamma[/math] at time [math]t\in\mathbb{R}[/math] is given by the vector field [math]v_\gamma = \frac{d \gamma_\mu}{dt} \partial^\mu \in T_* M[/math] which satisfies [eqn]\frac{dx}{dt} = \frac{d\gamma_\mu}{dt} \partial^\mu x \equiv v_\gamma(x).[/eqn] On vector fields, [math]V[/math] pushes forward to a map [math]V_* :T_* M\rightarrow T_*N[/math] with the property that the tangent vector on [math]N[/math] along the curve [math]V \circ \gamma[/math] is given by the vector field [math]V_*(v_{\gamma})(x) = v_\gamma(V(x)) = \frac{d\gamma_\mu}{dt}\partial^\mu V.[/math]

>>9675001
This is mostly correct, though the gauge group need not be [math]U(1)[/math], which corresponds to the requirement of probability conservation in QM. For a Hilbert space [math]\mathcal{H}[/math], this can be circumvented by descending down to the ray space [math]\mathcal{H}/S^1[/math] where you get Wigner's theorem.
Let [math]\xi:P\rightarrow M[/math] be a [math]G[/math]-principal bundle with local sections [math]s_\alpha : U_\alpha \rightarrow P[/math], and let [math]G[/math] be equipped with a differential algebra [math]\mathcal{A}[/math]. Let [math]\theta \in \mathcal{A}^1[/math] be the [math]G[/math]-equivariant connection, then given a representation [math]\rho:G\rightarrow \operatorname{Hom}(V)[/math], [math]\theta[/math] induces a connection on the associated vector bundle [math]P\times_G V \rightarrow M\times_G V = M_G[/math] where [math]V[/math] is the representation space of [math]G[/math]. On [math]M_G[/math], there is a canonical isomorphism [math]\mathcal{A} \cong \Omega(M,G)\otimes \operatorname{Lie}G[/math] on the affine space of Lie algebra-valued Cartan equivariant forms [math]\Omega(M,G)[/math], which, by Thom, is isomorphic to the de Rham differential complex, and which in turn, given [math]M[/math] is Riemannian metrizable, is isomorphic to the affine space of connections with isomorphism class [math]H_G(M)[/math], the equivariant cohomology of [math]M[/math]. After all that, WLOG [math]\theta \in \Omega^1(M)\otimes \operatorname{Lie}G[/math] can be pulled back to the universal bundle by local sections [math]A_\alpha = s_\alpha^*\theta[/math]. It is in this form that [math]A_\alpha[/math] is called a "gauge field" by physicists.
On the other hand if you just have a principal bundle, then all of the above is automatic if you take [math]G = \operatorname{End}_F(P)[/math], from which [math]A \in \operatorname{Lie}(\operatorname{Map}(M,G))[/math], but this means you have to pick [math]M[/math] carefully to get the symmetries you want.

>>9671169
Think about the cyclotomic field [math]\mathbb{Q}(a_m)[/math] acting on itself where [math]a_m = \exp(i2 \pi \frac{1}{m})[/math]. [math]a^{m} = e[/math] implies [math]a^{m+1} = a[/math], which looks like [math]a[/math] "twists back onto itself". A module is torsion then means that there are non-zero elements that "twists back onto itself", i.e. has torsion.
>>9671183
By Radon-Nykodym theorem, if [math]\nu \in \mathcal{M}[/math] is a measure that's absolutely continuous with respect to another measure [math]\mu[/math] on a Borel set [math]B[/math], then there exists a measurable function [math]f[/math] such that [eqn]\nu(A) = \int_B fd\mu[/eqn]. Since continuous functions are measures, absolutely continuous functions [math]r(t)[/math] such that [math]r(t) - r(t') = \int_{t}^{t'}d\tau r'(\tau)[/math] with a weak derivative [math]r'[/math] are absolutely continuous with respect to the standard measure [math]d\tau[/math], and hence automatically satisfies the Radon-Nikodym theorem with [math]f = r'[/math]. In addition, curves [math]\gamma \subset \mathbb{R}^n[/math] are Borel, hence
[eqn]E = \int_\gamma {\bf F}\cdot d{\bf r} = \int_\gamma {\bf F}\cdot {\bf r}' dt = \int_\gamma {\bf F}\cdot {\bf v} dt.[/eqn]

>>9651551
The Hessian is the curvature tensor of a function in flat space. Its determinant is precisely the curvature.
>>9651563
This is the correct geometric interpretation. The determinant of a Jacobian [math]J_{ij} = \frac{\partial x_i}{\partial y^j}[/math] for a coordinate transition [math]\{x_i\}_i \in U \rightarrow V \ni \{y_i\}_i[/math] between patches of a manifold [math]M \supset U,V[/math], say, is the ratio of the volumes of each of them: [eqn]\operatorname{det}J = \frac{\operatorname{vol}U}{\operatorname{vol}V}[/eqn].
>>9652093
It's correct away from [math]x = 0[/math]. Don't trust computer-generated graphs for something as pathological as [math]\sin(1/x)[/math].