FindRoot blocked by Jacobian in multi-valued function

Clear[t, r, z] c = 1; gam = 1.; tm = 16.; r[t_] = 2 c ArcTanh[Tan[t/(2 Sqrt[1 - Cot[gam]^2])]];  z[t_] = r[t] Cot[gam]; plr = PolarPlot[r[t], {t, 0, tm}, GridLines -> Automatic,    PlotStyle -> {Blue, Thick}] rad = Plot[r[t], {t, 0, tm}, GridLines -> Automatic,    PlotStyle -> {Red, Thick}] pp3 = ParametricPlot3D[{r[t] Cos[t], z[t], r[t] Sin[t]}, {t, -tm, tm},    PlotStyle -> {Magenta, Thick}] FindRoot[r[t] = 1.25, {t, 1.2345}] 

The last line FindRoot does not work due to Jacobian singularity. Can there be some work around? Thanks for help.

Tangent space and Jacobian

I’m reading John Willards Topology with a differential view point and an confused about tangent spaces.

To define the notion of derivative $ df_x$ for a smooth map between smooth manifolds we introduce a tanget space at each point $ x$ in the manifold $ M$ . The tangent space is denoted $ TM_x$ . If $ M$ is an $ m$ -dimensional manifold then $ TM_x$ is the $ m$ -dimensional hyperplane through the origin parallel to the hyperplane that that best approximates $ M$ at $ x$ . Similarly one things of the nonhomegeneous linear mapping from the tangent hyperplane at $ x$ to the tangent hyperplane at $ y$ which best approximates $ f$ .

My confusion lies the following sentence: Translating both hyperplanes to the origin, one obtains $ df_x$ .

Is this saying $ df_x$ is a map between these two hyperplanes? If so, how should I think about this map what is getting mapped to what?

Sequence of functions with bounded jacobian determinant

If $ (f_n)_{n \in \mathbb N}$ are smooth maps $ f_n \colon \mathbb R^d \to \mathbb R^d$ with Jacobian matrix $ \nabla f_n$ equi-bounded (say in in uniform norm) then by Arzela-Ascoli we have that, up to subsequences, they converge locally to some continuous function $ f_n \colon \mathbb R^d \to \mathbb R^d$ .

What happens is we weaken the assumption by only requiring that some minors of the matrix $ \nabla f_n$ are uniformly bounded? And what if we only have $ \sup_n |\det \nabla f_n | < \infty$ ?

Instead of using Arzela-Ascoli one could rely on compactness of functions of bounded variation in $ L^1$ , but this would in any case require a $ L^1$ -bound (not $ L^\infty$ ) on the full Jacobi matrix $ \nabla X_n$ .

Problem with Jacobian criterion and regular local ring

Take $ X=\operatorname{Spec}(k[x,y]/(y^2-x^p+t))$ with $ k=\mathbb{F}_p(t)$ and $ p\neq2$ .

The Jacobian ideal is $ J=(y,y^2-x^p+t)=(y,x^p-t)$ which is maximal ie a closed point of $ X$ so Jacobian criterion work to say that $ \mathfrak{p}=(y,x^p-t)$ is singular.

Let’s $ A=k[x,y]/(y^2-x^p+t)$ then in $ A_\mathfrak{p}$ one have $ \mathfrak{p}A_\mathfrak{p}=(y)$ because $ x^p-t=y^2\in(y)$ so $ \mathfrak{p}A_\mathfrak{p}$ has so many generator as $ \dim A_\mathfrak{p}$ (which is 1 because $ \mathfrak{p}$ is maximal in $ A$ so $ \dim A=\dim A_\mathfrak{p}$ ). So $ A_\mathfrak{p}$ is regular ie $ \mathfrak{p}$ is regular.

Where is my stupid mistake?

Jacobian between $TM$ and $M\times M$

Let $ M$ be a closed riemannian manifold and $ \phi: \begin{array}{ccc} TM&\to &M\times M \ (x,v)&\mapsto & (\exp_x(v),\exp_x(-v)) \end{array}$

I need an asymptotic expression for the jacobian of $ \phi^{-1}$ as $ \|v\|\to 0$ , but I’m unable to compute it … I guess the order $ 0$ term is $ 2$ , but even this I’m unable to compute correctly… and I’m also interested in the quadratic term (the one of order $ \|v\|^2$ ), wish I guess depends on curvatures at $ x$ . I guess it’s someway linked to Jacobi field… If that simplify thing, the manifold $ M$ has dimension $ 3$ in my problem.

I’m interested as much in the result itsef as in the way to obtain (and understand) it.

Thanks a lot, have a good day

Determining numerical class of divisors inside Jacobian

Let $ C$ be a smooth projective curve of genus $ g$ , let $ c$ be a point in $ C$ . Let $ (n_1,\dots, n_{g-1})$ be a $ (g-1)$ tuple of nonzero integers.

Consider the image $ f_{(n_i)}\colon C^{g-1}\to \mathrm{Pic}^0(C)$ given by $ $ (x_i)\mapsto \mathcal{O}_C\left(\sum n_i(x_i-c)\right),$ $ the image $ \mathrm{Im}(f_{(n_i)})$ is a divisor in $ \mathrm{Pic}^0(C)$ .

If $ C$ is general, then $ NS(C)$ is generated by theta divisor $ \Theta$ , so we can write numerical classes as a multiple of the theta divisor, $ $ \mathrm{Im}(f_{(n_i)})=d(n_1,\dots,n_{g-1})\Theta.$ $

Is there a reference what are the functions $ d$

(In geometry of algebraic curves vol I, page 223, the case (1,..1,-1,..,-1) was considered, but there seems to be a typo)

Analogue of identity of Jacobian theta function.

Let \begin{align*} \theta(z, \tau) = \sum_{n \in \mathbb{Z}} e^{2\pi i n z + \pi i n^2 \tau }, \end{align*} where $ z \in \mathbb{C}$ , $ \tau \in \mathcal{H}$ , be the Jacobian theta function. Is there some identity for $ \theta(z, \tau)$ which is similar to the following identity: $ \sin (z) = \frac{e^{iz}-e^{-iz}}{2i}$ ? Thank you very much.

Jacobian and area differential

A transformation T (u, v) is said to be a conformal transformation if its Jacobian matrix preserves angles between tangent vectors. Consider that the vector $ \langle 1,0\rangle$ is parallel to the line $ r=\pi$ and that the vector $ \langle 1,1 \rangle$ is parallel to the line $ r=\theta$ . Also, notice that $ r=\pi$ and $ r=\theta$ intersects at $ (r,\theta)=(\pi,\pi)$ at a $ 45^{\circ}$ angle. For $ (r,\theta)$ for polar coordinates, i want to calculate $ v=J(\pi,\pi)\bigg[\begin{matrix}1\0\end{matrix}\bigg]$ and $ w=J(\pi,\pi)\bigg[\begin{matrix}1\1\end{matrix}\bigg]$

Is the angle between v and w a $ 45^{\circ}$ angle? Is the polar coordinates transformation conformal?

I also want to find the jacobian and repeat the previous exercise for the transformation

$ T(\rho,\theta)=\langle e^{\rho}\cos{(\theta)},e^{\rho}\sin{(\theta)}\rangle$

I also want to calculate the area differential of the transformation $ T(\rho,\theta)=\langle e^{\rho}\cos{(\theta)},e^{\rho}\sin{(\theta)}\rangle$ both computationally and geometrically.

Jacobian of the action of a matrix on a Grassmannian

I’m looking for a reference concerning a calculation found in Furstenberg’s 1963 paper “Non-commuting random products”.

Lemma 8.8 of this paper states that if one takes a $ d\times d$ invertible real matrix $ A$ , then the Jacobian of its action on the space $ (d-1)$ -dimensional subspaces at a subspace $ S$ of $ \mathbb{R}^d$ is given by

$ J_A(S) = |det(A)|^{d-1}/|det_S(A)|^d$ ,

where $ det_S(A)$ is the determinant of $ A$ restricted to the subspace $ S$ (and one uses the inner-product from the ambient space to define the $ (d-1)$ -dimensional volume on $ S$ and its image under $ A$ ).

For example if $ d = 2$ and $ S$ is a 1-dimensional subspace of $ \mathbb{R}^2$ genereted by a unit vector $ v$ then

$ J_A(S) = |det(A)|/\|Av\|^2$ .

The proof is left to the reader.

While, I am able to reproduce the result, I’m looking for a reference where these types of calculations are treated systematically.

Does anyone know of such a thing?

Thanks in advance!