## 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?