1
vote
0answers
38 views

What is the explanation of the equation 1-4.6 in the book “Applied Exterior Calculus”?

If the $n$ function $\{f^i(x^m)\}$ are of class $C^{\infty}$ in some neighborhood of $0$, then system of autonomous ODEs $$\frac{d\bar{x}^i(t)}{dt}=f^i(\bar{x}^m(t)),\quad\quad i = 1,\cdots, n$$ has a ...
1
vote
2answers
43 views

Solving a certain differential equation when assuming a surface of revolution is minimal

The problem is the following: Consider the surface of revolution $$ \textbf{q} (t, \mu) = (r(t)\cos(\mu),r(t)\sin(\mu),t) $$ If $\textbf{q}$ is minimal, then $r(t) = a\cosh(t)+b\sinh(t)$ for $a,b$ ...
3
votes
1answer
47 views

Recovering a frame field from its connection forms

I have a faced a research problem where I would need to recover a frame field given its connection forms. More precisely, I begin with an orthonormal frame field (given by data) in $\Re^3$ written as ...
2
votes
0answers
19 views

non-smooth minmal surfaces and differenteial equations

This is the equation for a function $u(x,y)$ whose graph is a minimal surface (its mean curvature is $0$): $$(1+u_x^2)u_{yy}-2u_xu_yu_{xy}+(1+u_y^2)u_{xx}=0$$ My question is if there are non-smooth ...
0
votes
2answers
68 views

Can we solve backward heat equation?

I am reading a paper. In the proof, there is the following claim: For any $t_0>0$, let $h$ be any positive function. We solve the backward heat equation starting from $t_0$ with initial data ...
5
votes
1answer
114 views

Rank of a jet bundle of a vector bundle.

I am trying to understand the jet bundles but currently I am stuck on the following questions: Let $\pi: E\rightarrow X$ be a smooth (holomorphic) vector bundle of rank $k$ over a smooth (complex) ...
3
votes
1answer
77 views

Conditions on a $1$-form in $\mathbb{R}^3$ for there to exist a function such that the form is closed.

What are the conditions on a $1$-form in $\mathbb{R}^3$ for there to exist a function such that the form is closed? More precisely, given a point, $p$, what are conditions on the coefficients of a ...
2
votes
2answers
57 views

Solving a differential equation $\displaystyle \frac{d \alpha}{dt}=w \times\alpha$

Let $\alpha$ be a regular curve in $\mathbb{R}^3$ such that $\displaystyle \frac{d \alpha}{dt}=w \times\alpha$ for $w$ a constant vector. How can we determine $\alpha$ ? $\displaystyle w ...
0
votes
0answers
37 views

Existence and uniqueness on this semi-linear parabolic PDE

I want to know whether the existence and uniqueness of a classical solution can be found about this question: Find a classical solution $u : [0,T]\times [0,\infty] \rightarrow {\mathbb R}$, such ...
0
votes
0answers
29 views

what do the higher differentials act graphically?

I wanna know what do actually the second order and higher order differentials mean. what are their geometrical interpretation, ya of course we solve them , use them but how we show them graphically, ...
3
votes
0answers
52 views

Integral curves in neighborhood of a minimum point of the potential

Let $X: \mathbb R^3 \rightarrow \mathbb R^3$ be a smooth vector field, and $f: \mathbb R^3 \rightarrow \mathbb R$ be its potential (i.e. $X= grad\ f$). When $f$ has a minimum point $p_{min}$ and all ...
0
votes
1answer
53 views

Finding the Boundary Conditions for a Laplace's Equation in Polar Coordinates

I have solved Laplace's equation in Polar Coordinates for the scalar electric potential in a circle of radius R and have the solution $$ \phi(r,\varphi) = \phi_{0} + ...
1
vote
2answers
126 views

Differential Equations; A prerequisite to Differential Geometry?

Is thorough knowledge in ODEs/PDEs and solution techniques to be considered a prerequisite to the study of Differential Geometry (Specifically Riemannian Geometry)? If not, how would one describe the ...
2
votes
1answer
125 views

Question about Lie derivative

$(M,w)$ is symplectic manifold. $f_t : M\to M$ is a symplectic isotopy between $f_0=id$ and $f_1$. Let X_t be the vector field on M satisfying $d(f_t)/dt=X_t(f_t)$ Now I differentiate $(f_t)^*w$. Here ...
3
votes
1answer
107 views

Laplace's Equation with Neumann BC

Hi fellow math enthusiasts, I am currently working on some research to do with the electric field induced within the brain via magnetic stimulation. I am trying to solve the partial differential ...
1
vote
2answers
358 views

Solving (Frenet-Serret) differential equation system in Matlab

I'm about (trying) to solve the Frenet-Serret equation given by the known formulas, finding $e(s)$, $n(s)$, $b(s)$, where $e'(s) = \kappa(s)v(s)n(s)$ $n'(s) = -\kappa(s)v(s)e(s) + \tau(s)v(s)b(s)$ ...
3
votes
1answer
65 views

Uniqueness of integral curves.

Suppose we have a smooth compact manifold $M$ with boundary and a vector field $X$ on $M$. The maximal integral curves on $M$ are unique and hence their images give a partition of $M$. Let $p\in M$ ...
0
votes
0answers
30 views

arbitrary patch in terms of orthogonal patch

Let $\mathcal{M}$ be a $2$ dimensional differentiable manifold (a geometric surface). Let $\mathbf{y}$ be an arbitrary patch in $\mathcal{M}$. How can one prove that $y$ can be expressed as ...
1
vote
0answers
50 views

Smoothing operator on manifolds

I am reading John Roe "Elliptic operators, topology and asymptotic methods". On page 79 there is the definition 5.20 of smoothing operators. The definition is the following: "A bounded operator on ...
0
votes
3answers
124 views

top journals in analysis

as an undergraduate I find analysis as my favorite.I want to read journals regarding that. give me top 5 journals in analysis(real,complex)? top 5 journals in differential geometry? and generally some ...
3
votes
2answers
188 views

Show that the curves are circle.

For example, this question is also similar to the previous question I have asked, which is the following link; How to show the curves are conics. Question: Solve the equation ...
0
votes
1answer
67 views

How to show that a given vector field is not complete in $\mathbb{R}^2$

Suppose $X=(y^2,x^2)$ is a vector field in $\mathbb{R}^2$, show that there is an integral curve starting from some point $(c,c)$ is not defined on all $\mathbb{R}$, that is, $X$ is not complete in ...
2
votes
2answers
44 views

how to calculate frenet serre eqautions

how to calculate frenet serre eqations of the helix $$\gamma : \Bbb R \to \ \Bbb R^3$$ $$\gamma (s) =(\cos (\frac{s}{\sqrt 2}), \sin (\frac{s}{\sqrt 2}), (\frac{s}{\sqrt 2}))$$ i know the ...
2
votes
1answer
36 views

Help me Verifying that the equation is integrable and finding its solution

How can I verify that the equation is integrable and that find its solution; $$2y(a-x)dx+[z-y^2+(a-x)^2]dy-ydz=0$$ Honestly, I tried too much, but I got too strange results,thus I couldnt show my ...
0
votes
0answers
63 views

how to determine the curve

I have to determined the curve which passes through the point $(1/2, \sqrt3 /2)$ and cuts to each member of the family of circles $x^2+y^2=a^2$ forming a angle of $60º$ My idea is to create a ...
0
votes
1answer
42 views

problem of differential equations

Considering a family of curves $k(x,y,\lambda)=0$ defined in a domain $\omega$ of $R^2$ with $\lambda$ real, I have to calculate the differential equation of the curves intersect those under a ...
1
vote
1answer
83 views

Geometric problem-differential equations

I need to solve the following problem: Consider the stright lines that pass through origin. Find the equation of the trayectories that intersect those straight lines at a constant angle w (use polar ...
15
votes
1answer
364 views

When does gradient flow not converge?

I've been thinking about gradient flows in the context of Morse theory, where we take a differentiable-enough function $f$ on some space (for now let's say a compact Riemannian manifold $M$) and use ...
0
votes
1answer
85 views

Transport theory basics: can't understand solid angles

I don't understand something in transport theory: $$P(x,\vec{w})=p(x,\vec{w}) \cos\theta \, dw \, dA$$ This is the number of particles flowing across a differential surface element in the direction ...
0
votes
1answer
105 views

How to compute Bochner laplacian $\Delta=\nabla^*\nabla=\sum \nabla_{e_i}$?

I'm struggling with proving that Bochner laplacian can be described by the following formula similar to the standard laplacian formula from calculus: $$\Delta = \sum_i \nabla_i^2,$$ where $\nabla_i = ...
2
votes
0answers
141 views

A probable inspiring proof to Poincare lemma

Poincare lemma says if a smooth $p$-form $\omega$ is closed, then $\omega$ must be exact. Let's put it in another way, it says the solution of $d\omega=0$ is $\omega=d\eta$ for some $(p-1)$-form ...
4
votes
1answer
256 views

Example of commuting vector fields generating globally noncommuting flows

Recently, I discovered that a theorem from my differential geometry lecture is false due to too big generality - it stated that for vector fields $X,Y$ we have the equivalence: Incorrect! $[X,Y] = ...
2
votes
0answers
398 views

Hard Differential Equation. Please help.

first of all I'm not a mathematician, so I apologize if any of my understanding and terminology isn't up to par. Also, I've never used this website (or any of these kind of question/answer) websites ...
1
vote
1answer
79 views

Expressing pushforward of a flow in integral form

Let $\phi(t,x)$ be a flow of a vectorfield $V$ on some compact domain $\tilde{U} = U\times I \in R^n \times R$. Let X be a vector field. If one wants to write $(\phi(t,x))_{*}X)(\phi(t,x)(q)) = ...
1
vote
1answer
57 views

Complete vector field and metric

Let $M$ be a manifold, and $X$ a (smooth) vector field on $M$. If there is a metric $\rho$ on $M$ such that $\rho(M)$ is bounded, then $X$ is complete (no blow-up in finite time). But I don't see how ...
8
votes
1answer
112 views

Differential forms: The authors of a paper define $d(u\times du)$, but what is $u \times du$ supposed to mean?

I'm reading [1] recently and have another question about a remark in this paper. I tried to solve it myself (see below) but did not succeed. It could be just a notation problem. The Setup: Let $u ...
1
vote
1answer
86 views

Singular solution of DE problem

Find the singular solution of the given de $$y=x{dy\over dx}+a{dy\over dx}\left[1+\left({dy\over dx}\right)^2\right]^{-1\over 2}$$ My attempt : this is a clairaut's form of DE of the form ...
1
vote
2answers
211 views

Application of the implicit function theorem

Assume that the equation $F(x,y,p)=0$ defines a regular submanifold $M$ of $R^3$. Consider the projection $\pi :M \rightarrow R^2$, given by $\pi (x,y,p)=(x,y)$. By the implicit function theorem, in ...
10
votes
0answers
164 views

Invariant submanifolds

Let $M$ be a smooth manifold, and let $N$ be a submanifold. Let $V$ be a smooth vector field on $M$ which generates a flow $\Phi_t$ on $M$. My intuition tells me (perhaps modulo some technical ...
0
votes
0answers
71 views

Geodesic equation for a 2D manifold

I am having trouble understanding how the following statement (taken from some old notes) is true: For a 2D manifold such that $$ds^2=\frac{1}{u^2}(-du^2+dv^2)$$ If we assume that $$\dot x^a\dot ...
4
votes
1answer
64 views

Smoothness in Banach space

I need a reference about a definition. Let $n$ be an integer and $G$ be a group of $H^n$(Sobolev) automorphisms of a vector bundle $E$ on some manifold $M$ and $C$ be the space of connections of class ...
2
votes
0answers
45 views

Orientation-preserving diffeomorphism [duplicate]

Can you help for solving this please. Although I study this subject I could not solve this question please help me ı am willing to learn this question.
11
votes
0answers
175 views

Kähler Geodesics

Consider the Kähler manifold in coordinates $(a,b)$ given by the complex Riemannian metric $$\begin{pmatrix} ...
3
votes
1answer
169 views

Lemme 2.4 in Morse theory by Milnor

This is lemma 2.4 from "Morse theory" by Milnor ,with the prove I have some questions about this prove : 1) why $\displaystyle\frac{dc}{dt}(f)=\lim_{h\rightarrow 0} \frac{fc(t+h)-fc(t)}{h}$ and ...
1
vote
0answers
49 views

Cohomologies of $\mathbb R^n$ with rational differential forms

We can consider de Rham complex $0 \to \Omega^1 \to \Omega ^ 2 \to...$ on $\mathbb R^n$, where $\Omega ^r$ are $r$-forms on $\mathbb R^n$ with rational coefficients. What are homologies of this ...
1
vote
0answers
35 views

Unique continuation for elliptic operators

Consider the following system of linear elliptic equations: $ \Delta s_i = \sum_{j=1}^{d} l_{ij} s_j $ for $ i=1,\ldots d $ where $ l_{ij} = l_{ji} $. It should be true that the following unique ...
4
votes
0answers
105 views

Clarification in a paper

This is regarding a clarification in page 384 of a paper published in Annals of Statistics by Amari. In page no. 384, he defines $$R_i(t)=\frac{\partial}{\partial \theta_i} ...
2
votes
1answer
108 views

Does there exist a vector field tangent to a given curve?

Let $\gamma : \mathbb{R} \to \mathbb{R}^2$ be an injective $C^2$ curve. Does there exist a $C^1$ vector field $X : \mathbb{R}^2 \to \mathbb{R}^2$ tangent to $\gamma$, ie. such that $\gamma'(t)= ...
11
votes
1answer
283 views

When does a vector field admit orthogonal fields?

My question is: Let $\,X$ be a nonvanishing smooth vector field over an open subset $U \subset \mathbb{R}^3$. Which conditions on $X$ guarantee the existence of a smooth nonvanishing vector field ...
1
vote
0answers
28 views

Boundaries- regularity and local parametrization

Suppose we have a bounded domain $\Omega \subset \mathbb{R}^3$ with $C^2$ boundary.Let $x_0 \in \partial \Omega$. We choose a $X_1,x_2,x_3$- coordinate system such that the $x_1,x_2$-plane is ...