0
votes
1answer
29 views

Definite Integral theorem validity :- $\int_{0}^{L} \left( \int_{s}^{L}p(t)\ dt \right) \ ds =\int_{0}^{L} \ p(s) \ ds$?

Can we write $\int_{0}^{L} \left( \int_{s}^{L}p(t)\ dt \right) \ ds =\int_{0}^{L} \ p(s) \ ds\tag 1$ ? In other words, is this result valid? If so, could you help me to get the proof it NB :: ...
1
vote
0answers
31 views

Dirichlet Eigenvalues of Laplace-Beltrami operator in Hyperbolic space

Consider the hyperbolic half-plane $\mathbb{H}=\lbrace (x,y)\in\mathbb{R}^2: y>0 \rbrace$ with standard Riemannian metric. The Laplace-Beltrami Operator can be written as ...
2
votes
1answer
32 views

The curvature of a Cycloid at its cusps.

My lecturer proposed a question to particular result regarding the curvature of a Cycloid (generated by circle of radius 1) at its cusps. Having left it as an open problem, I thought it'd be ...
3
votes
2answers
81 views

Symmetry group of the vector field $V=x \partial /\partial x + y \partial /\partial y$

I was trying to solve an exercise in one of Arnold's book that asks for the symmetry group of the vector field $V=x \partial /\partial x + y \partial /\partial y$, that is the diffeomorphisms $g$ of ...
2
votes
0answers
78 views

Quaternion conversion

We have a normalized orthogonal co-ordinate frame travelling through the curve as in figure 1 below, from one end to other. Let us call starting end as A and ending end as B. What we know is initial ...
1
vote
0answers
38 views

Equation of a curve with a local minimum fixed at $x=a$ when we rotate the curve about the origin.

We have a strangely curved plank. If we place a round weighted object on it, it will rest itself at one point of it, when we incline the plank slowly, the object will gradually move towards a ...
0
votes
1answer
36 views

Local isometric embedding

Every $n$-dimensional smooth Riemannian manifold admits a local isometric embedding of class $C^1$ into $\mathbb R^{n+1}$ by the Nash-Kuiper theorem. An example by Nadirashvili and Yuan shows that in ...
1
vote
0answers
57 views

Integrability of 1-forms and Stokes' Theorem

Let $\alpha$ be a $1$-form defined on a manifold $M$ and $\Delta = ker (\alpha)$. The classical theorem of Frobenius says that $\Delta$ is integrable if $\alpha \wedge d\alpha =0$ i.e if $d\alpha$ is ...
0
votes
1answer
21 views

Reducible to Separable First Order Differential Equation Word Problem in Analytic Geometry 1.4-29

I completed near all problems om a differential equations text chapter on reducing non-separable first order differential equations to separable by using an appropriate substitution for example u = ...
1
vote
1answer
18 views

Reducible to Separable First Order Differential Equation Word Problem in Analytic Geometry 1.4-28

I completed near all problems of a differential equations text chapter on reducing non-separable first order differential equations to separable by using an appropriate substitution for example $u = ...
1
vote
2answers
37 views

Is it possible to write the curl in terms of the infinitesimal rotation tensor?

Is it possible to write the curl in terms of the infinitesimal rotation tensor? Basically, we can write the curl as a matrix operator $$ curl=\begin{bmatrix} 0 & -\partial z & \partial ...
2
votes
0answers
25 views

Derivative of terminal state w.r.t. the inital conditions.

Let $x\in R^n$ and consider the system $$ \dot{x}=f(t,x) \;\;\mbox{with}\;\; x(0)=x_0 $$ and suppose that we know it's exact or very accurate solution $x(t)$ for the time interval $[0,T]$. I'm ...
0
votes
0answers
14 views

Numerical solution of first order ODE

I have an in-homogeneous ODE. $R'(x)-(C_1 +C_2 x) R(x) = R_1-C_1 R_0\, x \tag 1$. What I know is the constant matrix $ R(0)$ as initial condition. Question:- how to find out R(1) by numerical ...
4
votes
1answer
115 views

Function whose gradient is of constant norm

Let $f:\mathbb R^n\rightarrow \mathbb R$ be a smooth function such that $\|\nabla f(x)\|=1$ for all $x\in \mathbb R^n$ and $f(0)=0$. I would like to prove that $f$ is linear. I first looked at the ...
1
vote
0answers
75 views

Computation of the Frenet-Serret trihedron in $\Bbb L^3$ (Lorentz-Minkowski space)

Consider $\Bbb L^3 = (\Bbb R^3, \langle , \rangle)$, with the convention $$\langle (x_1,y_1,z_1), (x_2,y_2,z_2)\rangle = x_1x_2+y_1y_2 - z_1z_2$$ and $\| v \| = \sqrt{|\langle v, v \rangle|}$. Let ...
1
vote
1answer
44 views

Prove the formula for the Lie derivative of a differential form

If $X$ is a vector field then by $\mathcal F^t_X$ I will denote it's flow. If $\alpha \in \Lambda^k$ then by definition $$ \mathcal L_X \alpha = \frac{d}{dt}(\mathcal F^t_X)^*\alpha \, ...
2
votes
1answer
53 views

Problem with a pushforward of vector field formula (Michael Taylor, “Partial Differential Equations”)

Let $X$ denote a vector field and let $\mathcal F^t_X$ denote its flow. If $X$ and $Y$ are two vector fields we denote by $\mathcal F^t_{X\#}Y$ the vector field satisfying $$ \mathcal ...
7
votes
1answer
172 views

Special conformal killing fields - solving for integral curves.

For each $b\in\mathbb R^d$, let a vector field $X_b:\mathbb R^d\to\mathbb R^d$ be defined as follows: \begin{align} X_b(x) = 2(b\cdot x)x - x^2 b, \end{align} where $x^2 = x\cdot x$. This is the ...
0
votes
2answers
35 views

Solving $\left\{\begin{matrix}u'v''-u''v'=0 \\ R^2u'u''+v'v''=0 \end{matrix}\right.$.

Given that $u,v$ are functions of $t$, $R$ constant, solve $\left\{\begin{matrix}u'v''-u''v'=0 \\ R^2u'u''+v'v''=0 \end{matrix}\right.$. When trying to find geodesic on cylinder, I get this ...
3
votes
1answer
83 views

Monodromy representation of Airy equation

Let $K=\Bbb{C}(z)$ with the usual derivation and consider the Airy dierential equation $y^{(2)}-zy$=0. How to determine the monodromy representration? Airy equation is not Fuchsian diferential ...
2
votes
0answers
26 views

Derivation of the prolongation formula for finding symmetries of diff equations from Olver

I am having a problem with the derivation of the prolongation formula from PJ Olver's text :"Applications of Lie groups to differential equations" Page 105,106. Considering a differential equation ...
1
vote
1answer
69 views

number of points of tangency of the zero divergence vector field and the equator of the sphere.

Let $V$ be vector field on the sphere $S^2$ and $\operatorname{div} V=0$. What is the minimum number tangency points of this vector field and the equator of the sphere?
1
vote
2answers
61 views

Need help on books on diff. equations/geometry and theoretical computer science

I am looking for recommendation of 3 different books on the following topics: 1.Differential Equations -Ordinary diff. equations -Vector field, transport equations -Equation of wave and heat -Use ...
2
votes
2answers
137 views

Finding the Asymptotic Curves of a Given Surface

I have to find the asymptotic curves of the surface given by $$z = a \left( \frac{x}{y} + \frac{y}{x} \right),$$ for constant $a \neq 0$. I guess that what was meant by that statement is that surface ...
1
vote
0answers
41 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
57 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
55 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
1answer
167 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 ...
6
votes
1answer
128 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
97 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
64 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
72 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
31 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
57 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 ...
1
vote
2answers
265 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
139 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 ...
1
vote
2answers
506 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
83 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
55 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
140 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
196 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
78 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
56 views

How to calculate Frenet-Serret equations

How to calculate Frenet-Serret equations of the helix $$\gamma : \Bbb R \to \ \Bbb R^3$$ $$\gamma (s) =\left(\cos \left(\frac{s}{\sqrt 2}\right), \sin \left(\frac{s}{\sqrt 2}\right), ...
2
votes
1answer
39 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
66 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
107 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 ...
16
votes
1answer
509 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 ...