# Tagged Questions

30 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 ...
31 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 ...
80 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 ...
72 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 ...
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 ...
35 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 ...
56 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 ...
18 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 = ...
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 = ... 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 $$curl=\begin{bmatrix} 0 & -\partial z & \partial ... 0answers 25 views ### Derivative of terminal state w.r.t. the inital conditions. Let x\in R^n and consider the system$$ \dot{x}=f(t,x) \;\;\mbox{with}\;\; x(0)=x_0 $$and suppose that we know it's exact or very accurate solution x(t) for the time interval [0,T]. I'm ... 0answers 14 views ### Numerical solution of first order ODE I have an in-homogeneous ODE. R'(x)-(C_1 +C_2 x) R(x) = R_1-C_1 R_0\, x \tag 1. What I know is the constant matrix R(0) as initial condition. Question:- how to find out R(1) by numerical ... 1answer 115 views ### Function whose gradient is of constant norm Let f:\mathbb R^n\rightarrow \mathbb R be a smooth function such that \|\nabla f(x)\|=1 for all x\in \mathbb R^n and f(0)=0. I would like to prove that f is linear. I first looked at the ... 0answers 74 views ### Computation of the Frenet-Serret trihedron in \Bbb L^3 (Lorentz-Minkowski space) Consider \Bbb L^3 = (\Bbb R^3, \langle , \rangle), with the convention$$\langle (x_1,y_1,z_1), (x_2,y_2,z_2)\rangle = x_1x_2+y_1y_2 - z_1z_2$$and \| v \| = \sqrt{|\langle v, v \rangle|}. Let ... 1answer 43 views ### Prove the formula for the Lie derivative of a differential form If X is a vector field then by \mathcal F^t_X I will denote it's flow. If \alpha \in \Lambda^k then by definition$$ \mathcal L_X \alpha = \frac{d}{dt}(\mathcal F^t_X)^*\alpha \, ... 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\#}Ythe vector field satisfying \mathcal ... 1answer 166 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 ... 2answers 34 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 ... 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 ... 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 ... 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? 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 ... 2answers 137 views ### Finding the Asymptotic Curves of a Given Surface I have to find the asymptotic curves of the surface given byz = a \left( \frac{x}{y} + \frac{y}{x} \right),$$for constant a \neq 0. I guess that what was meant by that statement is that surface ... 0answers 41 views ### What is the explanation of the equation 1-4.6 in the book “Applied Exterior Calculus”? If the n function \{f^i(x^m)\} are of class C^{\infty} in some neighborhood of 0, then system of autonomous ODEs$$\frac{d\bar{x}^i(t)}{dt}=f^i(\bar{x}^m(t)),\quad\quad i = 1,\cdots, n$$has a ... 2answers 57 views ### Solving a certain differential equation when assuming a surface of revolution is minimal The problem is the following: Consider the surface of revolution$$ \textbf{q} (t, \mu) = (r(t)\cos(\mu),r(t)\sin(\mu),t) $$If \textbf{q} is minimal, then r(t) = a\cosh(t)+b\sinh(t) for a,b ... 1answer 55 views ### Recovering a frame field from its connection forms I have a faced a research problem where I would need to recover a frame field given its connection forms. More precisely, I begin with an orthonormal frame field (given by data) in \Re^3 written as ... 0answers 19 views ### non-smooth minmal surfaces and differenteial equations This is the equation for a function u(x,y) whose graph is a minimal surface (its mean curvature is 0):$$(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 ... 1answer 160 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 ... 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) ... 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 ... 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 ... 0answers 71 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 ... 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, ... 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 ... 2answers 254 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 ... 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 ... 2answers 499 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) ... 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 ... 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 ... 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 ... 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 ... 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 ... 1answer 76 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 ... 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), ... 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 ... 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 ... 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 ... 1answer 104 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 ... 1answer 488 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 ...
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 ...