For question about the differential of a map from an open set of a vector space to a vector space.

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33 views

How to rewrite a differential operator when I eliminate the variable?

I have a system with three variables: $x_a$, $x_b$, and $x_c$. I have an operator that depends on differential operators of these variables $$\mathcal{F}=a \frac{\partial^2 }{\partial x_a^2} + ...
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1answer
21 views

Differentiation and PDE Theory

I have been given the following two definitions: 1) $D^ku$ is the set of all derivatives of order k of u 2) Let $\Omega$ be a non-empty subset of Euclidean space $\mathbb{R}^N.$ An expression of the ...
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0answers
24 views

Understanding the Jacobian past calculus

What's taught in calculus: In the calculus of multiple variables I learned that the Jacobian $$\textbf ...
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0answers
23 views

Recover a surface's equation from its curvature

Can the equation of a surface in Euclidean 3 space be recovered from the equation of its Gaussian curvature?
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0answers
35 views

integral equation into differential equation

I have the equation $$ E = \alpha \int \int_S E dS $$ and I need to find a solution for E. My first instinct is to re-arrange it into a second order differential equation, but because dS is an area, ...
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2answers
27 views

limit of a function with a matrix exponential

I spent too many time trying to solve this problem...and finals are coming. Please help me! I just can't see a method to do this demonstration: "For an $A_{n \times n}$ matrix, demonstrate that a ...
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0answers
8 views

compute next frame curvature torsion based on existing tangent, normal, binormal, position, curvature and torsion

I have a question that is making me headache. Suppose I have $r_{s}$, $T_{s}$, $N_{s}$, $B_{s}$, $\kappa_{s}$ and $\tau_{s}$ for position of the sample, tangent, normal, binormal, curvature and ...
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0answers
24 views

How can I solve this equation analytically?

How can I solve this equation analytically? $$f''[y] - \frac{3}y\cdot f'[y] - \log\left[\frac{Y}{y}\right]\cdot f[y] = 0$$ I know that for $Y >> y,\ \log \left(\frac{Y}{y}\right)$ can be ...
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1answer
29 views

First order differential equation problem

Suppose we have $$ \frac{dy}{dx} +f(x)y = r(x) $$ and it has two solutions $y_1(x)$ and $y_2(x)$ then how to prove that solution of differential equation $$ \frac{dy}{dx} +f(x)y = 2r(x) $$ Will be ...
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1answer
25 views

Differential Equation: Find the instability criterion

My attempt: Question (d) I took the derivative of the original differential equation, $$dI/dt = BI(N-I) -uI = g(I)$$ $$g'(I) = BN - 2BI - u$$ Set $$g'(I) = 0$$ Isolate $$I = Ro$$ $$I = Ro = ...
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1answer
50 views

Solve $\frac{dx}{dt}=1$, $\frac{dy}{dt}=\cos(x(t))$

Question: Solve $$\frac{dx}{d t}=1$$$$\frac{dy}{dt}=\cos(x(t))$$ Where $x(0)=x_0$ and $y(0)=y_0$ Answer: I have gotten $x(t)=t+x_0$, cant seem to get $y(t)$ should be a simple problem, but just cant ...
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1answer
26 views

Compute $L_\mathbb{X}\beta$

Given information: $$\alpha+(x+y)dy+(x^2-y^2)dz$$$$\beta=zdx\wedge dy+xzdx\wedge dz$$$\mathbb{X}$ is the vector field given by $$\mathbb{X}=(0,-x,-1)$$ I have found $i_\mathbb{X}\beta=2xzdx$ ...
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1answer
32 views

Compute the contraction of $i_\mathbb{X}\beta$

Question: Let $\beta=zdx\wedge dy+xzdx\wedge dz$, and let $\mathbb{X}$ be the vector field on $\mathbb{R}^3$ given by $\mathbb{X}=(0,-x,-1)$. Compute $i_\mathbb{X}\beta$, combining terms where ...
3
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1answer
73 views

Need help solving this differential equation

$$p^3 - 2xyp + 4y^2 = 0$$ where $p = \mathrm dy / \mathrm dx$. I don't know which type of equation it is or how to simplify it. Though there is an observation that $$(xy^2)′=y^2+2xyy′$$
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0answers
32 views

Evaluating first order differential equation

I was given $$(x+3y^2)\frac{dy}{dx}=y$$ and also $y>0$ so I wrote it as $$\frac{dx}{dy}=\frac{x}{y} + 3y$$ now substituting $x=y.t$ and $$ \frac{dx}{dy}=t+y.\frac{dt}{dy}$$ I am finally left with ...
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1answer
34 views

Inequality with a differentiable function + diffeomorphism

Assume that $h:\mathbb{R} \to \mathbb{R}$ is a differentiable function for which there is a number $\lambda \in \mathbb{R}_+^n$ so that: $$\lvert ((dh)(x)(t)\rvert \ge \lambda \lvert t \rvert, \forall ...
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0answers
19 views

Clarifications about SDEs, Differentials & Derivatives

A general SDE look like the following: $$ \mathrm{d}\psi=a\mathop{}\!\mathrm{d}t+b\mathop{}\!\mathrm{d}W,\tag{1} $$ where $\psi:t\mapsto y = \psi(t)$ is the solution, while $a$ and $b$ can be both, ...
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0answers
19 views

Condition for integrability of relation between differentials of coordinates of a mechanical system

Let $q_1, q_2, q_3$ be generalized coordinates of some mechanical system. We have a differential relation between the coordinates $A_1dq_1+A_2dq_2+A_3dq_3=0 $, where the $A_k$ are functions of the ...
4
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1answer
48 views

Question on how to work with “differential”

I have the following equation which is from a famous paper in economics: $$ \sum_{i=1}^n x_i \, dx_i=\frac{1}{2} d\Big[\sum_{i=1}^n x_{i}^2\Big]=XH \, dX+\frac{1}{2}d\big[X^2H\big] $$ Can you tell me ...
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1answer
17 views

Differential equations - maximal domain

I was solving an exercise about differential equations, and i really don't get how can I determinate the maximal domain of solution. Example: $$(dy/dx) = x - y/(1+x), y(0) =-1$$ The solution is ...
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1answer
31 views

Computing the differental of an orthogonal projection

I am having trouble computing the differental of a map. This is the context: Let $S\subset \mathbb R^3$ be a regular surface, and fix a point $p\in S$. Let $\pi:\mathbb R^3\to T_pS$ be the ...
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0answers
13 views

Strong derivative of a compound map

I find the strong Fréchet derivative of $\Phi(h,\psi(h))$, where $\Phi:T_0\times T_\xi\to Y$ with $T_0, T_\xi, Y$ Banach spaces and $\psi:T_0\to T_\xi$ is strongly differentiable in $0$, evaluated in ...
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0answers
24 views

What type is this differential equation and how to solve it?

This is the rocket equation with drag and gravity: $$dv = -\frac{V_e}{M}dM - k \frac{v^2}{M}dt - g\cdot dt $$ where $V_e, k,$ and $g$ are constants.
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2answers
71 views

y' = y^3 - y stable points

So I have this differential equation: $$y' = y^3 - y$$ And I need to find out which of the points (0, 1, -1) are stable. So here we go: $$y = \pm \frac{1}{\sqrt{e^{2x+c}+1}}$$ (solution to ...
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0answers
11 views

Diffrerential geometry: singular point

Problem:Let C be a regular closed simple curve on sphere. Let v be a differentialble vector field on sphere such that the trajectories of v are never tangent to C. prove that each of the two regions ...
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1answer
28 views

Solving a 2nd order ODE & phase lag computation

I'm reviewing differential equations, and came across this problem. In the MIT OCW lecture, the professor utilizes the trig formula $A\cos t + B\sin t = C\cos(t - \phi)$ where $C$ is the amplitude ...
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0answers
17 views

Can some components of metric be Fisnlerian while the others be Riemannian?

A Finsler metric reduces to a Riemann metric in case it loses its dependence on velocities. Now, my question is this: Can we have a Finsler metric in which some components of the metric have velocity ...
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1answer
21 views

Solving ODE first order by Laplace transformation?

given is the following first order ODE: $\dot\epsilon(t) = \frac{1}{\eta}\cdot\sigma(t) + \frac{1}{E_1}\cdot\dot\sigma(t)$, where $\eta$ and $E_1$ are constants. The initial conditions are: ...
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0answers
19 views

Maple Interactive Plot Builder

I'm using Maple and have made several plots in my document. When I remove output and run it through (by clicking !!!), the "interactive plot builder: specify expressions" dialog box keeps on popping ...
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1answer
22 views

condition of plane to touch a given surface

Q. Show that the plane $ax+by+cz+d = 0$, touches the surface $px^2+qy^2+2z=0$, if $a^2/p + b^2/q +2cd = 0$. How to start to solve this problem?
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1answer
35 views

Geometric interpretation of ${\partial f\over \partial x}= {\partial f \over \partial y}$

I know that $${\partial f\over \partial x}= {\partial f \over \partial y}$$ iff there exists a differentiable function $g$ (of one variable) such that $g(x+y)=f(x,y)$ (where $f : D\subseteq \mathbb ...
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2answers
30 views

Finding complete general solution of differential equation with repeated roots (undetermined coefficents)

How do you get a complete general solution for a differential like this? $y^{\prime\prime}+6y^{\prime}+9y=14e^{-3x}$ This is what I have so far for the first part of the problem: $yp=Ce^{-3x}, ...
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1answer
68 views

Differentiating with respect to a vector

Hello i'm new to this forum and this is my first post. I was going over the transport theorems in fluid mechanics and there is one way in which you can convert reynolds transport theorem into a single ...
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1answer
36 views

Whitney's Embedding

The Whitney embedding theorem says that any smooth manifold of dimension $n$ may be embedded in $R^{2n}$. I am just beginning to study differential geometry for application to physics (general ...
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1answer
24 views

Undetermined Coefficient for 2 first order differential equation.

I could not understand the textbook clearly. When you are trying to find a particular solution of x' = -2x + y + 2e^(-t) y' = x -2y + 3t I understand that 2e^(-t) would have a form of ate^-t + ...
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1answer
27 views

Classifying phase portait regarding zero eigenvalue.

When you have two equations $x' = 4x -3y$ $y' = 8x -6y$ The solution turns out to be $x = c_1e^{-2t} + 3c_2$ $y = 2c_1e^{-2t} + 4c_2$ and I understand how the phase portrait is visualized. ...
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0answers
16 views

Solutions of the form with eigenvalue that has algebraic multiplicity of more than 2?

Let's say I have 3 x 3 matrix with X' = AX and for instance I have a 3 eigenvalues that are the same. It's interesting and I could not find any information on textbook How would the third ...
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1answer
24 views

Can you help me with this ODE?

Can you help me solving this differential equation? I am at the beginning of studying them and I have many doubts in understanding the good method to choose. Thanks!! $$\frac{d^2x}{dt^2}= 1-(ax)^4$$
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2answers
42 views

What does constant terms for complex roots represent when drawing a phase portrait?

For example if I have ODE of $x$ is a matrix of $x_1$ and $x_2$ $x_1 '$ = $x_1$ - $5\cdot x_2$ $x_2 '$ = $x_1$ - $3\cdot x_2$ and I found the solutions which are $x_1$ = $c_1 \cdot e^{-t}\cdot ...
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4answers
95 views

Global Coordinates in Differential Geometry?

In trying to learn a bit about differential geometry I have hit a puzzler. Most texts emphasize that one coordinate system will not suffice in general, but the reasoning is never given. After all, if ...
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4answers
32 views

Total derivatives explanation

If you have a function with multiple variables, lets say $f(x,y)$, then the total derivative for small changes would be $$\Delta f = f_x\Delta x + f_y \Delta y$$ And because of that we can assume that ...
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1answer
32 views

Inverse laplace transform with complex roots

hello I am having some trouble finding the solution to this inverse laplace transformation $$ I(s)= \frac{6s+24}{s^2 +4s+8} $$ The solution is solved using Euler identity and partial fractions, $$ ...
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1answer
32 views

strictly decreasing on interval, establishing inequalities

Establish the inequality $2/\pi < \sin x/x$ for $0 < x < \pi/2$ by showing that the function $f(x)= \sin x/x$ is strictly decreasing for $0 < x ≤ \pi/2$. this is all i have, dunno if im ...
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0answers
13 views

Order notation problem

I have got stucked for order notation and expansion. Can you give me some clear hint?
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0answers
19 views

Application of the Mean Value Theorem in Real Analysis in R^n

Let $f\in C^2(a,a+2h),a\in \mathbb{R}$,and $h>0$. Show that there exists $c \in (a, a + 2h)$ such that $f(a + 2h) − 2f(a + h) + f(a) = h2f''(c)$. Hint: Introduce the auxiliary function $\varphi: ...
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0answers
37 views

Solution of a nonlinear ODE with boundary conditions on infinity

\begin{align*} &f'''(x)+f(x)f''(x)-(f'(x))^2-M~f'(x)=0\\ &\text{boundary conditions for $x=0$ are}\\ &f'(x)=1,~~~~f(x)=N\\ &\text{also}\\ &\lim_{x->\infty}f'(x)=0\\ \end{align*} ...
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3answers
82 views

Basis of a Kernel

How would i find the basis of the kernel of the differential operator below $$8y'' + 3y' + 7y$$ We know the equation was homogenous and i believe the basis is two dimensional
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1answer
44 views

How can I solve this differential equation? please help

How can I solve this differential equation $$y'= \sin(x+y) ,\ \ y(0) = -\frac{\pi}{2}, -\infty < x < \infty$$ I tried to denote $z=x+y$ And I got unfamiliar integral. Please help. Thanks
2
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3answers
63 views

Can a first derivative of a function have more roots than the original function?

This is a general question. Function is to be considered differentiable on some domain. More specifically, I am given a function $f(x)$ which is twice differentiable and has three distinct real ...
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1answer
36 views

Please help me understand the concept of variable, and differentiation of variables.

`I am in the first year of college and know mathematical analysis in a very rigorous context, from high school/ math olympiads Imo's etc. But the concept of $df$ seems totally weird and ...