Questions tagged [differential]

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

Integrate $\frac{dR}{dt}=\frac{t^3+R^3}{t^2R+tR^2}$ and find the solution for $R(1) = -1$

Yesterday I asked for some help regarding this question which I found from here but I'm stuck again and I'll show you my working so far with my homework First I got this equation separated ...
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2answers
33 views

Variable Separable Form for $\frac{dR}{dt}=\frac{t^3+r^3}{t^2r+tr^2}$

I just need a hint or a method on how to separate the equation $$\frac{dR}{dt}=\frac{t^3+R^3}{t^2R+tR^2}$$ so I can integrate it and get a general solution in terms of $R(t)$
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13 views

Derivate by parameter or by initial conditions

Denote in $y(x, τ, ξ)$ solution to the Cauchy problem: $$y' = −y + (x − 3)(1 − e^{−y}), y(τ) = ξ$$ Find: $$\frac{∂y(x, 2, ξ)}{∂ξ}|ξ=0$$ How to do it? Any notes?
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Differential of the sine function

I know this is very basic, but it's a doubt I have since a couple years. What is the differential of a function like $w = q_1 \sin(\frac{x}{L})$ ? Is it the following? \begin{equation} \delta w = ...
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2answers
29 views

What's the differential of the cross product wrt a vector?

$$ w= u(x) \times v(x), \qquad u(x), v(x), x \in R^3$$ How to calculate the $ \frac{\partial w}{\partial x}$? Thanks very much!
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1answer
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Solution of first order linear Differential equations (solving integrating factors) [closed]

An integrating factor of the differential equation - $$(y+\frac{y^3}{3}+\frac{x^2}{2})dx+\frac{x}{4}(1+y^2)dy=0$$
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1answer
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Regarding equivalent definitions of Euclidean Submanifolds in Gallot, Hulin and Lafontaine's book Riemannian Geometry

In the book Riemannian Geometry by Gallot, Hulin and Lafontaine, a proposition which characterises equivalent definitions of submanifolds is given as follows: 1.3 Proposition The following are ...
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26 views

Differential Equation, Mice in corner of a square problem [closed]

I'm supposed to solve the mice in the corners of a square or triangle problem, where in a triangle, in every corner is a mouse, and from t = 0s, they start moving towards each other. It is a ...
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vertical bundle [closed]

Why the vertical bundle equals to ker(dπ)? such that π is the projection space from the bundle E to the manifold M for the fibre bundle. In another book, this is a definition of a vertical bundle. but ...
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1answer
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Stability of the origin of the linear system $\dot{x} = Ax$ given $A^2 = I$

Suppose that $A$ is an $n\times n$ matrix such that $A^2 = I$. What can you say about the stability of the origin of the linear system $\dot{x} = Ax$? Are there nontrivial stable and/or unstable ...
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2answers
55 views

Differential of a matrix function: $f:\mathbb{R}^{n \times m} \rightarrow \mathbb{R}^{m \times 1}$ $f(A) = A^T\cdot \vec{v}$

I wish to calculate the differential of a function: $f(A) = X^T\cdot \vec{v}$ when $A\in \mathbb{R}^{n \times m}$ with respect to $A$. Since this is a linear function, if we think about $D\in \...
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30 views

Proving that the Lagrangian is time translation invariant

I know that the context is Physics, but my problem actually is related to Math: In my textbook, the author deduce the expression of the lagrangian $L(q_i(t), \dot q_i(t), t )$ of a free particle only ...
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DIFFERENTIAL EQUATIONS QUESTION EMERGENCY [closed]

Suppose that 𝐵 is a 5𝑥5 matrix with eigenvalues 2 with multiplicity 2 and 4 with multiplicity 3. Also suppose that 𝑣1 and 𝑣2 are the corresponding eigenvectors of 𝜆 = 2 and (𝐵 − 4𝐼)𝑣3 = 0, ...
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61 views

Stability conditions of discrete dynamic systems

I have a system of 6 difference equations with three state variables. Let $X_t = [x_{1t}, x_{2t}, x_{3t}, x_{4t}, x_{5t}, x_{6t}]^T$. The general form of the problem is $g(X_{t+1}, X_t) = 0$. The log-...
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1answer
57 views

In differential equations, why is ln|x|+c = ln(cx)

So, let's say I get the expression $ln|x|+c, c \in R$. My reasoning is that I can do the following: let $c_1 = e^c$, then $c_1 > 0$ $ln|x|+c, c \in R$ becomes: $ln(c_1|x|), c_1 >0$ How I can ...
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1answer
36 views

Find a differential equation that corresponds to the given direction field. [closed]

How do we find the differential equation that corresponds to the given direction field ??? Do not know where to start this question.
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Does square of differential of argument is the same as differential of differential of square of argument?

Well, I have a problem: if x is an independent argument so is $$dx^2 = (dx)^2$$ ? Or if $z = z(x, y)$ is $dz^2 = (dz)^2$? If not, why? I really can't understand it.
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38 views

What does a vertical line mean in calculus notation? [duplicate]

I am having trouble understanding what the horizontal bar with the r=R on the bottom means in this equation (cylindrical coordinates): I have not seen this notation before.
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21 views

Solving this Laplacian Equation?

How do I solve the following? $\frac{∂^2\phi}{∂x^2}$ + $\frac{∂^2\phi}{∂y^2}$ + $\lambda^2\phi$ = $0$ To start of with, is it right to consider using the method of separation of variables, or is ...
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1answer
32 views

What are some “derivative notations” worth to memorize? Like $\frac{x''}{x'}$ is equal to $\left(\ln\left(x'\right)\right)'$ (x depends on t)

It can make the calculation easier especially in differential equations but I only know this and $-\frac{x'}{x^2}$ = $\left(\frac{1}{x}\right)'$
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42 views

Is $p=x'$ always the best substitution to reduce the order of a differential equation?

I came across some exercises and it doesn't work with $p=x'$. The exercises our teacher has sent to us: 1.1. $x'' = \operatorname{arctg} t$, 1.2. $x'' = (x')^2 + x' + \frac1{t^2}$, 1.3. $...
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What is natural way to local trivialize TM (tangent bundle) via normal coordinate?

Suppose dimension n=2 amd M be a Riemannian manifold. We have normal coordinate around p in small ball. In my opinion there are two way to trivialize. Let $ (x_1,x_2)$ be a normal coordinate such ...
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0answers
18 views

Laplace equation over the square mesh of side four units

Solve the Laplace equation $\nabla 2u = 0$ over the square mesh of side four units satisfying the following boundary conditions round to three decimal digits for first three iterations $u(x, 0) = 0 \...
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2answers
35 views

show that $\int_0^1\frac{(dx)}{\sqrt{1-x^2}}=\frac{(1)}{(n)}*\beta({\frac{(1)}{(2)}},\frac{(1)}{(n)})$

Show that $\int_0^1\frac{(dx)}{\sqrt{1-x^2}}=\frac{(1)}{(n)}*\beta({\frac{(1)}{(2)}},\frac{(1)}{(n)})$ ,when $n\in\mathbb{Z}^+$. It was hard for me to prove it . I integrated the left hand side and ...
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1answer
22 views

How to find the general solution of $u''(a)+u(a)=0$?

When I get to know the solution of $u''(a)+u(a)=0$, I always come with the solution with exponential but to get to the solution $u(a)=A \cos(a-b)$ how should I proceed? And the where this $b$ has come ...
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0answers
30 views

Solving differential equation with linear algebra

problem2 Hi all, I am trying unsuccessfully to solve problem 2. Those exercises are part of the gilbert strang's book on linear algebra. This chapter covers eigenvalues/vectors and differential ...
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1answer
32 views

Differential product

I'm trying to convert formally the cartesian differential product $dxdy$ into its equivalent in the cylindrical coordinates which is $rdrd \theta$ . Doing the change of variable: $x=rcos\theta$ and $y=...
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1answer
29 views

Flows are Stable Under Diffeomorphism

In chapter 2 of this book (entitled: The Simplicity Of Diffeomorphism Groups) the author says that given any compactly supported smooth vector field $V$ on a simply connected and connected (finite-...
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0answers
12 views

How do I use an indicial equation to show solutions of ODE? (Frobenius Method)

I'm given this equation:$$xy''+(1-x)y'+\alpha y=0$$, within the indicial equation $$r(r-1)+b_0 r+ c_0 =0$$ I don't understand how should I use the indicial equation to show the two solutions of this ...
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2answers
43 views

Nonlinear Differential Equation of High Degree

Any help, please? How can I start to solve them? I tried to use $y'=p$. Also I tried $x=e^x$ and so many methods, but I couldn't reach them to the end. I always got blocked in the middle. Thanks in ...
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1answer
33 views

Manifold definition of $C^1$ vs $p \mapsto dF_p$ continuous

I don't know much about differential geometry but as I understand it for a map between manifolds $F : M \to N$ to be $C^1$ around $p$ means that there exists charts $(U,\phi)$ containing $p$ and $(V,\...
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1answer
39 views

Compactly Supported Flow [closed]

Consider the following initial value problem $$ \begin{cases} \frac{d}{dt} Y_t = \rho(Y_t)\\ Y_0 = 0 \end{cases} $$ where $\rho(x)$ is a bump function supported near $0$ on $\mathbb{R}^1$. Let $f:t\...
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2answers
59 views

Is d(sinx) and sin(dx) same?

Is d(sinx) and sin(dx) same? If it's not then is there a way to represent sin(dx) or for that matter any function f(x) where x is put to be some differential? Examples are most welcome if needed.
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27 views

Recovering a function from its partial differential

If I have a function of two variables, say $f=f(x,y)$, I know that its differential is computed as $$df=\frac{\partial f}{\partial x}dx + \frac{\partial f}{\partial y}dx$$ But is it possible to ...
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23 views

In at 3-dimensional system, if two of my axes are the center subspace, would my center manifold be the plane of these two axes?

I was doing extra problems for my dynamics class and I came across this system after I transformed to Jordan normal form: $$\begin{bmatrix}\dot{u}\\\dot{v}\\\dot{w}\end{bmatrix}=\begin{bmatrix}-1&...
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Derivation: How do I derivate this integral with different parameter

I have this integral and I want the producer to come in. Which THEOREM should I use to solve? the derivation I want to be made with respect to the variable r!! ...
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1answer
33 views

Fundamental theorem on existence and uniqueness of solutions of differential equations

Given two differential equations 1) $\frac{dy}{dx} = x^{\frac{2}{5}}$ 2) $\frac{dy}{dx} = y^{\frac{2}{5}}$ with the initial condition $y(0) = 0$ find which lacks uniqueness of the solution, and ...
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2answers
35 views

Differentiate $(x^3 + xy^2 + a^2y) dx + (y^3 + yx^2 – a^2x) dy =0$

Differentiate $$(x^3 + xy^2 + a^2y) dx + (y^3 + yx^2 – a^2x) dy =0$$ Is the above equation an exact differential equation? because it doesn't follow the necessary condition of exact differential ...
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1answer
24 views

argmax of a function involving summation of max functions

I am trying to calculate argmax of the following function w.r.t. $x$: $$ f(x) = \sum \limits_{i=1}^{N} \Big( max (-||y_i - x||_2^2 + \alpha, 0)\Big) $$ where $x$ ...
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0answers
26 views

differentiate wrt time

I came across these equations in a mechanics textbook and wish to differentiate the first equation w.r.t. time to obtain that second equation. Any help is much appreciated! $c(f_1+f_2)=a^2(\frac{1}{...
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3answers
21 views

What is the laplace transform of the following function

$f(t)= t^{10}\cosh(t)$ Is there another method that doesn't include differentiating the Laplace transform of $\cosh t$ ten times?
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1answer
24 views

Finding the Values of a Constant For Which A Curve has Local Maximum and Minimum Values

Use calculus to find the values of the constant $c$ for which the curve has local maximum and local minimum points. $g(x) = 4x^3 +cx^2 +10x$. Show that the graph always has one inflection point for ...
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2answers
34 views

Calculus Problem Regarding Voltage, Differentials, and Linearizations

I don't know how to begin solving this problem. If anyone could help me get the equations that I need and small hints, I would hugely appreciate it! The voltage across a resistor is given by $V(t) = ...
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0answers
22 views

Looking for analytical solution methods to PDE in radial coordinates (reaction-diffusion equation)

I am attempting to find a solution for a PDE (reaction-diffusion equation) in radial coordinates with a specific set of boundary conditions. I have been looking for a solution in literature in order ...
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1answer
45 views

Derivative of $\frac{1}{\sqrt{x-1}}$

I want to check whether if my way is correct or not, and why? \begin{align*} f'(x)&=\left(\dfrac{1}{\sqrt{x-1}}\right)'\\ &=\dfrac{-\left(\sqrt{x-1}\right)'}{\left(\sqrt{x-1}\right)^{2}}=\...
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0answers
24 views

To find class K infinity bounds on given radially unbounded function

Let $f = x_1^2 + x_2^4$. How to find class $\mathcal{K_{\infty}}$ functions $\alpha_1(||x||)$ and $\alpha_2(||x||)$, such that, $\alpha_1(||x||) \leq f \leq \alpha_2(||x||), \forall x \in {R}^2.$ $\...
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0answers
16 views

Problem on the Invertibily of a operator

Define $L:C^{1}_{p}[0,T]\to L^{1}[0,T]$ like this $L(y)=y'-qy$ where $q\in L^{1}[0,T]$. Since $e^{\int_{0}^{T}q(s)ds}\neq1$ prove that $L$ is invertible. Recall: $C^{1}_{p}[0,T]$ is the set of all ...
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0answers
15 views

What is the relation between De Sitter space and the Forward sheet of Hyperboloid Model?

The forward sheet of hyperboloid is given by, $-z_0^2 + z_1^2+ ... z_n^2 = -a^2$. The induced distance function is 1) $d(x,y) = arcosh(-\langle x,y \rangle)$. The de Sitter space is defined as - $-...
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1answer
13 views

Proving the definition of differentiability in higher dimensions.

Prove that a function $f:\mathbb{R^n}\to\mathbb{R^m}$ is differentiable at $p$ with $Df(p)=A\in M_{m\times n} \mathbb{(R)} \iff f(x)=f(p)+A(x-p)+r(x) $ when $r=o(x-p) $ Hi everyone. I am trying to ...
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1answer
19 views

How do we get the test for exactness in differential equations

If the solution to the differential equation $P(x,y)dx + Q(x,y)dy =0$ is $\Phi(x,y)=c,$ then that would mean that $$\frac{\partial\Phi}{\partial y}= Q(x,y)\hspace{1cm} (1)$$ and $$\frac{\partial\Phi}{\...

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