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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questions with no upvoted or accepted answers
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L'Hôpital's rule in topological vector spaces
Let $E$ be a (separated) topological vector space over $\mathbb{R}$, $f\colon [0,1]\to E$ continuous. Assume that for every $t \in (0,1)$ we have a derivative
$$f'(t) = \lim_{h\to 0} \frac{f(t+h)- f(h)...
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Continuous function nowhere Differentiable
Let $D(x):\mathbb{R}\rightarrow\mathbb{R}$ be:
$$ D(x)=\sum^\infty_{k=1}\frac{1}{k!}\sin((k+1)!x)$$
Prove that $D(x)$ is nowhere differentiable.
What I've done is that I supposed that exist a $x\in \...
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Solving a differential equation with a square root
I am trying to solve the differential equation
$
A(x)\frac{d^{2}f(x)}{dx^{2}}+B(x)\frac{df(x)}{dx}=\frac{1}{3}\frac{1}{\sqrt{f(x)}},
$
where
$
A(x)=\frac{x}{x+1}
$
and
$
B(x)=\frac{2x+1}{(x+1)^{2}}
...
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Exercise about Schwarz derivative
If $f$ is three times differentiable and $f'(x)\neq 0$, the Schwarz derivative of $f$ at $x$ is defined by $$\mathscr{D}f(x)=\dfrac{f'''(x)}{f'(x)}-\dfrac32\left( \dfrac{f''(x)}{f'(x)}\right)^2.$$ (a) ...
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A differentiable function satisfying an inequality
Let $f$ be a continuously differentiable function from $\Bbb{R}^n$ to $\Bbb{R}^n$ satisfying
$||f(x)-f(y)||\geq||x-y||$ for all $x,y$.
Can we conclude that f is an open map?
What about closed map?
...
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"Distorted" differential equation
Are there any results on differential equations of the type $y'(t)+y(g(t))=f(t)$? I am discarding Delay Differential Equations for which a big body of work already exists.
As an example, we could ...
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Exact Differential: Integrating Factor in Higher Dimensions
In two dimensions, we can turn every inexact differential $f(x,y)dx+g(x,y)dy$ into an exact version by multiplying both functions $f(x,y)$ and $g(x,y)$ by an additional function $I(z)$, i.e. $I(z) f(x,...
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Confused about the definition of tangent and conormal sheaf on a scheme
I have recently been learning about sheaves of differentials, and in particular the conormal sheaf. That is, the sheaf of differentials corresponding to the diagonal morphism. I have become rather ...
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If $\sum_ix^i\partial_if=0$, then $f$ is constant?
Let $f$ be a real-valued function defined on a neighborhood $D$ of $0\in\mathbb R^n$ - is it true that $\sum_ix^i\partial_if=0$ implies that $f$ is constant? Intuitively, I'd say yes:
Consider the ...
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Lie derivative of a coordinate form into its coordinate direction
I have to calculate the Lie derivative of $\alpha = d x_1$ with respect to the vector field $X = \partial_1$, but I cannot use Cartan's magic formula (which would immediatly show that $L_X\alpha= 0$).
...
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Meaning of the differential of a function, and its relationship to the derivative
After having sat through a semester of Calculus, (Physics-oriented), I decided to work on my basic math and tried to learn Real Analysis (from "Analysis Course Volume II",Lima, Elon. L). I ...
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Notation of Derivative and Differential
Let $\gamma:[a;b] \to U, \gamma(t)=\big(x(t),y(t)\big)$ be a path. The integral of $\omega$ along $\gamma$ is defined as
$$ \int_\gamma \omega := \int_a^b \omega_x\big(x(t),y(t)\big)\dot{x}(t)+\...
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Solving a nonlinear (vector or multi-variable) ODE
I am interested in solving the following differential equation:
$$\frac{d\mathbf{v}}{dt}=A\mathbf{w}, \qquad \mathbf{v}=\left[x,y\right]^T, \mathbf{w}=[x^2,y^2,xy]^T, A\in\mathbb{R}^{2\times 3}$$
...
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Cartan Differentiable calculus. Show $g(x,y)= \frac{f(x)-f(y)}{x-y}$ is differentiable at $(x_{0},x_{0})$
I'm doing problem 8 of Cartan Differentiable Calculus book. The problem says as follow:
Let $f$ assume its values in a Banach space $E$, an let it be of class $\mathcal{C}^1$ in an open interval $I$. ...
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prove the identity for a differentiable function
Let $f$ be a smooth real function and $f(0)=0, f (1)=1$, prove that exists various $x_1$, $x_2$ $ \in [0;1] $ such that $$\frac{1}{f'(x_1)}+\frac{1}{f'(x_2)}=2$$
I tried to use the mean value theorem ...
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Differential of Riemannian exponential map
Let $(M^n,g)$ be a Riemannian $n$-manifold. Let $p\in M$, and let $v\in T_pM$. By the existence and uniqueness theorem (of ODEs, hence of geodesics), there is a unique geodesic $\gamma$ on $M$ such ...
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Parametrization of a Cone such that $E=G$ and $F=0$
How would we parametrize a regular conical surface such that $\dfrac{x^2+y^2}{c^2} = z^2$ to have the first fundamental form $E=G$ and $F=0$?
I'm asking this so that we can ensure the existence of a ...
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Can you help me solve this 2nd order ode?!
$$\frac{d^2y}{dx^2}+\frac{1}{x}\frac{dy}{dx}-ye^{-x}=0$$ subjected to $y'(1)=-1$ and $y'(a)=0$. I was trying to fit it to the general bessel differential equation. But no luck. :( :(
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Have I shown $\nabla^2$ is invariant under rotation?
In a homework problem I was asked to show that Laplacian operator (3D) is invariant under rotation. The below is my reasoning, i don't know whether it is correct.
For the 3D rotation R, I can ...
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Why is $dx$ an infinitesimal?
In standard calculus, how can dx be an infinitesimal if infinitesimals do not exists in standard calculus
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Integration of $e^\frac{-s^2}{4k}$
currently I'm trying to exercise function integrals and got stuck at a particular function which I found on the internet:
$\int e^{\frac{-s^2}{4k}} ds$
Sadly, there is no solution online. This my ...
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answer
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Coupled second order differential equations
I would like to solve this coupled second order differential equations
$$x^2E''(x)+xE'(x)-(p_2x^2+p_0\nu^2)E(x)+s\nu \,xG'(x)=0,$$
$$x^2G''(x)+xG'(x)-(q_2x^2-q_0\nu^2)G(x)+s\nu \,xE'(x)=0,$$
where $...
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answers
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"Division" of an inexact differential form by an exact differential form
In thermodynamics I have come across a peculiar expression whose mathematical nature eludes me. Maybe one of you can help me finding out. First of all let us assume that in this scenario all premises ...
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Understanding the Jacobian past calculus
What's taught in calculus:
In the calculus of multiple variables I learned that the Jacobian
$$\textbf J=\frac{\partial(x_1,\ldots,x_n)}{\partial(t_1,\ldots,t_n)}=\left(\begin{array}{ccc}\frac{\...
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Combining two differential equations
I have two differential equations that are connected by an equation,
$L_1\frac{d^2I_1}{dt^2} + \frac{1}{C_1}I_1=\frac{dV}{dt}$
$L_2\frac{d^2I_2}{dt^2} + \frac{1}{C_2}I_2=\frac{dV}{dt}$
$I_1+I_2=I$
...
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Differential of the log of a SPD matrix
I am interested in finding the differential of the logarithm of a SPD matrix, where each component can be taken potentially in a different basis, i.e., for a diagonal matrix $D$, let $\log_\alpha(D) = ...
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How do I find the constants in this differential equation that describes the time taken for a bubble to rise to the surface.
I'm trying to model the behaviour of bubbles in a water column as accurately as I'm able to, and I modelled a Differential equation using F = ma and a free body diagram. The equation of motion I ...
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How to treat Lie-Algebra valued differential forms in a variational calculation
In the WZW model, you have a 2 dimensional manifold $\Sigma$ and it's 3 dimensional extension $\tilde{\Sigma}$. In the WZW action, there is a term of the form: $S_{1}[g] = \alpha\int_{\tilde{\Sigma}} ...
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How do I find A in $y = Ax^2 + x + 7000$ (differential calculus, Leibniz's notation)
So the problem I'm trying to solve is prefaced with this:
Earlier we mentioned that NASA claims that the Vomit Comet can make passengers experience weightlessness for about 25 seconds. Let’s check on ...
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Shortest path connecting two points in the plane constrained by tangents at endpoints, maximum curvature and change of curvature.
I have two points and corresponding tangents in the plane. I'm looking for the shortest path that connects the points and is aligned with the tangents at these points. At any point on the path, the ...
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Suggested books on Leibniz notation (differentials)?
I'm a freshman in Computer Engineering (but will probably switch to Math very soon) and I've followed some course in Analysis 1 and 2. I found the concept of differentials really interesting, but I've ...
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Differential of Determinant - Differential Geometry
Working from the definition:
Let $F:M \to N$ a smooth map between smooth manifolds , for each $p\in M$ the function
$$ dF_p : T_pM \to T_{F(p)}N$$
is called the $\textit{differential of $F$ at $p$}$. ...
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views
Polar coordinate differentials
We all know that when switching to polar coordinates we have:
$$
x = r\cos\theta,\qquad y = r\sin\theta
$$
and either by a geometric argument or using the Jacobian we have:
$$
dx\,dy = r\,dr\,d\theta
$...
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votes
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answers
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Is there a limit definition and english definition of $\text{d}x$?
Is there a limit definition of a differential? I came up with this but I would like some feed back.
\begin{align*}
\text{d}x & = \lim_{x \to c}(c - x)\\
\text{d}x & = \lim_{\Delta x \to 0} \...
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Inconsistent solution to nonlinear first order differential equation.
I am trying to solve the following first order differential equation,
$$
y'(x)=-\frac{a}{x}-\frac{b}{y(x)}
$$
with boundary condition of,
$$
y(c)=d
$$
The above equation is the Abel's equation ...
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In the differential $da_{ring}=\pi(r+dr)^2-\pi r^2 = 2\pi r dr+\pi(dr)^2$ why can we ignore $(dr)^2$ as $dr \to 0$?
Consider a thin uniform disc of mass M and radius R.
I am interested in the calculation of moment of inertia about an axis passing through the center of mass, perpendicular to the disc, specifically ...
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How do I prove the following differential equation is homogenous of degree 0?
I understand that if I have a differential equation that I can write in the form of
$$
A(x,y)dx+B(x,y)dy=0
$$
, where $A(x,y)$ and $B(x,y)$ are both homogenous functions of the same degree, then I ...
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votes
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answers
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Change of Coordinates - John Lee's Smooth Manifolds, eq. 3.11
I am trying to understand John Lee's derivation of equation 3.11, which shows how tangent basis vectors on a smooth manifold are transformed under a change of coordinates.
He starts with the following ...
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answers
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Arguments to show that $\lim_{\iiint d\tau\to 0}\frac{\iint d\vec{\sigma}\times\vec{V}}{\iiint d\tau}=\nabla\times\vec{V}$
So, I was solving the problem 1.10.6 from "George B Arfken, Hans J Weber - Mathematical Methods For Physicists- Sixth edition" and the problem was to show that:
$\lim_{\iiint d\tau\to 0}\...
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Legendre Polynomials proving a relation using Rodrigue's Formula
I have been trying to prove the relation$\left (P'_n(1)=\frac{1}{2}n(n+1)\right )$ and have proved it directly from the Legendre differential equation as $P_n(x)${the Legendre polynomial} is the ...
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Find a parametric equation of a plane curve
The task is to find a parametric equation of a curve, if you are given its curvature in arclength parametrisation. I know that I need to integrate but I don’t get the idea of angle theta(what’s it?) ...
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How to solve this Smoluchowski PDE?
I am trying to solve the Smoluchowski equation for diffusing particles under the influence of a potential $W(x)$.
$$\frac{\partial P(x,t)}{\partial t} = D\frac{\partial^2 P(x,t)}{\partial x^2} + \beta ...
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votes
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answers
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Deriving a method of solving Pursuit curves
I am currently studying ODEs and became interested in pursuit curves. However I am having trouble simplifying the equations into solvable DEs. Here is my work:
If mouse (m) moves along a curve then ...
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votes
1
answer
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Submanifold separates the manifold
I believe that if we delete the small neighbourhood of a submanifold with codimension more than 2 from the ambient manifold, it won’t separate the manifold. Is there any topological argument I can use?...
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Strange ordinary differential equation
I am new to the world of differential equations. I found this strange one on a PDF online. The excercise itself says that "Wolfram-Alpha won't help" and I cannot find any explanatory solution of it. ...
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What is the $\int{w^a(1-w)^b\mathrm d w}$?
I recently came across the $$\int{w^a(1-w)^b\mathrm d w},$$ which looked ridiculously simple at first, but I subsequently discovered that I could not reduce it to some elementary form.
To clarify, I ...
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Second order approximation in rocket equation
The simple rocket equation with no external forces can be derived in multiple ways.
Assume a rocket with mass $m$ and velocity $v$ at time $t$ that looses the mass $-dm > 0$ with relative ...
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Doubt about differentia operatorl in polar coordinates
The differential operator $d$ can be written as $$d=\partial_i dx^i$$ where $\partial_i\equiv\frac{\partial}{\partial x_i}$. If I have $d$ expressed in cartesian coordinate and I want to obtain the ...
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What does 'els' mean?
What would you understand by the term 'els' used in. $$B_1, B_2 \textrm{ els} Q(z).$$ The context is differential algebra and the Risch integration algorithm. My best guess is 'elementary elements of' ...
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Solution of Differential equation as an integral equation
I was looking for the solution of the following problem.
Prove that if $\phi$ is a solution of the integral equation
$$y(t) = e^{it} + \alpha \int\limits_{t}^\infty \sin(t-\xi)\frac{y(\xi)}{\xi^2}d\...
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