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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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Differential Equations riddle: $f=(f’)(f’’)(f’’’)(f’’’’)\dots$ [closed]

$$f=(f’)(f’’)(f’’’)(f’’’’)\dots$$ I found this question someone posted in a group chat, and no one has solved it yet
Brian Li's user avatar
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Taking the Differentials of $u = x+y$

To solve $\int_{}^{}\cos(x+y)dx$ by substitution, we use $u = x+y$ to get $\int_{}^{}\cos(u)du = \sin(u) = \sin(x+y)$. This means that $du = dx$, my question is why? How do we take the differentials ...
Gavin Chan's user avatar
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Order reduction problem with one solution [closed]

how can i solve this edo: $y'' + \frac{3y'}{x} = 0$ with the solution : $y_1(x) = 1$ and i have to use order reduction
Yan Lacerda's user avatar
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A question about the definition of derivative in different coordinate systems

The definition of the derivative goes like this: If $x$ is an interior point of a set $E \subseteq {\Bbb R}^n$, then a function $f: {\Bbb R}^n \rightarrow {\Bbb R}^m$ is said to be differentiable at $...
WhyNót's user avatar
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Determinant form of Curl of a vector field on a differentiable manifold

I was trying to compute the curl of a vector field on a differentiable manifold in terms of a determinant like in calc 3. I want to show $\nabla \times v=\frac{1}{\sqrt{det(g)}}\det\begin{pmatrix}\...
Alexander C's user avatar
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Calculate differential $dF_j$ of $F:j\mapsto j^2+I$ on $O(n)$

Given the map $F:j\mapsto j^2+I$ on the orthogonal group $O(n)$, what is the differential $dF_j$ ? How do I calculate this? I am trying to understand Example 7 in Lecture Notes on Symmetric Spaces by ...
Andrius Kulikauskas's user avatar
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Comparing the definitions of Derivative in Guillemin Pollack and in other differential topology books.

In standard differential geometry and topology books I have seen the authors defining the derivative/differential in the following way: Let $f:M\to N$ be a smooth map between two smooth manifolds.Then ...
Kishalay Sarkar's user avatar
1 vote
2 answers
45 views

Computations in coordinates of Hamiltonian vector fields

I just need someone to snap me out of a (hopefully small) misunderstanding: in p.574 of (the 2nd edition of) Lee's Introduction to Smooth Manifolds it is written that a Hamiltonian vector field $X_f$ ...
Sam's user avatar
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Differential Approximation of $f(x,y)\:=\:x\cdot \:e^{x^2-y^2}$ - Multivariable Calculus Problem

I have a differential approximation of a multivariable calculus problem of my academy that I can't figure out. Here's the question: Given the result of the calculation of $3.02\cdot e^{3.02^2-2.9^2}$, ...
Yuval Yanay's user avatar
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How to show $x'''(y)=\frac{3y''(x)^2}{y'(x)^5}-\frac{y'''(x)}{y'(x)^4}$?

The question states that you have to show: $$\frac{d^3x}{dy^3}=\frac{3}{(\frac{dy}{dx})^5} (\frac{d^2y}{dx^2})^2 - \frac{1}{(\frac{dy}{dx})^4} \frac{d^3y}{dx^3}$$ I have tried so far to rewrite them ...
Deimos's user avatar
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2 votes
1 answer
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does dx equals to (x+h)-x? If it is, why isn't it explained like this?? [duplicate]

While I was looking at a question regarding derivatives, I suddenly got enlightened when I realized, $dx=\lim_{h\rightarrow 0} (x+h)-(x)$ I noticed this while considering on the equation $\frac{df(x)}{...
Emin Bedir's user avatar
2 votes
1 answer
58 views

Chain rule for Clarke-derivatives

The Clarke-gradient is often introduced to extend ideas from convex analysis to non-convex functions, see [Clarke, Sec 2.1]. In particular, given $f:\mathbb{R}^n\rightarrow \mathbb{R}$ Lipschitz in $x$...
Bazinga's user avatar
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Clarke regularity of piecewise linear functions

Let $f:X\rightarrow\mathbb{R}$, with $X\subseteq\mathbb{R}^n$, be a Lipschtiz continuous piecewise linear function. Is this sufficient to guarantee that $f$ is Clarke-regular (Definition 2.3.4 in [1])...
Bazinga's user avatar
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1 answer
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Studying stability of pde

I have a problem with studying the stability of this PDE. $$U_t = U_{xx} + f(U).$$ Let $U^{*}(x)$ be a solution for this equation. As conditions we have ${U^{*}}'(x) > 0$ for $x < x_{0}$, ${U^{*}...
Dan's user avatar
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How to compute the differential of a function

Let $f(x,y)=x^2$ on $\mathbb{R}^2$ and let the vector field be \begin{equation} X=\text{grad}f=2x\frac{\partial}{\partial x} \end{equation} Compute the coordinate expression for $X$ in polar ...
Superunknown's user avatar
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Calculating the Derivative of a Complex Vector Function

I am trying to calculate the derivative of the function $$f(x) = x^T a (x^T a)^* = x^T a x^H a^* = x^T a a^H x^*$$, where $(.)^T$, $(.)^H$, and $(.)^*$ represent the transpose, Hermitian (conjugate ...
Omid Abasi's user avatar
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Differential 1-form with vanishing z-component function

Let $\omega $ be a closed 2-form in $\mathbb R^3$. Prove that there is $\eta\in \Omega^1(\mathbb R^3)$ of the form $\eta=\eta_1dx+\eta_2dy$ such that $\omega =d\eta$. I know that if I write $\omega =...
Runyang Wang's user avatar
1 vote
1 answer
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The intuition behind the differential/pushforward

First, I apologize for any english mistakes, it's not my first language and it has been while since the last time that I needed to write in english. To get on the core of my question, lets recap the ...
Paulo Estêvão's user avatar
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1 answer
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Differential of a Smooth Map Notation in Lee's Introduction to Smooth Manifolds

I am reworking through Chapter 3 of Lee's Introduction to Smooth Manifolds, and am looking at proposition 3.8: Let M be a smooth manifold with or without boundary, $p \in M, v \in T_pM$. If $f,g \in ...
PotusOtis's user avatar
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About the chain rule of the exponential entropy

In the paper unifying framework of information measures the conditional exponential entropy (see equation 29) is defined as: $\mathcal{E}_{\alpha}(X|Y) = E_y\left(\int_{\mathbb{R}} f^{\alpha}(x|y)\,...
Upax's user avatar
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Using differentials to calculate error isn't precise

Exercise: The radius r of a circle is measured with an error of at most 2%. What is the maximum corresponding percentage error in computing the circle’s area. So what we know is that $\frac{dr}{r}=0....
Stanislav Bashkyrtsev's user avatar
-1 votes
1 answer
31 views

Integrand must equal zero [closed]

Apprently I have to write some addtional wordage to give "some context" to this question.... as follows... In my fluid dynamics, "Notes on CFD, General Principles, Greenshields" on ...
Nick's user avatar
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3 votes
1 answer
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differential equations, substitution suggested by the equation

I took my examination in differential equations earlier; i would've gotten a perfect score, but i tripped in this problem: $$(1+5y\sin x)dy + y^4\cos xdx = 0$$ I used substitution suggested by the ...
lawrencium21's user avatar
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Interpreting $\int_\gamma f = \int P\ \text{d}x + Q\ \text{d}y$ via differential forms?

I've often seen the expression $\def\g{\gamma}\def\dx{\text{d}x}\def\dy{\text{d}y}\def\td{\text{d}}\def\br{\textbf{r}}\def\dt{\text{d}t}\def\RR{\mathbb{R}}$ $$\int_\g f\ \td\br = \int P\ \dx+Q\ \dy$$ ...
Sam's user avatar
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prove a property about absolutely continuous function

Suppose $f:[a,b]\to \mathbb{R},f \ is \ absolutely \ continuous \ function,$ Prove:$$If \ m(Z)=0,then \ m[f(Z)]=0,"m" \ denotes \ the \ Lebesgue \ measure,Z\subseteq[a,b] $$(I am not sure it is ...
MathNoob's user avatar
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1 answer
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Approximating solution of a second degree ODE

Consider the second-order DE $y'' + p(x) y = 0$, such that $\int_{}^{\infty} t|p(t)| dt < \infty$. Show that, for any solution $y(x)$, $\lim_{x\to\infty} y'(x)$ exists, and every nontrivial ...
R_Squared's user avatar
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0 answers
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Why is this construction of the differential coordinate-invariant?

In Lee's Smooth Manifolds the differential of a function $f\in\mathcal{C}^\infty(M)$ is defined as covector field $$df:v\mapsto vf$$ where $v$ is some element of the Tangent space at a point on $M$. ...
John Doe's user avatar
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Why general solution of system of differential equations or simultaneous equations are given in pair of relation?

So i am confused or interested in knowing why general solution for system of differential equation ( say system of 2 equation) is given by 2 ralations or a pair say u1(x,y,z)=c1 and u2(x,y,z)=c2 . Is ...
kaushal trada's user avatar
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1 answer
37 views

Solving PDE using Lagrange's equation and multipliers such that the numerator is differential of the denominator

The given PDE is $$y^2(x-y)p+x^2(y-x)q=z(x^2+y^2)$$ I know that the the lagrange equation will be $$\frac{dx}{y^2(x-y)}=\frac{dy}{x^2(y-x)}=\frac{dz}{z(x^2+y^2)}$$ And that the solution will be $F(u,v)...
Rafae Farrukh's user avatar
1 vote
0 answers
21 views

Differential manifold connection with different notation

We aim to prove that the set {V(n, 2) = {(a, b) \in R^n \times R^n : |a|^2 = |b|^2 = 1 \text{ and } <a,b> = 0} is a smooth (2n-3)-submanifold of R^n \times R^n. This will be demonstrated by ...
Russell Hua's user avatar
1 vote
0 answers
12 views

Differential Manifold about constructing an disjoint ball segments

Let $B \subset \mathbb{R}^n$ be an open ball in $\mathbb{R}^n$. Find a $C^\infty$ function $f : \mathbb{R}^n \rightarrow \mathbb{R}$ such that: $f^{-1}(0, \infty)$ and $f^{-1}[0, \infty)$ are $B$ and ...
Russell Hua's user avatar
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1 answer
31 views

Prove that a function is differentiable and its derivative function is integrable

Suppose $f:D \subset \mathbb{R} \to \mathbb{R}$ is continuous and satisfies the Lipschitz condition,that is $$\exists M>0, \forall x,y\in D:|f(x)-f(y)|\leq M|x-y|.$$ I want to know whether $f'(x)$ ...
MathNoob's user avatar
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1 vote
1 answer
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Need Help with Particle Motion Analysis in $\mathbb{R}^2$

I'm currently working on a problem related to particle motion in the plane $( \mathbf{R}^2 )$ and could use some guidance. The particle moves along a curve, and at any given time $( t )$ seconds after ...
tarek hankir's user avatar
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1 answer
31 views

Identifying Hidden Initial Values in Integral Equations: Insights Needed

I've been pondering over a question sparked by two math problems: How can one assert from the outset that a problem contains hidden initial values, and what methods can be employed to discern this? My ...
郑远程's user avatar
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0 answers
59 views

Is f(x)dx in integral a differential?

Is $dx$ in $\int f(x)dx$ a differential of the identity function? In Zorich analysis, it is said that if $dF = f(x)dx$, then $\int dF=\int f(x)dx$. This only makes sense if $dx$ actually represents a ...
juekai's user avatar
  • 155
3 votes
1 answer
75 views

How to calculate the value of the n-th derivative of this function at this point

$$\forall n\in \mathbb{N},\left.\frac{d^{2n+1}}{dx^{2n+1}}(x^{\ln x})\right|_{x=e^{\frac{n}{2}}}=0$$ How to prove this equation? I find that if $n=0$ or $n=1$, the answer is $0$. But I can't prove ...
MathNoob's user avatar
  • 331
0 votes
2 answers
167 views

Counterexamples about the differentiability of several variables

I've learned about differentiability of several variables. If $f(x,y)$ is differentiable then we can use chain rule on it. But I suspect the converse of this proposition is not right. So, is there a ...
Andrews's user avatar
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0 answers
27 views

Pushforward of canonical form

I am currently reading Arkani-Hamed et al. on Positive Geometries and Canonical forms (https://arxiv.org/abs/1703.04541). There, they define the pushforward of a canonical form by The canonical forms ...
max_121's user avatar
  • 769
1 vote
1 answer
45 views

Find the equation of the tangent to a system using implicit function theorem

Here is my system of equations: $C: \begin{cases}x^2 + y^2 +z^2 = 14\\ x^3+y^3+z^3=36 \end{cases}$ Firstly I managed to show that for all $a \in C$, the implicit function theorem applies to express $...
Alex's user avatar
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0 answers
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$\left(y+y \sqrt{x^{2} y^{4}-1}\right) d x+2 x \ d y=0$ differential equation

I am solving the differential equation $$\left(y+y \sqrt{x^{2} y^{4}-1}\right) d x+2 x \ d y=0$$ and I have to use the substitution $u=xy^{2}$ to find the generic solution: I have expressed $$y'=dy/dx ...
Allegrina's user avatar
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0 votes
1 answer
53 views

Identity regarding the differential of a smooth action of a Lie group

Hey I am currently following a course on Lie groups and I have followed a course on smooth manifolds. The question I have is the following, Let $G$ be a Lie group acting smoothly on a manifold $M$ by ...
user avatar
2 votes
1 answer
38 views

Calculate the differentiate of $ Q: \mathbb{R}^n \rightarrow \mathbb{R}$ defined by $Q(x)=B(x;x)$

I have a question on the differentiable of a function and I want to be sure that my understanding of this concept is correct. Question: Calculate the differentiate of $ Q: \mathbb{R}^n \rightarrow \...
OffHakhol's user avatar
  • 719
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1 answer
51 views

Why can we change indices when defining covariant derivative formula?

For $ \overrightarrow{V} = V^i \overrightarrow{e_i} = V_i \overrightarrow{e^i} $, $ \frac{\partial \overrightarrow{V}}{\partial x^j} = \frac{\partial V^i}{\partial x^j} \overrightarrow{e_i} + V^i \...
posfn0319's user avatar
1 vote
1 answer
34 views

Cancelling terms out of differential equation

I haven’t had my differential equations class yet and I’m not sure if it’s mathematically okay to do this (originally I’ve obtained this equation from some thermodynamics law but it’s not SUPER ...
Emmannuelle_Legolas's user avatar
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0 answers
48 views

Exact an Inexact Differntials.

How can the addition of two(or more) inexact differential give an exact differential? Moreover, if an exact differential represents a linear map, what does an inexact differential represent(...
Aditya Krishna Panickar's user avatar
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0 answers
74 views

Existence and unique solution to a linear PDE

I'm doing an exercise with no solution, the question says $u_x+xu_y=0$ for $x,y\in \mathbb{R}$, with initial value $u(x,0)=f(x)$, where $f$ is a real function. Now the question ask me to impose some ...
kkk's user avatar
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0 votes
0 answers
23 views

Is the differential of an integral an even or odd function?

If I have an integral of the type $\int \,d^3x$, is $\int \,d^3(-x)=-\int \,d^3x$? Does it even make sense to ask something about this? I was wondering about this but I can't understand if it is a ...
Salmon's user avatar
  • 315
1 vote
2 answers
107 views

Solve this PDE $xu_x+yu_y=u$ with "condition" $x=\cos t, y=\sin t, u=1$ (problem with condition/particular solution)

Find the particular solution for the part differential equation $$xu_x+yu_y=u \\ x=\cos t, y=\sin t, u=1$$ It's the first time I've encountered this PDE with this type "condition" (and I don'...
Gregory99's user avatar
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0 answers
31 views

Explicit example of differential forms in manifolds

Let $\omega=xdx\wedge dz-zdy\wedge dz$ and $X=f\frac{\partial}{\partial x}+g\frac{\partial}{\partial y}$, for some $f,g\in C^\infty(\mathbb{R}^3).$ I want to compute $\omega(X,\frac{\partial}{\partial ...
hugh_maths's user avatar
3 votes
1 answer
53 views

Right Differentiability of $\|f(x)\|$

Problem: Let $E$ be a normed space, and $f:[a,b]\to E$ be continuous. Define $g:[a,b]\to\mathbb R$ by $g(x)=\|f(x)\|$. Prove that if $f'_{+}(t_0)$ exists for some $t\in[a,b)$, then so does $g'_{+}(t_0)...
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