# 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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### Prove $\delta L(y;v) = L(v)$, for all $y, v$ belong to $y$. [closed]

I need help for the following exercise: If $L:\mathcal Y\to R$ is a linear function on a linear space $\mathcal Y,$ prove that $$\delta L(y ;v) = L(v),\forall y,v\in\mathcal Y.$$
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### Differential of a linear map

I need help for the following exercise: If $L:\mathcal Y\to R$ is a linear function on a linear space $\mathcal Y,$ prove that $$\delta L(y ;v) = L(v),\forall y,v\in\mathcal Y.$$
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### Change of 'role' of variables in an ODE

I would appreciate some help understanding a variable change in an equation a textbook I'm reading is working through. The book is trying to illustrate the role of changing the roles of x and y in a ...
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### What does the absolute value sign mean when used around the differential part of an integral?

The radon transform wikipedia shows an equation like $$\int_L f(\mathbf{x})|\mathbf{dx}|$$ What does absolute value around $$|\mathbf{dx}|$$ mean in an integral? I understand that for the Radon ...
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### How to solve this problem integer? [closed]

$$\int_0^{s}z^{1-2\alpha}(z_s)^qds$$ or
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1 vote
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### Motivation and intuition about differential forms [duplicate]

So I have tried to get motivation behind the formal definition of differential forms and that what I understood and I want to make sure that I’m on the right track: So we want to integrate over a ...
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1 vote
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### Solving a differential equation $y''+2y'=x$

I've been stuck on this equation. $y''+2y'=0$ is fairly simple, the general solution comes out as $y=c_{1}+c_{2}e^{-2x}$. But when I try to solve the particular equation including the $x$ on the right ...
1 vote
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### Deriving a composed matrix function [closed]

I'm interested in deriving the following function, with respect to the matrix $W$, \begin{align} R\left({G}; H\right) = \log \bigg|{I} +s \cdot \big({G}^H{G}\big)^{-1}{G}^H{H}{H}^H{G}\bigg|. \...
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### Derivation of Euler-Lagrange's equation by Susskind : " the change in x_i when I change v_i a little bit is $1/ \epsilon (= 1/ \Delta t)$".

Pr. Susskind tries a " easy" derivation of Euler-Lagrange's equation in this video : https://www.youtube.com/watch?v=3apIZCpmdls&t=4086s . His method is to turn the equation for the ...
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### Why can I integrate two differentials on the two side of an equation? [duplicate]

Suppose there is an ode: $$\frac{dx}{dt} = a(t)x$$ Then we can solve it by following steps: $$\frac{dx}{x} = a(t)dt$$ integrate on both side and we get: $$\int \frac{dx}{x} = \int a(t)dt + C$$ ...
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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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### A different (more intuitive?) definition of the tangent space and the differential. Why are they not used?

Probably a silly question, but the following seem to me intuitive definitions to make: Throughout let $M$ and $N$ be smooth manifolds of dimension $m$ and $n$ respectively. Definition: the tangent ...
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### Clarification of some concepts - calculus of multiple variables

I am reading from two books a few definitions and a few theorems related to differentiability, differential, directional derivative, etc., and I got a bit confused so I want to clarify if I am ...
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### Element in normalizer of Lie algebra gives rise to differential with invariant subspace

I'm studying Lie Algebras and Lie Groups from the book "Lie algebras and Lie groups" by Bourabki Nicolas. There he claims that if $H$ is a connected closed subgroup of a Lie group $G$, then ...
1 vote
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### Solution to an ODE in the form $y'' + \frac{m}{mx+b} y' + k(mx + b)y = 0$ [closed]

I am trying to find a solution to a second order ODE in the form $y'' + \frac{m}{mx+b} y' + k(mx + b)y = 0$ where $m,a,b$ are constants. I cannot find a method to solve such form. Any tips would be ...
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### Derivating a polynomial equation from two others

In an old article of J. Cockle, Sketch of a Theory of Transcendental Roots, pages $146$/$147$, the author obtains an expression for a cubic equation I cannot reproduce. This equation, when you solve ...
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1 vote
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### In a differential equation, would you call "y" a variable or a function?

In a differential equation, would you call "y" a variable or a function? Or can you call it both things?
1 vote
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### Differential of a function as a limit

Based on intuitive definitions of the differential of a function, it seems to me that for $f:\mathbb{R}^n \to \mathbb{R}^m$ something like $$df(a) = \lim_{|r| \to 0} \frac{|f(a+r)-f(a)|}{|r|}$$ ...
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### Limits in Z transforms time delay property

In my university the z transfrom's time delay property also called shifting property is shown as follows $$Z[f(k - n)] = z^{-n}F(z) = z^{-n} \sum\limits_{k=-\infty}^{\infty}f(k)z^{-k}$$ but according ...
1 vote
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### Definition of Variables in Fluid Flow Differential Equation?

I'm a student who is familiar with Calc 2 and is starting to learn MultiVariable calc. I came across this article which seemed interesting for Fluid Flow: http://www.ipt.ntnu.no/~kleppe/TPG4150/...
1 vote
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### local diffeomorphism.

Let $x: U \subset R^2 \to S$ be the parametrization of a surface of revolution S: $$x(u,v)=(f(v)cos(u),f(v)sin(u),g(v)), f(v)>0$$ $$U=\{(u,v) \in R^2; 0<u<2\pi,a<v<b\}$$ a) Show that ...
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### Relation between flow and exponential map

In Proposition 1.7.12 in Hamilton's Mathematical Gauge Theory he states Let $G$ be a Lie group and $X$ a left-invariant vector field. Then its flow $\phi_t(p)$ through a point $p \in G$ is defined ...
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### Unsure of an implicit differential equation

I don't want a full solution per se, just an appropriate hint/guidance to nudge me in the right direction if possible. The problem at hand: $$(dy+dx)^3 = 27(x+y)^2(dx)^3$$ Thus far I've tried using a ...
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### How do I compute the differential of the Gauss map?

Let $\sigma:V\rightarrow S$ be a regular patch. Then for all $p=\sigma(u_0,v_0)$ we define the positive normal vector \textbf{N}_p:=\frac{\sigma_u\times \sigma_v}{\|\sigma_u\times \sigma_v\|}(u_0,...
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### second-order total differential

Let $f$ be a twice-differentiable function. Find the second-order total differential of the function $\phi(x, y, z) = f(u)$ if $u = xyz$. (Please use $f'(u), f''(u)$ signs.) So what I was trying to do:...
1 vote
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### Is the set $\{ x,y \in \mathbb{R} \,|\,x^3+y^3 = 1\}$ a manifold?

I think $\{ x,y \in \mathbb{R} \,|\,x^3+y^3 = 1\}$ is a manifold in $\mathbb{R}$, specifically using the function $f:\mathbb{R^2}\to\mathbb{R}$ defined as $f(x,y)=x^3+y^3 -1$. It has the property ...
1 vote
Suppose $\Omega \subset \mathbb{R}^{m}$ is open and let $f: \Omega \to \mathbb{R}$ be differentiable. The derivative of $f$ at $x$, denoted by $df(x)$, is a linear functional, and it can be written as:...
### Finding special feature in a solution of a differential equation $y'' = x^{2}y$
Given a differential equation $y'' = x^{2}y$ and its solution with $y(0)=1$ and $y'(0)=0$, how can I prove that this $y$ is an even function and has $y(x)>0$ for all x?