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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Maths differential calculus Taylor series [closed]
Let $S$ be the set of all continuous functions $f:[- 1, 1]\to R$ satisfying the three conditions:
1)$f$ is infinitely differentiable on the open interval $(-1,1)$
2)The Taylor series
$$f(0) + f'(0)x + ...
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confusion over statement: If $x' = f(x)$, $f$ is continuously differentiable, then all solutions are monotone or constant.
Here's the statement:
If $x' = f(x)$, an autonomous scalar dynamical system for $x\in\mathbb{R}$, where $f$ is continuously differentiable, then all solutions are either strictly monotone or constant....
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Prove the Form of the General Solution to a Linear Second Order Nonhomogeneous DE
I was trying to understand where the general solution for a linear second-order nonhomogenous differential equation comes from and I don't understand how/why $$Y1 - Y2$$ can turn into $$y(x) - Y_p(x)$$...
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Differential Equation of Parabolas with axis parallel to x-axis and strophoids $y^2=\frac{x^2(a+x)}{a-x}$ [closed]
Obtain the differential equation of the family of plane curves described.
Circles of radius unity. Use the fact that the radius of curvature is $1$.
Parabolas with axis parallel to x-axis.
The ...
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Optimal control propblem? How to formulated the parameters of following equation?
Recently, I have meet an interesting question? How to let the objective function must be non-negative? Here we have several parameters, for example,
$$\dot x(t)=u_1+u_2$$
$$k_1(x_0,u_1,u_2,T-t_0)=\...
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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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How to use different method to compute Laplace inverse
I was trying to find the inverse of the Laplace transform, which is $L^{-1}(\frac{3s+5}{s^2(s^2+4)})$.
I used Partial Fractions to compute and I got $\frac{3}{4}+\frac{5}{4}t-\frac{3}{4}\cos(2t)-\frac{...
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$\lim_{x \to 0}\frac{\sin^{-1}(x)}{\tan^{-1}(3x)} = \frac{1}{3}$ using epsilon delta definition of limit
To show that: $\lim_{x \to 0}\frac{\sin^{-1}(x)}{\tan^{-1}(3x)} = \frac{1}{3}$ using epsilon delta definition of limit
Attempt:
$ |f(x) - \frac{1}{3}| = |\frac{\sin^{-1}(x)}{\tan^{-1}(3x)} - \frac{1}{...
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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 ...
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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 ...
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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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Wronskian - independent solutions
Question based on Example sheet 3, Problem 2 from here: https://dec41.user.srcf.net/notes/IA_M/differential_equations_eg.pdf
Could somebody tell me if there's a mistake in the initial conditions for $...
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Integrating Differentials of Two Variables
In a thermodynamics text I'm reading, the author says the following,
"The integral of the perfect differential $dz = y\,dx + (x+c) \,dy \; (\mathrm{where} \; z = xy + cy)$ from point (0,0) to ...
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Differential Equations Problem Population with Birth and Death Rates [closed]
The death rate in a colony of bees is proportional to the population, and without any new births, half of the bees would die in 20 days. However, the actual population of bees doubles in 30 days. ...
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First order distribution differential equation
I am struggling to solve this differential equation where T is a distribution.
$$ T' -T = \delta_0$$
I know a method to solve differential equations of the form
$ T' +bT = T_f$ where $T_f$ is a ...
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Orientation preserving diffeomorphism $\iff$ positive determinant
I am stuck on a step in a proof, so I will write out the statement and the proof (it is not too long). I would like someone to explain the last 2 steps of the proof, if possible.
Statement: Let $U,V \...
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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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How to prove the existence of maximum and minimum of this set?
In Analysis I by Herbert Amann, Chap IV, Sec 2, Remarks 2.2 (b), there is a proposition as follows:
Let $f$ be continuous on $[a,b](\subseteq\mathbb{R})$ and differentiable on $(a,b)$. Then
$$\max_{x\...
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about a PDE question
imagine a D-sphere in a D dimensional Cartesian coordinate system, $r^2=\sum_{i=1}^{D}(x^i)^2$. Differentiating the aforementioned equation we have
(1): $rdr = \sum_{i}x^idx^i$
(2): $\frac{\partial{r}}...
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Reversible Function Composition
Based on the previous questions discussed in the link provided Find $g(x)$ midpoint.
Given $f_k(x)$, such that $f_k(x)$ is reversible and the graph that was generated by the points $(x,f_k(x))$ is the ...
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Differential Equations: What is the difference between a linear and nonlinear centre?
In 2nd year differential equations, we define a centre to be the graphical phenomena we witness around a critical point with complex eigenvalues in form $\pm bi$.
But, we also throw around this term &...
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What kind of object is the (second) exterior derivative of a moving point $R$ in $\mathbb R^n$? What does $dR\wedge\omega \mathbf e$ mean?
TL;DR: Given $f:\mathbb R \to \mathbb R^n$ smooth enough, what are $df$ and $d^2f$ and how are they computed wrt possibly moving bases $\{e_i(t)\}$? I (believe that I) understand the answers in the &...
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Derivative of a trace (Graph regularization) with Hadamard product
Let us assume that $A,M,L\in \mathbb{R}^{n \times n}$. The symbolic $ \circ $ represents Hadamard product. I am trying to partial derivative the following expression:
$$
F=Trace((A \circ M)L(A \circ ...
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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?
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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 ...
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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/...
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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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Solving the differential $\frac{y'y'''}{y''} = x$
I've been trying to solve the differential equation: $\frac{f'(x)f'''(x)}{f''(x)} = x$, $x\in \mathbb{R}$
My initial attempt was to integrate by parts, something like this:
$$\int_{}^{} \frac{f'(x)f'''...
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How is the E coefficients calculated or generated for the Radau IIA 5th Order?
With regard to Scipy, the E coefficients are written out. I've seen this from different sources as well. However, I cannot find any reference to how the E coefficients were computed.
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Inverse Jacobian vs Pullback
Part of inverse Function Theorem tells us that for
$f$: M $\rightarrow$ N, $J(f^{-1})( \textbf{y}) = [(Jf)(f^{-1}( \textbf{y}))]^{-1}$.
That is the Jacobian of the inverse function at $ \textbf{y}\in ...
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Tractrix as catenary evolvent
I’m trying to show that the involute of a catenary, calculated to be
$$\gamma(t)=(t-\tanh(t),\cosh(t)-\sinh(t)\tanh(t))$$
is a reparametrization of the tractrix
$$\beta(s)=\left(\cos(s)+\log\left(\tan\...
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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:...
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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 ...
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What is the meaning of a differential?
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:...
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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?
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