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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Show that the drainage is defined by this ode

A water tank with a rectangular cross-section and one trapezoidal side as shown in the figure contains a volume of water $𝑉(ℎ)$ [$\text{m}^3$] when the depth of the water in the tank is $ℎ ~\text{m}$ ...
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Utility of the coordinate free definition of the derivative on manifolds.

Preface: I am not an expert on the topic of smooth manifolds, nor do I have the perspective gained from knowing many theorems proven on smooth manifolds. Please try to look at the problem from the ...
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Modeling body temperature in a continuous framework [closed]

I am reviewing for an exam and was reviewing last year's exam. Since our professor doesn't want to solve it in class, I come here to see if someone is so kind to solve it. The problem has to be ...
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Question about write differential as sum of partial derivatives

I learned that if i have a function $f:R^p\to R^q$ i can write differential of function like $df(a)(u)=\frac{\partial f}{\partial x_1}(a)u_1+\frac{\partial f}{\partial x_2}(a)u_2+...+\frac{\partial f}{...
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Solve differential equation $yy''=2x(y')^2$ [closed]

I can solve it when $y$ is not there on left-hand side. Not getting any approach how to do it. Please help.
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Diagonalization DE Problem [closed]

How to solve this systems of DE by Diagonalization? Since $y'_3$ is missing here. Equations: $$y’_1=y_1+y_2+2y_3$$ $$y’_2=2y_1-y_2+3y_3$$
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How would this non-linear second order simultaneous differential be solved?

How would this non-linear second order simultaneous differential be solved? Everything other than $r(t)$ and $s(t)$ are constants and independent of time $t$. Can the runge-kutta method be applied ...
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(Diffusion and decay) Randall J. LeVeque

This is the question that I could not solve it enter image description here
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1 vote
1 answer
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Prove that $\frac{dl}{ds}=\sqrt{κ^{2}+\tau^{2}}$ for a biregular curve where $l$ and $s$ are arclengths [closed]

Can you help me with this exercise, I'm just getting to know the differential geometry course, I've tried to do it but I can't get anywhere. Let $\sigma : I → \mathbb{R}^{3}$ be a biregular curve of ...
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Is there any software to help with systems of differential equations?

I am wondering if anybody knows of a computer software that can help calculate equations like: $$\frac{dx}{dt} + \frac{d^2y}{dt} = e^{4t}$$ $$\frac{dx}{dt} + x + \frac{dy}{dt} - y = 4e^{2t}$$
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DIfferentiable function and generalized mean value theorem [closed]

Let f be continuous on [a,b] and differential on (a,b). Prove that if a >= 0 there are x1, x2, x3 $\in$ (a,b) such that $$f'(x_1) = (b+a){f'(x_2)\over 2x_2} = (b^2 + ba + a^2){f'(x_3)\over3x_3^2}$$ ...
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Changing $\Delta x$ with $\Delta x^2$ in the derivative definition.

We know definition of derivative is $\frac{\Delta f(x)}{\Delta x}$ when $\Delta x$ approaching 0. But what happens when we change $\Delta x$ by $(\Delta x)^2$. $\lim\limits _{\Delta x\to0}\frac{f(x+\...
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2 votes
2 answers
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Can every symmetric Jacobian matrix be a Hessian matrix?

Say we have a function $\mathbf{f}(\mathbf{x})$ where $\mathbf{x}\in\mathbb{R}^n$ and $\mathbf{f}:\mathbb{R}^n\rightarrow\mathbb{R}^n$ with a Jacobian matrix $\mathbf{J} = \partial \mathbf{f}/\partial ...
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Prove the following subset of $X = \mathbb{T}^2$ is not a submanifold of $X$

I am having trouble understanding the proof of the following statement: Let $X = T^2$ be the 2-dimensional torus defined as $$T^2 := \mathbb{R}^2 / \sim $$ where $(x,y) \sim (x',y')$ iff $x-x', y-y' ...
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Supremum norm on the space is not differentiable.

Prove the supremum norm on the space $C[0,1]$ is not differentiable at any element $x$ for which there are two point $t$ in $[0,1]$ where $|x(t)|=\|x\|$. My attemp: Suppose that $f:C[0,1]\to\mathbb{R}$...
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Differential of a multivariable function involving $\ln$

I am reading about Polytropic processes in Thermodynamics where the governing equation is $pV^n =$ constant. The author of the book wants to derive an expression and describes that he is taking the ...
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Solve $(\frac{x}{y} + 1 )dx + (\frac{x}{y} -1 )dy = 0$. Is my solution correct?

I tried to solve: $$(\frac{x}{y} + 1 )dx + (\frac{x}{y} -1 )dy = 0$$ $(y+1)dx+(x+1)dy=0$ $M(x, y) = \frac{x}{y} + 1$ $N(x, y) = \frac{x}{y} -1$ Then, we can multiply both $M$ and $N$ by $y$ to get: $...
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Functions differentiable and its derivative [closed]

Where is the function $f(x) = \sqrt[3]{x}e^{\sqrt{x}}$ defined, where is it differentiable, and what is its derivative Could anyone help me with this? I am not good at this part of math theory?
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Finding the path of a light ray using differential geometry

Hi I am trying to solve a calculus of variations problem I would like to solve it using differential geometric approach but i am not sure. As an example how would I go back doing it for the example ...
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Show that the curves $r^2 = a^2\cos^2(\theta)$ and $r = a(1+\cos(\theta))$ intersect at an angle $3\arcsin((3/4)^{1/4})$. [closed]

Show that the curves $r^2 = a^2\cos^2(\theta)$ and $r = a(1+\cos(\theta))$ intersect at an angle $3\arcsin((3/4)^{1/4})$. I know how to solve this question but I am not getting the angle to be the ...
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How do you define a point on the sphere in terms of the coordinates of its stereographic projection within the complex plane through the equator?

It’s usually understood that having a stereographic projection, you can define points in the complex plane passing through the equator in terms of the coordinates of its stereographic image on the ...
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Solve the minimization problem using subdifferentials on a non-differentiable continuous function subject to two constraints

This is a desmos plot of the problem I am exploring: https://www.desmos.com/calculator/0rxekqcj26 My first question is how to find the subdifferential of $f(x,y) = \max\{|x|, y + 4\}$. From my ...
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1 vote
1 answer
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Proof that Jacobian-Matrix is proportional to an orthogonal matrix.

Let $$f: \mathbb{R}^n \setminus \{ 0\} \to \mathbb{R}^n \setminus \{ 0\}, x \mapsto \frac{x}{\| x \|^2}$$ be the inversion of the unit sphere, which is differentiable. I want to show, that the ...
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Is the Differential Operator Matrix not Isomorphic? If so, why is it not when the derivative is a linear operator?

Is the Derivative operator not isomorphic? I can see by the proof that: D[cx] = cD[x] and D[x + y] = D[y] + D[x] but when I create the map of the derivative of of polynomial space $ \{1, t, t^2, t^3\} ...
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Proof clarification of Laplace transform heaviside stepfunction and timeshifted function

I am having trouble with the derivation of the Laplace transform of the heaviside step function and a time-shifted function. $$u_c(t) = \Big \lbrace_{1, \; t \geq c}^{0, \;t \leq c}$$ Deriving: $$ L \...
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Using calculus to determine drainage time for iV bags [closed]

This question got me thinking really hard, but still couldn't solve it. It concerns the drainage of an IV bag. Initially, the example model the output velocity of the bag using Bernoulli's Extended ...
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Construct a curve that is non-planar, such that it's a Frenet curve at all except at 1 point, yet the torsion is zero at every point except that one.

This problem is from Wolfgang Kühnel. The original statement is "Construct a non-planar C∞-curve which is a Frenet curve except for a single point, and outside of this point satisfies τ ≡ 0"....
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Show/"sketch" which area f is strictly positive.

I have the following question: Let $c>0$ and consider the following function $f:\mathbb{R}^2\rightarrow \mathbb{R}$ given by: $$f(x,y)=\begin{cases} c y e^{-x} & \text{0 < x < $\infty$...
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A question while proofing the equation of linearization error proof $g(a)=\frac{1}{2}{f}''(c)(x-a)^2$

I've got a question the linearization error proof part from the Book The calculus Lifesaver, we want to proof the error $g(a)=\frac{1}{2}{f}''(c)(x-a)^2$. Original derivation progress: The ...
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Using differentials (not limits!) to find the derivative of sqrt(x)

So I understand how to find the derivative of $f(x)=x^{1/2}$ using the power rule. I also know how to find it using the limit. $f'(x) = \lim_{h \to 0} \frac{(x+h)^{1/2} - x^{1/2}}{h}$ You could ...
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If the integral of a differential form is path-independent, then is the differential form an exact differential?

The statement "If it is an exact differential, then its integral is integral path-independent; it only depends on the both integral path endpoints regardless of which path between them is chosen.&...
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Differentiability of the functions $\mathbf{f}:\mathbb{M}_{n\times n}\to \mathbb{M}_{n\times n},\ \mathbf{f}(A)=A^2,\ \mathbf{f}(A)=A^T A$

I am trying to prove that the functions $\mathbf{f}:\mathbb{M}_{n\times n}\to \mathbb{M}_{n\times n},\ \mathbf{f}(A)=A^2,\ \mathbf{f}(A)=A^T A$ are differentiable. What I have done: We consider $\...
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Substitution of main function (y) into differential equation

I have a problem understanding how these substitutions are possible in examples (2) & (3), clearly this is not the main function as the place of the (A) & (B) parameters are switched in ...
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Differential of quadratic form

I'm having trouble understanding this: if $\phi(x)=x^{T}Ax$, where $A$ is a matrix of constraints. Then, the differential of $\phi$: $\mathrm{d} \phi=(\mathrm{d} x)^{T} A x+x^{T} A \mathrm{~d} x=x^{T} ...
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I have a question about 'Differential of function dy'. [duplicate]

I'v learnt that . . . If a single-variable and differentiable function $y=f(x) (f:R→R)$ is given, and if a two-varible function $dy$ is defined as $$dy = (f'(x))(dx),$$ and if an another two-variable ...
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1 vote
3 answers
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Derivative of $f:\mathbb{R}^n\to\mathbb{R},\ f(\vec{x})=A\vec{x}\cdot\vec{x}=\vec{x}^T A\vec{x},\ A\in\mathbb{M}_{n\times n}$

I ma trying to prove that if $A$ is an $n\times n$ matrix, then $f:\mathbb{R}^n\to\mathbb{R},\ f(\vec{x})=A\vec{x}\cdot\vec{x}=\vec{x}^T A\vec{x}$ is differentiable and $Df(\vec{a})\vec{h}=A\vec{a}\...
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Total derivative and partial derivative of an equation

Will an equation still be valid if we take total derivative of both side? What about partial derivative instead? For example, suppose we have $\varphi (s, \varphi(t,x)) = \varphi(s+t,x)$, does that ...
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Show that $\forall u\in L(h)$, $\|Df(u)-3Id\|\leq 6\|u-Id\|+3\|u-Id\|^2.$

Let $E$ be a banach space and let $f:L(E)\to L(E)$ defined by: $f(u)=u^{3},$ $u$ in the open ball $B(Id,\frac 1 3).$ Show that $f$ is $C^1$ and find its differential $Df(u)(h)$ , $ \forall u,h \in L(...
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2 answers
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Finding the inverse of the differential $d\pi_{i_p} : T_p( M_1 \times \dots \times M_k) \to T_{p_i}M_i$

Suppose that $M_1, \dots, M_k$ are smooth manifolds. Show that for each $i$ the projection $\pi_i : M_1 \times \dots \times M_k \to M_i$ is a smooth submersion. I've shown the smoothness of the ...
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What is the derivative of x^T e^-A x w.r.t. A?

What is the derivative $\frac{\partial}{\partial A} x^T e^{-A} x$ $\frac{\partial}{\partial A} x^T e^{A} x$ To be clear, $x \in \mathbb{R}^n$, $A \in \mathbb{R}^{n \times n}$ and $e^A = \sum_{n=0}^{\...
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2 votes
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Why is Leibniz Notation written this way for the second derivative? [duplicate]

Let $y = f(x)$. $$f'' = \frac{d^2y}{dx^2}$$ The explanation for this being that $$ \Bigl(\dfrac{d}{dx}\Bigr)^2 y = \dfrac{d^2}{dx^2}\,y = \dfrac{d^2 y}{dx^2};$$ Since there are two $d$'s in the bottom ...
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How to find the limit of $f(x)$ as $x \to 0$ where $f(x)=x\log(\sin x)$ [closed]

Find $\displaystyle \lim_{x \to 0} x\log(\sin x)$. From the graph of the function $$f(x)=x \log(\sin x)$$ it is evident that this limit is zero. Hope to get some formal way to find this limit.
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2 votes
1 answer
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Differential of the map $(f_1, f_2) : M \to N_1 \times N_2$.

Let $M, N_1$ and $N_2$ be a smooth manifolds and define the smooth maps $f_1 :M \to N_1, f_2:M \to N_2$. Let $p \in M$ and define the differential $d(f_1,f_2)_p : T_pM \to T_{(f_1(p), f_2(p))}(N_1 \...
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  • 129
1 vote
2 answers
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How to find the envelope of the family of curves $(x-t)^2 + (y-t^2)^2 = 1$?

How to find the envelope of the family of curves $(x-t)^2 + (y-t^2)^2 = 1$? Based on the given, the family of curves can be describe a set of circles whose centers are the points along the parabola $x=...
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Non Linear Equation unique solution

Can anyone help me prove that for every unique y, there exists unique a and b in the following equation: $$y = \frac{bm}{e^{(a+bm)} + 1}$$ Here m is constant.
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What is the Differential Of The Bernoulli Periodic Function.

I was trying to find the differential of the Bernoulli periodic function; e.g $$ P_4(x) = B_4(x - \lfloor x \rfloor) $$, so as to calculate the remainder integral of a certain euler maclaurin sum I a ...
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2 votes
2 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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transforming a differential equation through parametrization

In my book on differentiation, there is a transformation done that I cannot follow. I am sure that it includes the chain rule, but I do not know why the second equation equals the first (see picture ...
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2 votes
2 answers
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Verify that an implicit equation is the solution to the differential equation.

Verify that \begin{equation} a.\;x^3+y^3-3xy=0,\; \mathbb{R}_{x\neq2^{2/3}} \end{equation} is the solution to \begin{equation} b.\;(y^2-x)y' - y+x^2=0 \end{equation} As we know the function g(x) of a. ...
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2 votes
1 answer
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Find an equation of the curve that satisfies the condition.

My question is as follows; In the book Anton Calculus 12th edition, Question 56 in section 5.2 has a solution contradicting my own; I will break the reasoning down for my answer to aid a reader in ...
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