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
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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 ...
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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
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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 ...
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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 ...
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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 $X_f$ ...
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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 $3.02\cdot e^{3.02^2-2.9^2}$, ...
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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 ...
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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 $$X=\text{grad}f=2x\frac{\partial}{\partial x}$$ Compute the coordinate expression for $X$ in polar ...
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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 ...
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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 ...
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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 ...
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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 $\def\g{\gamma}\def\dx{\text{d}x}\def\dy{\text{d}y}\def\td{\text{d}}\def\br{\textbf{r}}\def\dt{\text{d}t}\def\RR{\mathbb{R}}$ $$\int_\g f\ \td\br = \int P\ \dx+Q\ \dy$$ ...
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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 ...
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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 ...
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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 $M$. ...
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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 ...
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