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    Euler equation in Polar coordinates

    Hello. I have 2D Euler equation for fluids. I cant derive it in polar coordinates. I defined functions u(x,y,t) = u'(r, theta, t) and v(x,y,t) = v'(r, theta, t). I started by computing derivatives \frac{\partial u'}{\partial r}=\cos\theta\frac{\partial u}{\partial...
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    Mass conservation

    Hello i want to solve \frac{\partial \rho}{\partial t}=\frac{\partial v_1\rho}{\partial x_1}+\frac{\partial v_2\rho}{\partial x_2} for v_1 = -x_2 and v_2=x_1 i obtain equation \frac{\partial \rho}{\partial t}+x_2\frac{\partial\rho}{\partial x_1}-x_1\frac{\partial \rho}{\partial...
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    First order PDE with two conditions?

    Hello, I have a problem in the form \frac{\partial u}{\partial t}+\frac{\partial u}{\partial x}+e^{x}u=0 with conditions u(x,0)=u_0(x) u(0,t)=\int_{0}^{\infty}f(x)u(x,t)dx Im confused, because in first order PDE i require only 1 condition. How to solve this for two conditions?
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    Navier stokes

    Hello, I have Navier stokes in 1D \rho\left(\frac{\partial u}{\partial t}+u\frac{\partial u}{\partial x}\right)=-\frac{\partial p}{\partial x}+\mu\frac{\partial^2u}{\partial x^2} Condition of imcompressibility gives \frac{\partial u}{\partial x}=0 So I have Navier stokes...
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    Solving specific PDE

    Hello, I derived a model in the form \begin{array}{rcl}\frac{\partial U(\vec{x},t)}{\partial t}&=&\gamma^2\Vert\nabla U(\vec{x},t)\Vert,\\\int_{\Omega}U(\vec{x},t)\, d \Omega&=&U_0,\quad\forall t\\U(\vec{x},0)&=&f(\vec{x}).\end{array} I don't know to solve that. THanks for help.
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    Wrong Time Dilation

    Hello ! Can you tell me what's wrong? I suppose two observers. They measure speed of light. One observer is in rocket of length L. Observer in rocket measured speed C'=\frac{L}{t'} and t' is time from observer's view. Second observer is outside. He measured C=\frac{L+vt}{t} where v is...
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    Proof - Substitution, Jacobian, etc.

    Hello! I recently tried to prove following theorem: Let \phi:B\to\mathbb{R}^2 be a diffeomorphism (regular, injective mapping). Then \int_{\phi(B)}f(\mathbf{x})\,\mathrm{d}x=\int_{B}f(\phi(\mathbf{t}))\left|{\mathrm{det}}\mathbf{J}_{\phi}\right|\mathrm{d}t With following I can't proof...
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    Open set (differential equation)

    Hello ! When I'm reading something about differential equations everywhere it's about open sets. For example when we define special kind of equation x'=f(t,x)\,;\;f:\Omega\subset\mathbb{R}\times\mathbb{R}\to\mathbb{R} Omega is open. Why Omega must be open? Thanks
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    Energy conservation law

    Hello, I nowhere find general form of energy conservation law, but in one book i found this (*)\hspace{1cm}E=E_{in}-E_{out}+E_{generated} where E(in) is energy flow into system, E(out) is energy flow out of system and E(generated) is energy generated. It was in sense of heat...
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    Convexity problem

    I'm reading book and there's proposition with convex function Function f is convex if and only if for all x,y (*)\quad f(x)-f(y)\ge\nabla f(y)^T(x-y) It's proven in this way: From definition of convexity f(\lambda x+(1-\lambda)x)\le \lambda f(x)+(1-\lambda)f(y) we have...
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    Inequality with maxs

    Hello, i've met during problem solving with inequality \max\{A+B,C\}\le\max\{A,C\}+\max\{B,C\} where A,B and C are real numbers. I don't know whether it holds, but I need to prove that. Thanks for reply...
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    Limit of arithmetic mean

    Homework Statement prove: lim x_n = L. Then \lim_{n\to\infty}\frac{x_1+\cdots+x_n}{n}=L Homework Equations The Attempt at a Solution i dont know abolutely. i tried definition \left|\frac{x_1+\cdots+x_n}{n}-L\right|=\frac{1}{n}\left|(x_1-L)+\cdots+(x_n-L)\right| Homework...
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    Continuum mechanics

    Hello! I read somewhere about intro to continuum mechanics. There was a vector \vec{\mu} and displacement vector \delta\vec{\mu}. As vector \vec{\mu} move, it will get new position \vec{\mu}'=\vec{\mu}+\delta\vec{\mu} \vec{\mu}'=\vec{\mu}+\frac{\partial\vec{\mu}}{\partial x_i}\delta...
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    Normal matrices

    Hello, can you help me with the proof? If A is normal A^TA=AA^T then A and A^T have the same nullspace (kernel). And ||Ax||=||A^Tx|| Thank you.
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    Proof of an inequality

    Homework Statement Prove that \frac{1-h}{2}<\sum_{k=1}^{n}x_{2k}(x_{2k+1}-x_{2k-1})<\frac{1+h}{2} where 0=x_1<x_2<\cdots<x_{2n+1}=1 such that x_{k+1}-x_{k}<h for 1\le k\le 2n Homework Equations How to prove? :-) The Attempt at a Solution I need to prove...
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    Functional equations

    Hello, could you explain me what's the right way to solve these equations. i've never solved it before. f(x+y)+f(x-y)=2f(x)f(y)\,\;\;\forall x,y\in\mathbb{R} f(x)+\left(x+\frac{1}{2}\right)f(1-x)=1\,\;\;\forall x\in\mathbb{R} thank you............
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    Elementary matrices theory

    Hello, I need to find some theory about elementary matrices. That are the matrices in the form \mathbf{E}(\sigma,\mathbf{u},\mathbf{v})=\mathbf{I}-\frac{1}{\sigma}\mathbf{uv}^{T} I can't find anywhere some theory about it. Can you give me some useful links? Thank you so much...
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    Gravitational potential and partial derivatives app's

    Hello, I'm a student of applied mathematics to economics. Basic course consists of all pure math subjects. We were talking about app's of differentiating the functions u:\mathbb{R}^{n}\to\mathbb{R}^m. We defined a gradient too. In my notes is written: Gravitational potential is a function...