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1. 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...
2. 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...
3. 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?
4. 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...
5. 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.
6. 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...
7. 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...
8. 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
9. 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...
10. 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...
11. 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...
12. 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...
13. 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...
14. 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.
15. 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...
16. 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............
17. 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...
18. 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...