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...
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?
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...
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.
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...
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...
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
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...
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...
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...
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...
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...
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...
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............
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...
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...