Intereting Posts

the elements of Cantor's discontinuum
Existence of exhaustion by compact sets
New proof about normal matrix is diagonalizable.
Solving $x\; \leq \; \sqrt{20\; -\; x}$
Additive basis of order n: Sets which allow every integer to be expressed as the sum of at most n members of that set.
Set of all unitary matrices – compactness and connectedness.
Closed form for integral of inverse hyperbolic function in terms of ${_4F_3}$
Prove that a function having a derivative bounded by 0.49 has a unique solution $\frac{2x+\sin(x)}{2}$
Problem evaluating $ \int_{0}^{\pi/2}\frac{x}{\tan x}dx $
Why do we assume principal root
A Polygon is inscribed in a circle $\Gamma$
Does $u v^T + v u^T$ have exactly one positive and one negative eigenvalue when $u \not \propto v$?
How can an ordered pair be expressed as a set?
Writing function as infinite Fourier sum with sine kernel
Explicit formula for Bernoulli numbers by using only the recurrence relation

Below is an image of my lecture notes explaining the idea behind the method of characteristics for quasilinear first order PDEs.

- Obtaining Positive Solutions by the Method of Characteristics for a First Order Linear PDE
- Treating shocks with conservation laws
- Why solving $\dfrac{\partial u}{\partial x}=\dfrac{\partial^2u}{\partial y^2}$ like this is wrong?
- Equivalence between norms in $H_0^1(\Omega)\cap H^2(\Omega)$.
- Can one obtain Neumann boundary conditions via Friedrichs extension?
- Steps to solve semi-infinite IBVP

However I don’t understand how the curve $C_s$ is defined and how it translates into the graph $Gr_u$. How does $C_s$ ‘start’ on the initial curve if there is no $f(s)$ or $g(s)$ in the equation for $C_s$? What does $x(t;s)$ mean, does that mean (say for the $x$ term) that for fixed $s$ one starts at $x(s-\epsilon_s)$ and finishes at $x(s+\epsilon_s)$? If so what is this ‘function’ $x$?

Thanks, I would really appreciate someone explaining this to me! :

- Derivation of weak form for variational problem
- Prove $\|v\|_{H^1(\Omega )}\leq C(\|f\|_{L^2(\Omega )}+\|v\|_{H^{1/2}(\partial \Omega )}+\|\partial _\nu v\|_{H^{-1/2}(\partial \Omega )})$
- Can we apply an Itō formula to find an expression for $f(t,X_t)$, if $f$ is taking values in a Hilbert space?
- Euler-Lagrange, Gradient Descent, Heat Equation and Image Denoising
- $\frac{\partial T}{\partial t} = \alpha \nabla ^2r$ for spherically symmetric problems
- Euler-Lagrange equation
- Consider $u_t - \Delta u = f(u)$ and $u=0$ on $\partial\Omega \times (0,\infty)$. Show if $u(x,0) \geq 0$, then $u(x,t) \geq 0$
- Sobolev space $H^s(\mathbb{R}^n)$ is an algebra with $2s>n$
- Best method to solve this PDE
- when does a separate-variable series solution exist for a PDE

- Prove these two conditional probabilities are equivalent
- Prove that $\frac{a^2}{a + b} + \frac{b^2}{b + c} + \frac{c^2}{c + a} \ge \frac{3}{2}$
- A curve whose image has positive measure
- On integrals related to $\int^{+\infty}_{-\infty} e^{-x^2} dx = \sqrt{\pi}$
- Show that $\mathbb{R}^n\setminus \{0\}$ is simply connected for $n\geq 3$
- Matrix on complex field.
- Congruence Modulo with large exponents
- Seeking a More Elegant Proof to an Expectation Inequality
- How can I find $\lim_{n\to \infty} a_n$
- Why doesn't the indirect proof of irrational roots apply to rational roots?
- Mathematical Induction divisibility $8\mid 3^{2n}-1$
- Matrix Norm set #2
- Prove that $5^{1/3}+7^{1/2}$ is irrational
- precise official definition of a cell complex and CW-complex
- Well-Ordering and Mathematical Induction