Articles of derivatives

Prob. 28, Chap. 5, in Baby Rudin: How to formulate the result on the uniqueness of the solution to a system of differential equations?

Here is Prob. 28, Chap. 5, in the book Principles of Mathematical Analysis by Walter Rudin, 3rd edition: Formulate and prove an analogous uniqueness theorem for systems of differential equations of the form $$ y_j^\prime = \phi_j \left( x, y_1, \ldots, y_k \right), \qquad y_j (a) = c_j \qquad (j = 1, \ldots, k).$$ Note […]

Any example of a function which is discontinuous at each point in a deleted neighborhood of a point at which that function is differentiable?

Let $f$ be a real (or complex) function defined on the segment $(a, b)$ of the real line, and let $p \in (a, b)$. If $f$ is differentiable at $p$, then of course $f$ is continuous at $p$ as well. (The converse may not hold.) But can we have an example where $f$ is differentiable […]

By using d'Alembert's formula, substitute $P(z)$ and $Q(z)$ into the general solution to obtain an expression for $u(x, t)$

We will define a motion that satisfies the equation: $$u_{tt} = c^2u_{xx}\qquad x ∈ (0, 1),\: t > 0$$We have the displacement of a string being $u(x, t)$, at position $x$ and time $t$, which is stretched between two fixed points at $x = 0$ and $x = 1$. where $c$ is a real positive […]

What is $\frac{dx}{d}?$

The operator $\frac{d}{dx}$ is common in calculus to denote a derivative. However, this also begs the question, what is the operator $\frac{dx}{d}$? Is this operator used commonly? If so, what is it called/what does it do? I have played aroud with it before, and found a natural way to define it seems to be that […]

How to obtain dimension of solution space of ODE?

We are given the equation, $$x^2y”-4xy’+6y=0$$ And we have to get the dimension of solution space in $(-1,1)$. My Attempt: I tried substituting $$x=e^z$$ and I get from that, the following, $$y=c_1x^2+c_2x^3$$ and concluded that dimension is $2$. But, I realize that at $x=0$, my ODE in standard form will face issues, and this solution […]

Sequence of funtions $f_n = n(f(x+ \frac{1}{n})-f(x))$ for the continous differentiable function $f$ on $\mathbb R$

Let $f$ be a continous differentiable function on $\mathbb R$. Let $f_n$ be a sequence of functions $f_n = n(f(x+ \frac{1}{n})-f(x))$. Then (a) $f_n$ converges uniformly on $\mathbb R$ (b) $f_n$ converges on $\mathbb R$, but not necessarily uniformly. (c) $f_n$ converges to the derivative of $f$ uniformly on$[0,1]$ (d) there is no guarantee that […]

For each $a \in U$ we have $Df(a):\mathbb R^n \to \mathbb R^n$ is a linear isomorphism

An invertible function $f: U \subseteq \mathbb R^n \to V \subseteq \mathbb R^n$ is given and we know that $f$ and $f^{-1}$ both are differentiable. Prove that for each $a \in U$ we have $Df(a):\mathbb R^n \to \mathbb R^n$ is a linear isomorphism. Then for $b \in V$, find $Df^{-1}(b)$. I can’t even understand how […]

Differential equation with separable, probably wrong answer in book

I have a differential equation: $$ \frac{dy}{dx} = y \log(y)\cot(x)$$ I’m trying solve that equation by separating variables and dividing by $y\log(y)$: $$ dy = y \log(y) \cot(x) dx$$ $$ \frac{dy}{y \log(y)} = \cot(x) dx$$ $$ \cot(x) – \frac{dy}{y \log(y)} = 0 $$ Where of course $ y > 0 $ regarding to division Beacuse: […]

Jacobian of $A (A^\top X A)^{-1} A^\top$

Let $A\in\mathbb{R}^{n\times m}$, $n\geq m$, be a full column rank matrix, and consider the function \begin{align} f&\colon \mathbb{R}^{n\times n} \to \mathbb{R}^{n\times n}\\ & X\mapsto A (A^\top X A)^{-1} A^\top, \end{align} where $\bullet^\top$ denotes transposition. Assuming that $(A^\top X A)^{-1}$ exists, I’m interested in the computation of the Jacobian matrix of $f$, i.e. $$\tag{1}\label{a} \mathbf{J}[f] = […]

Sum of exponentials with Fourier coefficient

Let $f$ be a continuous function with period $2\pi$. Define $$u(r,\theta)=\sum_{n=-\infty}^\infty r^{|n|}\hat{f}(n)e^{in\theta}$$ for $r\in[0,1)$, where $\hat{f}(n)$ is the $n$th Fourier coefficient of $f$. a) Express $u$ as a series in $z=x+iy$ and $\bar{z}=x-iy$. b) Show that $u$ is infinitely differentiable in $x^2+y^2<1$. For a), I want to substitute in $r=\sqrt{x^2+y^2}$ and $\theta=\tan^{-1}(y/x)$. But this doesn’t […]