2
votes
1answer
29 views

Differentiability implies continuity — possibly pedantic question about the common proof

The common proof that differentiability implies continuity arrives at this limit: $$\lim_{x\to a} [f(x) - f(a)] = 0$$ I'm failing to see the simple justification for moving to the next step, which ...
4
votes
1answer
51 views

Calculus and infinitesimals

In the definition of reimann integral, why do we put a 'dx' inside the integral sign when practically it serves no purpose except maybe telling what variable you are talking about. Then in some ...
2
votes
0answers
32 views

Estimate $\displaystyle\left|\int_{\frac{\pi}{k}}^{\frac{\pi}{2}} (\sin \theta)^{-1+\frac{i(n+1)y}{2}} d\theta \right|$

I have to estimate the following integral $$\left|\int_{\frac{\pi}{k}}^{\frac{\pi}{2}} (\sin \theta)^{-1+\frac{i(n+1)y}{2}} d\theta \right|,\quad \forall k,n\geq 2 $$ According to Sogge (Oscillatory ...
2
votes
1answer
38 views

Del on Riemannian manifolds

I was supposed to figure out what $grad(div(e_r))$ is, where $e_r$ is the unit vector in spherical coordinates. In the following I assume $g:=diag(1,r^2,r^2sin^2(\theta))$ be the metric tensor. My ...
1
vote
1answer
23 views

differential equation as taylor series

Consider the equation $\frac{\mathrm{d} x(t)}{\mathrm{d}t} = g(x(t))$ , with $x(0) = x_0$, where g is function that admits derivatives of all orders.If the solution of the equation can be written as a ...
1
vote
1answer
39 views

What is the limit of this function as $(x,y)$ approaches $(0,0)$?

Let the function $f \colon (\mathbf{R}^2 \setminus \{(x,y) \in \mathbf{R}^2 \colon x+y = 0 \}) \to \mathbf{R}$ be defined as follows: $$ f(x,y) \colon= \frac{xy}{x+y}$$ if $(x,y) \in \mathbf{R}^2$ ...
-2
votes
0answers
24 views

Convergence of series (putnam training) [on hold]

Does the series $\sum_{n=1}^{\infty} \frac {|\sin n|}{n} $ converge?
1
vote
0answers
32 views

To show a function differentiable

Let $A \in \mathbb{R}^n$ be a fixed vector and $T : \mathbb{R}^n \rightarrow \mathbb{R}^n$ a linear transformation . Define $f : \mathbb{R}^n \rightarrow \mathbb{R}$ by $$f(x) = \langle ...
0
votes
0answers
29 views

How to convert vector field from cartesian to spherical

I have a vector field $A ( r) = \omega \times r$, where $r=(x,y,z)^T$ and now I want to express this field in cylindrical coordinates. How do I do this?
11
votes
2answers
88 views

Second derivative of $f(f(\cdots f(x)\cdots )?$

For convenience, let's write $f_n(x)=f(f(\cdots f(x)\cdots )$ where $f$ is iterated $n$ times. Suppose: $$f(0)=0,\quad f'(0)=\alpha,\quad f''(0)=\beta$$ What is $f''_n(0)?$ I've found ...
0
votes
1answer
19 views

consequence of Mean Value Theorem

Let $f$ a continuous function on $[a, b]$ $a < b$ ,derivable on $(a, b)$ then there exist $c_1, c_2 \in (a, b)$ ,$c_1 \ne c_2$ such that $\frac{f (b) - f (a)}{b - a} = \frac{f '(c1) + f' ...
-1
votes
1answer
38 views

Advanced Calc proof help

Assume that for $a,b>0$ and any $0 < t< 1$ $$ a^tb^{1-t} ≤ ta+(1-t)b $$ Prove given $a_1,a_2,...,a_n ≥ 0$, $b_1,b_2,...,b_n \geq 0$ and $b_1+b_2+...+b_n=1$ We have $$ \left(\sum_{i = ...
2
votes
2answers
28 views

Show that the difference quotient of $1/x^n$ exists

Let $n>0$ be a positive integer. For all $x\not=0$, prove that $f(x) = 1/x^n$ is differentiable at $x$ with $f^\prime(x) = -n/x^{n+1}$ by showing that the limit of the difference quotient ...
0
votes
1answer
25 views

What are the values of the parameters that make the function differentiable at zero?

I think I might have found a way to solve this problem but I'm not sure if this is correct, if someone could tell me if this is the correct approach or not that would be nice. If it's not the correct ...
1
vote
2answers
24 views

The Lebesgue Integral and the Dirichlet function

I am considering the function $$f(x)=\begin{cases} 1 &\text{if } x\in [0,1]-Q \\{}\\ 0 &\text{if } x\in [0,1] \cap Q\end{cases}$$ I am trying to evaluate this using the Lebesgue integral. ...
-5
votes
1answer
50 views

Advanced Calc 1 Help [closed]

Suppose $u,v,u',v'$ are continuous on $R$ and $uv'-u'v$ is never zero. Show that between any two consecutive zeros of $u$ there is a zero of $v$, and similarly for $v$.
0
votes
0answers
10 views

Finite difference scheme and its stability

The Finite difference scheme: \begin{equation} y_{n+3}-y_{n+1}= \frac {h}{3}(f_{n}-2f_{n+1}+7f_{n+2}) \end{equation} Deduce that the scheme is convergent and find its interval of absolute stability(if ...
0
votes
0answers
6 views

Modified Symplectic Euler

Simple harmonic motion: $y'= -z $, $z'= f(y)$ and the modified Symplectic Euler equation are $$y'=-z+\frac {1}{2} hf(y)$$ $$y'=f(y)+\frac {1}{2} hf_y z$$ deduce that the coresponding approximate ...
0
votes
1answer
28 views

Implicit Euler Scheme and stability

Find the fixed points of the implicit Euler scheme \begin{equation} y_{n+1}-y_{n}= hf(t_{n+1},y_{n+1}) \end{equation} when applied to the differential equation $y'=y(1-y)$ and investigate their ...
0
votes
0answers
20 views

A predictor-corrector method

A predictor-corrector method for the approximate solution of $y'=f(t,y)$ uses \begin{equation} y_{n+1}-y_{n}=hf_{n} \tag P \end{equation} as predictor and \begin{equation} ...
0
votes
1answer
55 views

Complex number, strangely written

Find all the complex solutions of the equation: $$\frac{z^3}{i} = 1$$ I mean is this the same thing as $$z^3 = i$$? Because I don't understand why my teacher would put it like that on a test. At ...
0
votes
2answers
30 views

Finding numbers $a$ and $b$ for a complex number

Problem. Given a complex number $$z=2-2i$$ Find numbers $a$ and $b$ such that $$a+ib = \frac{1}{z}$$ I tried multiplying both sides by $z$ and got $$(a+ib)(2-2i)$$ $$= 2a-2ai+2bi-2bi^2$$ ...
2
votes
1answer
32 views

functions of two variables with one variable defined on a compact set uniformly converge to zero

Let $f$ be a holomorphic function on $[0,1]\times \mathbb{R}$. If for each $x\in [0,1]$ fixed, $\lim_{y\to\infty}f(x,y)=0$, prove that $f$ is bounded. My idea: I do not know how to prove and I also ...
0
votes
1answer
24 views

$\sum_{n=1}^{\infty}|a_{n} x^{n}| < \infty \implies \sum_{n=1}^{\infty} |a_{n}| (|x|^{n-1}+ |x^{n-2}y|+…+|y^{n-1}|) <\infty $?

Let $\{a_{n}\}_{n\in \mathbb N} \subset \mathbb C$ so that the series, $\sum_{n=1}^{\infty} a_{n} x^{n},$ converges absolutely for all $x\in \mathbb R$. Now, we let, $x, y \in \mathbb R$ with $x\neq ...
1
vote
3answers
43 views

$\sum_{n=1}^{\infty} a_{n}= \sum_{n=1}^{\infty}b_{n}$ and $\sum_{n=1}^{\infty} |a_{n}| < \infty \implies \sum_{n=1}^{\infty} |b_{n}| < \infty$?

Suppose that $\{a_{n}\}_{n\in \mathbb N}, \{b_{n}\}_{n\in \mathbb N} \subset \mathbb C$ such that both the series, $\sum_{n=1}^{\infty} a_{n}$ and $\sum_{n=1}^{\infty} b_{n}$ converges, and its sum ...
1
vote
0answers
24 views

Predictor-Corrector for Adams-Moulton

What is the order of the corrector of Adams-Moulton type required in order to apply Milne's method for estimating the error in PECE mode? Find the coefficient of the leading term in the truncation ...
0
votes
1answer
30 views

Can we write $f\in C^{1}(\mathbb R^{2})$ as $f(z_{1})-f(z_{2})= (z_{1}-z_{2})\cdot G(z_{1}, z_{2})$?

Mean-value theorem for one variable, tells us that if $f:\mathbb R \to \mathbb R$ is continuously differentiable, then we can write, $f(x)-f(y) = (x-y) G(x,y)$; where $x,y \in \mathbb R$ and actually ...
1
vote
3answers
37 views

$\frac{1}{a_n}\int_0^{a_n} f(x) \,dx \rightarrow f(0)$ if $a_n\rightarrow 0$

The full question I'm looking at: Suppose that $f:[0,1]\rightarrow\mathbb{R}$ is continuous. Suppose $a_n>0$ satisfy $a_n\rightarrow 0$ as $n\rightarrow \infty$. Prove that ...
20
votes
2answers
301 views

When does $(uv)'=u'v'?$ [duplicate]

In any calculus course, one of the first thing we learn is that $(uv)'=u'v+v'u$ rather than the what I've written in the title. This got me wondering: when is this dream product rule true? There are ...
0
votes
1answer
21 views

Analysis Proof of Inflection Points

We are supposed to prove this, and it seems relatively simple, but as per usual, I don't know where to start. I assume that a big factor is that the third derivative is not zero at $x_0$, which ...
0
votes
4answers
119 views

Does $xy\geq x+y$?

I just see the GM-AM inequality. But I would like to compare $xy$ with $x+y$ for any $(x, y)\in\mathbb{R}^2$. It looks like $xy>x+y$ since the first one is multiplication and the second one is ...
0
votes
1answer
60 views

Is this construction already a contradiction

Let's say we have $\ell^p$ for $p>2$ and $\ell^2 \subsetneq \ell^p$. Let $U$ be a closed subspace of $\ell^2$. Is it possible that $U$ is isomorphic to $\ell^p$ as a Banach space?- I hardly think ...
0
votes
0answers
17 views

conflictions of analytic functions to the boundary and Schwarz reflection principle

Let $\Omega$ be an open subset of $\mathbb{C}$ and $f:\Omega\longrightarrow \mathbb{C}$ be a holomorphic function. Then for any $z\in \Omega$ and any $r>0$ such that $D(z,r)\subseteq \Omega$, $f$ ...
1
vote
0answers
18 views

Cauchy integrals over a line

Can we generalize the Cauchy integral formula from a circle to a line? Since for real integrals, the following types of improper integrals do not converge, is it correct or not that for $z\notin ...
0
votes
0answers
16 views

Variation on Fubini's Theorem

My attempt: Let $P_1$ be a regular partition of $R_1$ and $P_2$ a regular partition of $R_2$. Denote by $P$ the corresponding regular partition of $R_1\times R_2$. Given a generalized rectangle ...
1
vote
0answers
17 views

Definition of a Regular Partition of a Closed Generalized Rectangle in $\mathbb R^n$

What the heck does this definition of a regular partition $P$ of $R$ mean? I follow what it is saying until we get to the last part, "the $k_1\cdot k_2\cdot \cdots \cdot k_n$ subrectangles of the ...
-1
votes
2answers
23 views

Equation for power of a number. [closed]

Is there an equation to find power of a number?? n^m while n,m are variable I see that is hard when coming to index numbers.. so without using the log book is there any way to come up with an ...
0
votes
1answer
37 views

Differentiable function strictly concave up $\iff f'$ strictly increasing

I feel like this is false, but I am stumped as to find a counter example. Would $f(x)=x^4$ be a candidate? Thanks!
0
votes
1answer
31 views

Integral-Fourier sum

I am trying to prove the following relation in (3) where $\alpha,\beta,\gamma,\delta,\omega \in \mathbb{R}$. Given the integral $$ I=\frac{1}{2}\int_0^\alpha dx \left( \beta ...
0
votes
1answer
32 views

Show that a function $\psi : \Bbb R^n \to \Bbb R$ is affine

Fix a point x in $\Bbb R^n$. Let c be a point in $\Bbb R^n$ and define the function $\psi : \Bbb R^n \to \Bbb R$ by $$\psi(\mathbf u) = \langle \mathbf c, \mathbf u - \mathbf x \rangle \text{ for } ...
2
votes
3answers
89 views

Why $\log xy=\log x+\log y$?

It is of course well known and basic formula. I am just curious. Is there a proof for it? How to prove that $\log xy=\log x+\log y$?
1
vote
2answers
38 views

Limit and integral properties of a continuous function

Let $f$ be a continuous function on $[0,\infty)$ such that $\displaystyle\lim_{x \to \infty}f(x)= c$. Show that $\displaystyle\lim_{x \to \infty} \frac{1}{x}\int_0^x f(s)\;ds = c$. I've tried ...
1
vote
1answer
35 views

Is $l^p$ closed in $L^p$?

Let's assume we have a subspace $X$ of $L^p$ and we know that $X \cong l^p$(this should just mean isomorph no isometry is assumed here). Can we infer from this that $X$ is closed? I have just read a ...
0
votes
1answer
14 views

Evaluate an integral over $\mathbb{R}^{3}$ and Green's Theorem

Let $p(x_{1}, x_{2}, x_{3})$ be a smooth function in $\mathbb{R}^{3}$ decaying sufficiently rapid as $|x| \rightarrow \infty$. Why is $$\int_{\mathbb{R}^{3}}p_{x_{i}}\, dx = 0?$$ By the Gauss-Green ...
0
votes
0answers
21 views

$\prod_{i = 1}^n x_i^{y_i} \leq y \cdot x$

I need to prove $\prod_{i = 1}^n x_i^{y_i} \leq y \cdot x$ where $x_i \geq 0$ for all $i$ and fixed $y$ where $\sum y_i = 1$. I have looked around and all the proofs I've found have used concavity of ...
1
vote
1answer
42 views

Basic sequence- what is so special about it?

Let $(x_n)$ be a Schauder basis of a vector space $X$. This means that the $span(x_n)$ is dense in $X$, right? Then wikipedia introduces the notion of a $\textbf{basic sequence} $ when $(x_n)$ is a ...
-1
votes
1answer
42 views

Real analysis: Continuity of a function

Define $f: [0,1) \cup [2,3] \rightarrow [0,2]$ by $$f(x)=\begin{cases} x & x \in [0,1) \\ x-1 & x \in [2,3] \end{cases}$$ Is the function continuous at $x=1$? Is the function continuous at ...
1
vote
1answer
42 views

How to find the Fourier transform of $\mathbf 1_{[0,2\pi]}(x)\sin(x)$?

How does one find the Fourier transform of $f(x):=\mathbf 1_{[0,2\pi]}(x)\sin(x)$? I have tried to use the definition from my text: \begin{align*} \hat f(\xi) & = \frac{1}{\sqrt{2\pi}} ...
1
vote
0answers
26 views

Notation for near optimal solution

Usually, $x^*$ is used to denote the optimal solution to a maximization problem. I need a notation to describe a solution that is not optimal but "good enough." In my case this solution is the first ...
0
votes
3answers
62 views

Can two function $f$ and $g$ have same values through out a given interval and different values outside that interval?

Is it possible that for two functions $f$ and $g$ and some interval $(a,b)$ we have $f(x)=g(x)$ for all $x\in(a,b)$ and $f(x)\neq g(x)$ for $x$ outside the interval $(a,b)$? $f$ and $g$ are ...