1
vote
0answers
25 views

If $\phi\in \mathcal{S}(\mathbb R) $ then $\phi_{t}(x)=\frac{1}{t} \phi(x/t)\in\mathcal{S}(\mathbb R)$?

Let $\phi:\mathbb R \to \mathbb C$ be a function; and define $$\phi_{t}(x):=\frac{1}{t}\phi (x/t), (t>0).$$ We note that, if $\phi\in L^{1}(\mathbb R),$ then $\phi_{t}\in L^{1}(\mathbb R);$ in ...
1
vote
1answer
41 views

How to evaluate this integral?

How to evaluate $$ \int_{a}^{b} [x] \ dx \ \ + \int_{a}^{b} [-x] \ dx \ ? $$ I know that $[-x] = -[x]$ if $x$ is an integer, whereas $[-x] = -[x] - 1$ if $x$ is not an integer. So is it about ...
5
votes
3answers
85 views

prove that $a^b\ge{b}^a$ where $a\le{b}$.

prove that $a^b\ge{b}^a$ for all $a,b\ge3$. given that $a\le{b}$. I was trying to solve the question by graph. Can anyone help me please?
2
votes
3answers
176 views

Derivative and integral of the abs function

I would like to ask about how to find the derivative of the absolute value function for example : $\dfrac{d}{dx}|x-3|$ My try:$$ f(x)=|x-3|\\ f(x) = \begin{cases} x-3, & \text{if }x \geq3 \\ ...
2
votes
1answer
122 views

Derivative of continuous function exists if limit of derivative exists

I'm stuck on this old qualifier problem. I suppose one could do it using the basic definitions of continuity and differentiability, but is there a simpler way? (For example, using DCT, FTC, Lebesgue ...
0
votes
3answers
58 views

On a certain series of complex numbers

Is it possible that the above infinite series is equal to ?
2
votes
3answers
109 views

Solve $\lim_{x\to 0} \frac{\sin x-x}{x^3}$

I'm trying to solve this limit $$\lim_{x\to 0} \frac{\sin x-x}{x^3}$$ Solving using L'hopital rule, we have: $$\lim_{x\to 0} \frac{\sin x-x}{x^3}= \lim_{x\to 0} \frac{\cos x-1}{3x^2}=\lim_{x\to ...
2
votes
3answers
63 views

value of an integral depending on a parameter in complex plane

For each $z\in\mathbb{C}$, evaluate the integral $$ \int_0^1\int_0^{2\pi}\frac{1}{re^{i\theta}+z}d\theta dr. $$ How to evaluate it? Thanks.
-2
votes
0answers
126 views

Can a curve via the edges of a staircase $\pi(x)$ exist?

Assuming the Riemann hypothesis, we can get a curve via the edges of a staircase $N(t)$. I expect that a curve via the edges of $\pi(x)$ will be the most important issue in the future analytic ...
1
vote
1answer
9 views

Bound the derivative norm of a convolution by the function norm

Is there a bound of the form $$ \|(f*\phi_\epsilon)'\|_{L^2}\leq C(\epsilon) \|f\|_{L^2}, $$ where $\{\phi_\epsilon\}$ are standard mollifiers, and $C(\epsilon)$ does not depend on $f$?
2
votes
1answer
44 views

Let $f$ be continuous and $U \subset \mathbb{R}^n$ open, if $f: U \rightarrow \mathbb{R}^m $ is injective then $n \leq m$?

I had intended to restrict the image then $f:U \rightarrow f(U) \subset \mathbb{R}^m $ is bijective. Therefore $\dim f(U) = n \leq m$. That's right?
0
votes
1answer
32 views

primitive function involving logarithm, square integrability

I want to ask if the following function, which is given by an integration $f(y):=\frac{1}{y}\int_0^y \frac{1}{x^{1/2}\log{x}}dx,$ is locally square integrable near $y=0$? Or equivalently, ...
1
vote
1answer
32 views

a question about multivariable integral!

If $\lfloor x \rfloor$ denotes the greatest integer in $x$, evaluate the integral$$ \iint_{R} \lfloor x+y \rfloor ~ \mathrm{d}x~ \mathrm{d}y$$where $R= \{(x,y)| 1\leq x\leq 3, 2\leq y\leq 5\}$. This ...
0
votes
1answer
40 views

Evaluate integral $\int\int xe^{xy} dx dy$, strange result after rearranging

I have to compute the following integral $$ \int_{-1}^0 \int_0^1 x\cdot e^{xy} dx dy $$ It exists according to WolframAlpha. Now I want to evaluate it, let $\varepsilon > 0$, then \begin{align*} ...
1
vote
1answer
54 views

Calculate area enclosed by curve

Calculate the area of the bounded surface enclosed by the curve $(x+y)^4 = x^2y$ with the help of the coordinate transformation $x = r\cos^2 t, y = r\sin^2 t$. As I see it the area is unbounded, so ...
0
votes
0answers
37 views

Integrating $xe^{a/x^2 - x^2}\text{Erfi}(x/\sqrt{2})$?

I want to solve any of the two integrals for the complex number $a$ \begin{aligned} I_1 & = \int\limits_{0}^{\infty} xe^{a/x^2 - x^2}\text{Erfi}(x/\sqrt{2}) dx\\ I_2 & = ...
1
vote
1answer
27 views

Verification of Stokes Theorem

I want to verify Stokes Theorem for the surface $$ \Phi = \{ (x,y,z) \in \mathbb R^3 : z = x^2 - y^2, x^2 + y^2 \le 1 \} $$ and the vector field $F(x,y,z) := (y,z,x)$. For this I use the ...
8
votes
3answers
70 views

Show that it is possible that the limit $\displaystyle{\lim_{x \rightarrow +\infty} f'(x)} $ does not exist.

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ a differentiable function with continuous derivative and the limit $\displaystyle{\lim_{x \rightarrow +\infty} f(x) }$ exists. Show with an example that it ...
0
votes
0answers
35 views

Newtonian potential is harmonic

I got this problem in page 321 from Advaced Calculus, written by Friedman: "Let $\sigma(x,y,z)$ be a continuous function, and let $S$ be a continuously differentiable surface. The Newtonian potential ...
0
votes
0answers
17 views

Conditional expectation for discrete random variables

Is it correct that for two discrete random variables $X,Y$ we just have $$E(X|Y \in A) = \sum_{x \in ran(X)} xP(X=x|Y \in A)?$$ This should follow from $$E(X|Y \in A) = \sum_{y \in A}E(X|Y = {y}) ...
2
votes
1answer
26 views

Newtonian potential at (0, 0, – a)

I found this problem in the book Advanced Calculus, written by Friedman. "Newtonian potential at (0, 0, – a) due to a mass with constant densinty $\sigma$ on the hemisphere S: $x^2 + y^2 + z^2 = ...
0
votes
1answer
26 views

relations between differential, partial derivative, directional derivative

I am a bit lost. Could you explain me relations between differential, partial derivative, directional derivative? I mean that I need some theorem and proofs that for example if differential exists ...
0
votes
3answers
109 views

Is speed a function of position?

Let $x$ be a smooth function from $[0,\infty)$ to $\mathbb{R}^n$ satisfying the following differential equation $x''(t) = f(x(t))$, where $f$ is a smooth function from $\mathbb{R}^n$ to itself. Then ...
3
votes
1answer
37 views

a function with infinity L^p norm

Let $1\leq p<\infty$, $1/p+1/q=1$. For a function $f$ with $||f||_q=\infty$, can we write $$ ||f||_q=\sup_{g\in L^p(\Omega),||g||_p\neq 0}\frac{\int_\Omega |fg|}{||g||_p}? $$ or $$ ...
5
votes
2answers
103 views

Prove that $\dfrac{0.5x^2 + x + 1}{x^2 + x + 1}$ is a strictly decreasing function.

This is part of an actuarial science problem. Unfortunately, the official solution of this problem takes the derivative of $$\dfrac{0.5x^2 + x + 1}{x^2 + x + 1}\text{, } \quad x \geq 0\text{.}$$ and ...
6
votes
1answer
72 views

Show there exists a unique $f$ (in $\mathbb R^+$) such that $\frac{d}{dx}f(x)=f^{-1}(x)$

Question: Show there exists a unique bijection $f:\mathbb R^+\to\mathbb R^+$ such that $\frac{d}{dx}f(x)=f^{-1}(x)$, where the right-hand side is the functional inverse. I figured I would start by ...
0
votes
1answer
21 views

Two notions of conditional expectation

For a randomn variable $Y$ and an event $B$ we can define: $$E(Y \mid B) = \frac{E(1_B\cdot Y)}{P(B)}$$ as the conditional expectation. Now, for a sigma algebra $\mathcal{B}$ and sets $B$ in it you ...
3
votes
1answer
39 views

$f:\mathbb{R^2}\setminus\{(0,0)\}\ \rightarrow \mathbb{R}$ of class $C^2$ for which $f_x(x,y)=\frac{y}{x^2+y^2}$ and $f_y(x,y)=\frac{-x}{x^2+y^2}$

Is there exists $f:\mathbb{R^2}\setminus\{(0,0)\}\ \rightarrow \mathbb{R}$ of class $C^2$ for which $f_x(x,y)=\frac{y}{x^2+y^2}$ and $f_y(x,y)=\frac{-x}{x^2+y^2}$ for all ...
0
votes
2answers
45 views

Period of $\frac{\sin(Ny)}{sin y}$ with $N$ odd?

The function $$f(y) = \displaystyle \frac{\sin(Ny)}{\sin y}$$ is periodic with period $2 \pi$ in general. But tracing the graphic of that function for $N$ odd it seems that for $0 \leq x < \pi$ ...
0
votes
0answers
34 views

Dense subsets in $L^1(\mathbb{R})$

Which of the following are dense subsets in metrical space $L^1(\mathbb{R})$? set of smooth functions $C_0^{\infty}(\mathbb{R})$ with compact supports; set of above-mentioned functions' derivatives ...
0
votes
1answer
19 views

Integral over smooth, closed curve of vector field

Why doesn't vector field $v:\mathbb{R^3}\rightarrow\mathbb{R^3}$ given by $v(x,y,z)=(x,\cos y,e^z)$ does not meet $$\int_{\gamma} \langle {v\frac{\gamma'}{\|\gamma'\|}\rangle}\ d\sigma_1=0$$for every ...
0
votes
1answer
15 views

Surface measure of $A={\{(x,y,z)\in\mathbb{R^3}:z=f(x,y),x^2+y^2<1}\}$

Function $f:\mathbb{R^2}\rightarrow\mathbb{R},\ f\in C^{\infty}$ is Lipschitz continuous with constant $1$ and $$A={\{(x,y,z)\in\mathbb{R^3}:z=f(x,y),x^2+y^2<1}\}.$$ Why does it imply that ...
2
votes
2answers
26 views

mean value property of derivatives in high dimensions

Let $E$ be a path-connected subset of $\mathbb{R}^n$ and $f$ a differentiable function on $E$. Prove or disprove: for any $x,y\in E$, there exists $z\in E$ such that $f(x)-f(y)=\nabla f(z)\cdot ...
3
votes
1answer
72 views

Solving the ODE $[(1-x^2)\frac{\partial}{\partial x} - \lambda]f = [\frac{\partial}{\partial x} - \frac{\lambda}{a}]g$

I want to solve $f(x)$ in terms of $g(x)$ in the following ODE $$\left[(1-x^2)\frac{\partial}{\partial x} - \lambda\right]f(x) = \left[\frac{\partial}{\partial x} - \frac{\lambda}{a}\right]g(x),$$ ...
-3
votes
0answers
37 views

Giving some hints [duplicate]

The sequence of $\begin{Bmatrix} {x}_{n} \end{Bmatrix}$ is strictly decreasing,$\quad \lim_{n\to\infty }{x}_{n}=0 \quad $,and$\quad \lim_{n\to\infty }{y}_{n}=0 .$ From the above ...
2
votes
1answer
60 views

A calculus problem

Question: Suppose that $u(x,t)$ is continuous, together with its first and second partial derivatives; suppose that $u$ and its first partial derivatives are periodic in $x$ of period $1,$ and ...
1
vote
3answers
76 views

Evaluate the limit $\lim_{t\rightarrow\infty}\left(te^t\int_t^{\infty}\frac{e^{-s}}{s}\text{d}s\right)$

$$\lim_{t\rightarrow\infty}\left(te^t\int_t^{\infty}\frac{e^{-s}}{s}\text{d}s\right)$$ I have no idea where to start. Any help will be appreciated!
4
votes
2answers
66 views

$f'$ is bounded and isn't continuous on $(a,b)$, so there's a point $y\in(a,b)$ such that $\lim_{x\to y}f'$ does not exist

Prove/disprove: $f$ has a bounded derivative and $f'$ isn't continuous on $(a,b)$, so there's a point $y\in(a,b)$ such that $\displaystyle\lim_{x\to y}f'$ does not exist. I think that if $f'$ ...
5
votes
0answers
85 views

Solve PDE by getting two ODEs

My goal is to solve this PDE for $f:[-1,1] \times \mathbb{R}_{\ge 0}\rightarrow \mathbb{C}$ $$ \partial_t f(x,t) = -\partial_x^2 f(x,t) + g(t)V(x)f(x,t).$$ I would consider this PDE to be solved if ...
0
votes
1answer
39 views

surface integral (curl F n ds)

Let $F$ be a vector field and let $n$ be normal vector of the closed surface $S$. Then show that $$\iint_S \mathrm{curl} \ F \cdot n\ ds=0. $$ I need help on this exercise.
5
votes
4answers
232 views

Prove no existing a smooth function satisfying … related to Morse Theory

i) Show that there does not exist a smooth function $f:\mathbb{R} \rightarrow \mathbb{R}$, s.t. $f(x) \geq 0$, $\forall x \in \mathbb{R}$, $f$ has exactly two critical points, $x_1,x_2\in\mathbb{R}$ ...
1
vote
1answer
42 views

proof of coarea formula for n dimensional hypersurface in $R^n$

$f:R^n \rightarrow R$ be continuous and summable. please give the proof for these formulas $\int_{R^n}f dx = \int_0^\infty(\int_{\partial B(x_0,r)}fdS)dr$ $\frac{d}{dr}\int_{ ...
5
votes
2answers
54 views

line integral: anticlockwise parametrisation in $\mathbb R^3$

Consider $\gamma$ given by the sides of the triangle with vertices $(0,0,1)^t$, $(0,1,0)^t$ and $(1,0,0)^t$. So $\gamma$ runs through the sides of the triangle. Let $f(x,y,z)=(y,xz,x^2)$. I want to ...
1
vote
3answers
26 views

a simple multivariable limit

$\lim_{(x,y)\to(0,0)}\frac{-x}{\sqrt{x^2+y^2}}$ I get confused finding this limit. I approach with lines $y=mx$ and i get $\lim_{x\to 0}\frac{-x}{\sqrt{x^2+m^2x^2}}$. How can i ended that this limit ...
1
vote
0answers
42 views

Most Suitable Book after Kline's Calculus?

I've been working through Morris Kline's Calculus: An Intuitive and Physical Approach and it's an absolutely excellent book for self-studying applied single-variable (and some multi-variable) calculus ...
0
votes
3answers
145 views

Baby Rudin without knowing multivariate?

I have read Spivak's Calculus and it has went well. I didn't have any problem with the rigorosity of the book at all. Now, I have never had any experience in multivariate. I only have experience with ...
2
votes
1answer
59 views

the series $\sum_{k=1}^\infty a_k$ converges implies the series $\sum_{k=1}^\infty a_k\sin (k\pi x)$ converges for $x$ irrational

Let $\sum_{k=1}^\infty a_k$ be a convergent series. Then can we obtain $\sum_{k=1}^\infty a_k\sin (k\pi x)$ converges for $x$ irrational? If $\sum_{k=1}^\infty a_k$ converges absolutely, then I can ...
3
votes
3answers
275 views

Limit - Could you help me with it

Can you help me with this limit? What do I have to do? I'm lost. $$\lim_{n\to\infty}n\left(\sum_{i=1}^{n}\dfrac{1}{(n+i)^2}\right)$$ The solution given is $\dfrac{1}{2}$.
-1
votes
4answers
145 views

Does $\sum_{j = 1}^{\infty} \sqrt{\frac{j!}{j^j}}$ converge? [closed]

I need to solve $$\sum_{j = 1}^{\infty} \sqrt{\frac{j!}{j^j}}$$ Does this converge or diverge and why?
0
votes
1answer
31 views

Help me how to find the limit. In this case something is strange.

A fn(x) is a sequence of function. fn:[0,1]→R defined by fn(x) = n(1-x) × x^n. And I want to know limit of this function as n→∞. In my opinion we can see this like this n(1-x) × x^n. So as n→∞, ...