All Questions
1,376
questions
0
votes
1
answer
61
views
How to show that the function: $ f: (0,1) \to \mathbb{R}^3, x \mapsto (\sin(x),\cos(x),x^2 )$ is injective?
How to show that the function: $ f: (0,1) \to \mathbb{R}^3, x \mapsto (\sin(x),\cos(x),x^2 )$ is injective?
To prove this, I was thinking to start by supposing that $f(x_1)=f(x_2)$; this implies that ...
- 1,211
0
votes
1
answer
70
views
Prove that a function between two topological spaces is a homeomorphism
Is the complement of the closed unit disk in the plane homeomorphic with $\mathbb R^2\setminus \{(0,0)\} $ ?
My question is about the above, somebody has defined a function and its fine. But for ...
- 4,624
2
votes
2
answers
228
views
Extension of a differentiable function $f$ to an open superset
This is a question the book Munkres-Calculus on Manifolds pg.144(Exercise 3-b)
If $f :S\to \mathbb R$ and $f$ is differentiable of class $C^r$ at each point $x_0$ of $S$,then $f$ may be extended to a $...
- 1,138
1
vote
1
answer
23
views
Help to write the domain in a forma correct way
The question is simple: the domain of $f(x, y) = \ln(\sqrt{xy} + 1)$.
Now this is just $xy \geq 0$, which means either $x \geq 0$ and $y \geq 0$ or $y \leq 0$ and $y \leq 0$. This is easy to say and ...
- 157k
2
votes
2
answers
128
views
How to evaluate double integral: $\iint \frac{y}{x} \, dx \, dy$ if it is in the first quadrant and is bounded by: $y=0$, $y=x$, and $x^2 + 4y^2 = 4$
I want to evaluate this double integral:
$$
\iint \frac{y}{x} \, dx \, dy \quad
$$
which is bounded by functions:
$$
y = 0 \quad
$$
$$
y = x \quad
$$
$$
x^2 + 4y^2 = 4 \quad
$$
And is in the first ...
- 8,455
3
votes
2
answers
82
views
Existence of $1/(e-1)$ value of a continous and differentiable function $f:[0,1]\to \mathbb R$ given $f(1)=1, f(0)=0$
Let $f:[0,1]\to \mathbb R$ be a continous function on $[0,1]$ and differentiable on $(0,1)$, $f(0)=0, f(1)=1$. Prove that there exists $c\in (0,1)$ so that $f(c)+\frac{1}{e-1}=f'(c)$ where $e$ is the ...
0
votes
0
answers
54
views
Customizing the bump function
I have the standard bump function below.
$$ \Psi(x) = e^{-\frac{1}{1 - \mathrm{min}(1, x^2)}} $$
How can I customize it to be like below:
Translate and scale
I can translate and scale by:
$$ \Psi(x) ...
- 271
0
votes
1
answer
1k
views
When right inverse of a surjective mapping is continuous?
It is proved that if $(X,d)$ is a metric space and $f: X\to X$ is surjective, there exists $f^\ast:X\to X$ such that $f\circ f^\ast x=x$ for all $x \in X$. Here, $f^\ast$ is called right inverse.
I ...
CommunityBot
- 1
1
vote
1
answer
19
views
Determine if a vector valued function has a root based on the behaviour on the boundary
Say $f: \mathbb R^n \to \mathbb R^n$ is a vector valued continuous function. Say we further know that $t^\top f(t) \ge 0$ $\,\forall \,\,||t||=1$. Does this imply that $f$ has a root in the unit ball.
...
- 32.7k
0
votes
0
answers
18
views
Functions of bounded variation - inequality
Let $u: \mathbb{R} \to \mathbb{R}$ such that $u \in C^1 (\mathbb{R}) \cap \mathrm{BV} (\mathbb{R})$. Prove that
$$\frac{1}{\varepsilon} \int_{-\infty}^{\infty} |u(x+\varepsilon) - u (x)| dx \leq TV (u)...
- 31
1
vote
0
answers
40
views
Parametric prolate epicycloid modelling and integration
I was trying to model an epicycloid for my math assignment but none of the parametric equations I found ended up helping me model it on desmos.
One of the more prominent equations I found on the ...
- 1,512
1
vote
1
answer
32
views
Searching functions that satisfy the differential inequalities
I'm trying to find two nonnegative continuous functions $f_1$ and $f_2$ defined on a bounded domain $[0,a]$ such that satisfies the next conditions:
\begin{align}
f_1(x) > & f_2(x),\quad\text{...
- 21.6k
5
votes
3
answers
21k
views
Increasing/Decreasing intervals of a parabola
I am being told to find the intervals on which the function is increasing or decreasing.
It is a normal positive parabola with the vertex at $(3,0).$ The equation could be $y = (x-3)^2,$ but my ...
- 1,512
1
vote
2
answers
72
views
Confusion about two branches of Lambert $W$ function
I am confused about the usage of the two branches $W_{0}$ and $W_{-1}$ of Lambert $W$ function. Suppose $\epsilon > 0$ and we have an equation for the form:
$$-xe^{-x} = -e^{-1-\epsilon}$$
Note ...
- 7,212
0
votes
0
answers
184
views
Show that there is a $c$, such that $f'(c)=0$
Given that $f\colon \mathbb{R} \to \mathbb{R}$ is a continuous and differentiable function satisfying
$$\int_a^b x^2f(x)dx ≠ \left(\frac{b ^ 3}{3} - \frac{a ^ 3}{3}\right)\cdot f(c) $$
for any $a, b \...
- 19