This is an exercise in Inverse Function Theorem http://en.wikipedia.org/wiki/Inverse_function_theorem we are given the function $f:\mathbb R^2 \to \mathbb R^2$, $f(x,y)=(e^x \cos y,e^x \sin y)$ 1) Show that $f$ is injective around every point in $\mathbb R^2$ – I managed to solve this 2) Find environments around the points $(0,\pi)$ and $(-1,\frac{\pi}{2})$ such that $f$ is […]

I can kind of understand the main direction (slope) of $y$ over the different $x$ intervals, but I can’t figure out why the values of $y$ take on the shape of straight lines and not curves looking more like those of sin, cos… EDIT: I understand that the derivative of Arccos(Sin(x)) gives 1 or -1 […]

If $\Phi(y)$ is a monotonic decreasing function is true that $$\frac{d\Phi^{-1}(y)}{dy} = \frac{1}{\Phi'(\Phi^{-1}(y))}$$ If so, how? It works for $y = \Phi(x) = e^{-x}, \quad \Phi^{-1}(y) = -log(y), \quad \frac{d\Phi^{-1}(y)}{dy} = \frac{-1}{y}, \quad $

I have found three cases: 1) If $f$ and $g$ are the same function. 2) If $f$ and $g$ are mutually inverse. 3) If both are polynomials of degree $1$ Maybe there are more.

Why is $x^{1/n}$ continuous for positive $x,n$ where $n$ is an integer? I can’t see how it follows from the definition of limit. And I don’t see any suitable inequalities so is this an application of Bernoulli’s or Jensen’s inequality?

I was wondering if this answer would be correct the multiplicative of $11$ modulo $59$ would be $5$ hence $5\cdot11 \equiv 4 \pmod{59}$. Is this correct?

Let $A$ be an invertible skew-symmetric $(2n \times 2n)$-matrix. Prove that $A^{-1}$ is also skew-symmetric. (You may assume that $(AB)^T = B^TA^T$). I did this with a $2 \times 2$ matrix and got that it worked, but I don’t know how to show it for a general $2n \times 2n$ matrix, as it is a […]

I am wondering if this holds in every single case: $$\sqrt{x^2} = \pm x$$ Specifically in this case: $$\sqrt{\left(\frac{1}{4}\right)^2}$$ In this one we know that the number is positive before squaring, so after removing the square and root shouldn’t we just have: $$\frac{1}{4}$$ Also in a case such as this: $$\sqrt{(\sqrt{576})-8}$$ we have $$\sqrt{\pm24-8}$$ which […]

Let $G$ a directed graph and $A$ the corresponding adjacency matrix. Let denote the identity matrix with $I$. I’ve read in a wikipedia article, that the following statement is true. Question. Is it true, that $I-A$ matrix is invertible if and only if there is no directed cycle in $G$?

Let $A$ be an $n \times n$ invertible matrix with \begin{align} \left(\begin{array}{ccc} a_{11} & \cdots & a_{1n} \\ \vdots & \ddots & \vdots \\ a_{n1} & \cdots & a_{nn} \end{array}\right)^{-1}= \left(\begin{array}{ccc} b_{11} & \cdots & b_{1n} \\ \vdots & \ddots & \vdots \\ b_{n1} & \cdots & b_{nn} \end{array}\right) \end{align} Is there an explicit formula […]

Intereting Posts

The free abelian group monad
Useful mathematical fora
Continuous function positive at a point is positive in a neighborhood of that point
The dual space of locally integrable function space
$1989 \mid n^{n^{n^{n}}} – n^{n^{n}}$ for integer $n \ge 3$
How are the elementary arithmetics defined?
converting a recursive formula into a non-recursive formula.
The proof of the Helmholtz decomposition theorem through Neumann boundary value problem
Why $\sqrt{-1 \times -1} \neq \sqrt{-1}^2$?
A contest math integral: $\int_1^\infty \frac{\text{d}x}{\pi^{nx}-1}$
for which values of $p$ integral $\int_{\Omega}\frac{1}{|x|^p}$ exists?
Show $S = f^{-1}(f(S))$ for all subsets $S$ iff $f$ is injective
Comparing the Lebesgue measure of an open set and its closure
If $F(\alpha)=F(\beta)$, must $\alpha$ and $\beta$ have the same minimal polynomial?
Convexity of functions and second derivative