Let $f : D \subseteq\Bbb R^3 \to \Bbb R$ be a $C^1$ function. A point $(a, b, c) \in \Bbb R^3$ is a critical point of $f$ if $$ \nabla f(a, b, c) = (0, 0, 0). $$ The corresponding number $k = f(a, b, c)$ is called a critical value: any real number $c$ that is not a critical value is called a re...

Check out this answer for a real method to evaluate this integral. First note that $$ \int_0^\infty e^{-x^2}\,\mathrm{d}x=\frac{\sqrt\pi}2\tag{1} $$ The integral $$ \int_\gamma e^{-z^2}\,\mathrm{d}z=0\tag{2} $$ over the curve $\gamma$ consisting of the line from $0$ to $R$ then counterclockwise...

Problem: Suppose $X$ is a topological space. Let $(A_n)_n$ be a sequence of connected subsets of $X$, for which, $A_n\cap A_{n+1}\neq \varnothing$. Then, $A=\bigcup_n A_n$ is connected. My attempt: Suppose $\bigcup_n A_n$ is disconnected, so there exists a non-constant continuous function $f:\b...

Let $A$ be a proper subset of $X$ and $B$ a proper subset of $Y$. If $X,Y$ are connected. Show that $X\times Y\backslash (A\times B)$ is connected. Lemma: Let $X$ be a space and $A_1,A_2...,A_n$ a finite sequence of connected subsets in $X$. If $A_j\cap A_{j+1}\neq \varnothing$ for each $j=1,2....