https://liberty.ioCurated star highlights with fresh energy.en-usTue, 18 Aug 2026 08:33:48 -0400https://liberty.io/operation-that-is-commutative-but-not-associative-duplicate.htmlhttps://liberty.io/operation-that-is-commutative-but-not-associative-duplicate.htmlDoes there exist an operation that is commutative but not associative? do you know what is operation?...Tue, 18 Aug 2026 12:32:32 -0400https://liberty.io/how-to-calculate-the-surface-area-of-parametric-surface.htmlhttps://liberty.io/how-to-calculate-the-surface-area-of-parametric-surface.htmlSuppose you have the surface $z=3xy$ and you want to find the area that lies within the cylinder $x^2+y^2\leq 1$. My homework is forcing me to use the parameterization $$\textbf{r}_1(s,t)= <......Tue, 18 Aug 2026 12:25:53 -0400https://liberty.io/rational-solutions-for-x-2-y-2-3.htmlhttps://liberty.io/rational-solutions-for-x-2-y-2-3.htmlAre there any rational numbers $x$ and $y$ such that $x^2+y^2=3$. I think there are no rational solutions, but I haven't been able to prove it....Tue, 18 Aug 2026 12:23:35 -0400https://liberty.io/why-is-sphere-non-euclidean-space.htmlhttps://liberty.io/why-is-sphere-non-euclidean-space.htmlThere are 5 axioms that define euclidean space and I believe that all hold also for a sphere. The definition of axioms from wikipedia: "To draw a straight line from any point to any point." Defi......Tue, 18 Aug 2026 12:13:25 -0400https://liberty.io/size-of-eigenvalues-and-diagonal-dominance.htmlhttps://liberty.io/size-of-eigenvalues-and-diagonal-dominance.htmlIf $X$ is a diagonally dominant matrix and if $Y$ is a diagonal matrix with the diagonal elements of $X$, then how can one show that $ Y^{-1}(Y - X)$ has eigenvalues whose magnitudes are all strictly ...Tue, 18 Aug 2026 12:03:01 -0400https://liberty.io/finding-roots-of-z-3-8.htmlhttps://liberty.io/finding-roots-of-z-3-8.htmlI am having trouble finding the cubed roots of $8$ as a complex number. $$z^3 = 8+0i$$ $$z^3 = r^3 e^{3\theta i}=8e^{2i\pi k},\quad k\in \Bbb Z$$ $$\implies r=2,3\theta = 2\pi k\implies \theta = ......Tue, 18 Aug 2026 11:56:00 -0400https://liberty.io/when-to-stop-checking-if-a-transition-matrix-is-regular.htmlhttps://liberty.io/when-to-stop-checking-if-a-transition-matrix-is-regular.htmlThe definition that I have of a Transition Matrix for a Markov Chain is: A transition matrix is regular if some power of it is positive. Doesn't this mean though that in theory, you could keep ...Tue, 18 Aug 2026 11:53:03 -0400https://liberty.io/how-to-show-that-if-f-2-x-is-uniformly-continous-function-then-f-is-un.htmlhttps://liberty.io/how-to-show-that-if-f-2-x-is-uniformly-continous-function-then-f-is-un.html$f:R\to [0,\infty)$ is function such that $f^2(x)$ is uniformly continuous on R then I have to show that f is uniformly continuous ? My attempt : $|f^2(x)-f^2(y)|<\epsilon$ for $|x-y|<\del......Tue, 18 Aug 2026 11:45:35 -0400https://liberty.io/what-is-the-limit-of-lim-x-1-x-1-x-when-x-approaches-infinity-without-.htmlhttps://liberty.io/what-is-the-limit-of-lim-x-1-x-1-x-when-x-approaches-infinity-without-.htmlthis is my first post. I've been struggling whit this limit for too long (without using l'Hôpital's rule): $$\lim_{x\to {\infty}} \left(\frac{x-1}{x+1}\right)^x$$ My answer is $\frac1e$, but the ...Tue, 18 Aug 2026 11:35:18 -0400https://liberty.io/how-to-find-net-torque-closed.htmlhttps://liberty.io/how-to-find-net-torque-closed.htmlFor the meter stick in Figure 10-4, what is the magnitude of the net torque due to both forces $F_1$ and $F_2$ about an axis perpendicular to the page through point $A$? Is it clockwise or ...Tue, 18 Aug 2026 11:31:13 -0400https://liberty.io/proving-lim-x-n-a-n-as-x-goes-to-a-from-definition.htmlhttps://liberty.io/proving-lim-x-n-a-n-as-x-goes-to-a-from-definition.htmlUsing the $\varepsilon-\delta$ definition show that $\displaystyle \lim_{x \to a} x^n = a^n$ What I've done? Pretty much stuck on the step where one usually constructs delta: $\displaystyle |x^......Tue, 18 Aug 2026 11:28:56 -0400https://liberty.io/proof-that-there-are-infinitely-many-primes-congruent-to-1-mod-3.htmlhttps://liberty.io/proof-that-there-are-infinitely-many-primes-congruent-to-1-mod-3.htmlNeed a quick check on this proof. Show that there are infinitely many primes congruent to 1 mod 3. Proof: Suppose by contradiction finitely many primes $p≡1\pmod{3}$ say $p_1,...,p_k$. Define $m......Tue, 18 Aug 2026 11:28:22 -0400https://liberty.io/notation-convention-for-integer-in-a-certain-range.htmlhttps://liberty.io/notation-convention-for-integer-in-a-certain-range.htmlI am wondering what notation I should use when writing that some variable is an integer within some range. What is the most common way to do this? Here are some ideas I have but I'm not sure what the ...Tue, 18 Aug 2026 11:12:06 -0400https://liberty.io/calculating-the-divergence-of-a-tensor.htmlhttps://liberty.io/calculating-the-divergence-of-a-tensor.htmlI am working through a fluid dynamics paper and came across this equation: $$ \frac{\partial \vec{v}}{\partial t} + \vec{v}\cdot\nabla\vec{v}=\nabla\cdot T - \frac{1}{\rho}\nabla \phi\tag1$$ where ......Tue, 18 Aug 2026 11:09:36 -0400https://liberty.io/find-the-derivative-of-y-2-x-as-a-function-of-y.htmlhttps://liberty.io/find-the-derivative-of-y-2-x-as-a-function-of-y.htmlFind the derivative of $$y^2=x$$ as a function of y. i have found for the function of x, it will be $$\pm\frac{1}{2\sqrt x}$$ however for the function of y will be $$\frac{dy}{dx}=2y$$ ? it loo......Tue, 18 Aug 2026 11:06:19 -0400https://liberty.io/where-does-b-2a-come-from.htmlhttps://liberty.io/where-does-b-2a-come-from.htmlIve came across the idea to find the minimum/maximum of a parabola , I keep running into the formula $-b/2a$ i wanted to know where does this actually come from ? I know that the parabola has to be ...Tue, 18 Aug 2026 11:03:44 -0400https://liberty.io/how-can-you-prove-that-1-5-9-cdots-4n-3-2n-2-n-without-using-induction.htmlhttps://liberty.io/how-can-you-prove-that-1-5-9-cdots-4n-3-2n-2-n-without-using-induction.htmlUsing mathematical induction, I have proved that $$1+ 5+ 9 + \cdots +(4n-3) = 2n^{2} - n$$ for every integer $n > 0$. I would like to know if there is another way of proving this result wi......Tue, 18 Aug 2026 10:50:29 -0400https://liberty.io/how-to-simplify-a-sum-of-exponential-equation.htmlhttps://liberty.io/how-to-simplify-a-sum-of-exponential-equation.htmlSuppose I have three constants $a, b, c\in R$. I have a formulation as $f=e^{ab}+e^{ac}$. Can I have some result like $f'=e^{a(b+c)}$. I know $f'$ does not hold. But I just want to combine the two ......Tue, 18 Aug 2026 10:49:36 -0400https://liberty.io/prove-that-if-f-n-in-o-g-n-then-g-n-in-omega-f-n.htmlhttps://liberty.io/prove-that-if-f-n-in-o-g-n-then-g-n-in-omega-f-n.htmlProve that if $f(n) \in O(g(n))$ then $g(n) \in \Omega(f(n))$. So I know that $$O(g(n)) \Rightarrow f(n) \leq c\cdot g(n)$$ and that $$\Omega(g(n)) \Rightarrow f(n) \geq c\cdot g(n)$$ but how ......Tue, 18 Aug 2026 10:47:54 -0400https://liberty.io/what-is-the-practical-application-of-factorials.htmlhttps://liberty.io/what-is-the-practical-application-of-factorials.htmlI'm trying to understand the practical application of factorial - in simple applications. I searched the math.stackexchange and could not find an answer. I understand that a factorial of n items......Tue, 18 Aug 2026 10:33:18 -0400https://liberty.io/natural-deduction-proof-problem-with-existential-elimination.htmlhttps://liberty.io/natural-deduction-proof-problem-with-existential-elimination.htmlI have problems with the following proof: $$ \forall x \exists y (Rxy \land Py), \forall x \neg Rxx \vdash \neg \forall x \forall y (Px \implies (Py \implies x=y)) $$ The problem is with the applic......Tue, 18 Aug 2026 10:30:16 -0400https://liberty.io/25-chance-of-happening-but-has-to-happen-4-times-in-a-row-what-is-the-.htmlhttps://liberty.io/25-chance-of-happening-but-has-to-happen-4-times-in-a-row-what-is-the-.htmlSay you need to win 4 times in a row at a 25% success chance for each event, how do I calculate the chance of winning? Overall, how does this calculation goes? given that 1st is a win, what is the ......Tue, 18 Aug 2026 10:16:05 -0400https://liberty.io/difference-between-state-space-and-sample-space.htmlhttps://liberty.io/difference-between-state-space-and-sample-space.htmlAren't they both the same? According to me they are just different terminologies used for different types of systems. What I understand is that sample space is used when we talk about static system......Tue, 18 Aug 2026 10:08:55 -0400https://liberty.io/which-value-goes-on-which-axis-in-plot-a-against-b.htmlhttps://liberty.io/which-value-goes-on-which-axis-in-plot-a-against-b.htmlWhen the question asks to plot $a$ against $b$, which value should go on which axis?...Tue, 18 Aug 2026 10:07:49 -0400https://liberty.io/can-039-t-understand-a-recursive-definition-of-concatenation-of-two-st.htmlhttps://liberty.io/can-039-t-understand-a-recursive-definition-of-concatenation-of-two-st.htmlI'm reading Rosen's Discrete Mathematics and its applications(6ed), but I can't understand a recursive definition about concatenation of two strings: Two strings can be combined via the operatio......Tue, 18 Aug 2026 09:44:44 -0400https://liberty.io/probability-of-getting-a-triple-when-rolling-12-fair-12-sided-dice-onc.htmlhttps://liberty.io/probability-of-getting-a-triple-when-rolling-12-fair-12-sided-dice-onc.htmlThere are $12^{12}$ = 8,916,100,448,256 total possible outcomes. I decided to approach it by calculating the probability of rolling the $12$ dice WITHOUT getting a triple and subtract from $1$. I f......Tue, 18 Aug 2026 09:27:04 -0400https://liberty.io/what-is-x-times-sqrt-x.htmlhttps://liberty.io/what-is-x-times-sqrt-x.htmlDoes $x \times \sqrt x = x$? I thought it was correct because sqrt is the opposite of multiplying by a number, so I figured by multiplying by a number it would balance out and be that number norm......Tue, 18 Aug 2026 09:26:15 -0400https://liberty.io/representing-ln-x-as-a-power-series-centered-at-2-without-computing-an.htmlhttps://liberty.io/representing-ln-x-as-a-power-series-centered-at-2-without-computing-an.htmlI am working through a calc book and one of the problems asks the above question. However, taylor and maclaurin series have not been introduced yet. In some worked examples, they leverage old ser......Tue, 18 Aug 2026 09:11:34 -0400https://liberty.io/what-is-the-number-of-numbers.htmlhttps://liberty.io/what-is-the-number-of-numbers.htmlFor a positive integer $n≥2$, let $f(n)$ denote the smallest prime divisor of $n$. Find the number of integers $n$, $2≤n≤1000$, such that $f(n)=5$. All the numbers I could think of were $5,25,35,55......Tue, 18 Aug 2026 09:07:29 -0400https://liberty.io/prove-the-trigonometric-identities-cot-x-cot-2x-csc-2x.htmlhttps://liberty.io/prove-the-trigonometric-identities-cot-x-cot-2x-csc-2x.htmlProve $\cot(x) - \cot(2x) =\csc(2x)$. I start to solve from LHS, and change all the terms into $\sin$ and $\cos$, but I could not prove it into $\csc(2x)$....Tue, 18 Aug 2026 09:03:45 -0400https://liberty.io/express-the-following-complex-numbers-in-the-form-of-a-bi.htmlhttps://liberty.io/express-the-following-complex-numbers-in-the-form-of-a-bi.htmlI'm struggling with the following exercises: I tried to use the reasoning as follows: $$(a+bi)^n=(re^{\theta i})^n=r^ne^{\theta in}=r^n(\cos(\theta n)+i\sin(\theta n))$$ So for the first one I d......Tue, 18 Aug 2026 09:00:07 -0400https://liberty.io/the-difference-of-two-consecutive-perfect-squares-is-always-odd.htmlhttps://liberty.io/the-difference-of-two-consecutive-perfect-squares-is-always-odd.htmlI am working on another homework assignment about proofs. The question is: Prove or find counterexample: the difference of two consecutive perfect squares is odd? There is no counterexample corre......Tue, 18 Aug 2026 08:35:54 -0400https://liberty.io/i-ab-be-invertible-leftrightarrow-i-ba-is-invertible-duplicate.htmlhttps://liberty.io/i-ab-be-invertible-leftrightarrow-i-ba-is-invertible-duplicate.htmlassume $A,B\in M_n(F)$ if $I-AB$ be invertible then how to prove $I-BA$ is invertible and how find inverse of $I-BA$ Thanks in advance...Tue, 18 Aug 2026 08:34:19 -0400https://liberty.io/do-the-taylor-series-of-sin-x-and-cos-x-depend-on-the-identity-sin-2-x.htmlhttps://liberty.io/do-the-taylor-series-of-sin-x-and-cos-x-depend-on-the-identity-sin-2-x.htmlI had this crazy idea trying to prove the Pythagorean trigonometric identity;$$\sin^2x+\cos^2x=1$$by squaring the infinite Taylor series of $\sin x$ and $\cos x$. But it came out quite beautiful, ...Tue, 18 Aug 2026 08:30:23 -0400https://liberty.io/help-with-proof-of-injection-and-surjection.htmlhttps://liberty.io/help-with-proof-of-injection-and-surjection.htmlFor the record, I am sorry, I haven't yet learnt how to use LaTeX I have a function $f(x) = 2x^3 - 1$ My proof of injection is as follows: $f$ is one to one for all $x_1,x_2$ element $X$, if $f(x......Tue, 18 Aug 2026 08:14:51 -0400https://liberty.io/relationship-between-primes-and-fibonacci-sequence.htmlhttps://liberty.io/relationship-between-primes-and-fibonacci-sequence.htmlI recently stumbled across an unexpected relationship between the prime numbers and the Fibonacci sequence. We know a lot about Fibonacci numbers but relatively little about primes, so this connect......Tue, 18 Aug 2026 07:58:37 -0400https://liberty.io/integral-definition-of-e.htmlhttps://liberty.io/integral-definition-of-e.htmlI know that $e$ can be defined via a convergent series: $$ e = \sum_{n=0}^\infty {1\over n!}$$ Or as a limit: $$ e = \lim_{n \to \infty} { \left(1 + {1 \over n}\right)^n }$$ Or as the value which ...Tue, 18 Aug 2026 07:54:49 -0400https://liberty.io/how-many-zeros-are-in-all-natural-numbers-from-1-to-2019.htmlhttps://liberty.io/how-many-zeros-are-in-all-natural-numbers-from-1-to-2019.htmlProblem : How many zeros are in all natural numbers from $1$ to $2019$ ? My attempt : From $1$ to $100$ we have $11$ zeros Now from $101$ to $110$ we have $10$ zeros So from $20$ zeros ...Tue, 18 Aug 2026 07:43:23 -0400https://liberty.io/optimizing-svd-factors.htmlhttps://liberty.io/optimizing-svd-factors.htmlProblem Description My aim is to find the SVD decomposition of the minimizer of $F:\mathbb{R}^{m\times n}\to\mathbb{R}$ $$ U_*\Sigma_* V^\top_* = A_*\qquad \text{where} \qquad A_* = \arg\min_A F(A)......Tue, 18 Aug 2026 07:37:53 -0400https://liberty.io/ratio-of-triangle-a-and-b-if-the-length-of-the-sides-are-a-25-25-30-an.htmlhttps://liberty.io/ratio-of-triangle-a-and-b-if-the-length-of-the-sides-are-a-25-25-30-an.htmlIf $A$ triangle's side length are $25,25$ and $30$ and $B$ triangle's length are $25,25$ and $40$ what is the ratio between the two areas of the triangles? #math...Tue, 18 Aug 2026 07:36:12 -0400https://liberty.io/i-have-a-doubt-in-roman-numericals.htmlhttps://liberty.io/i-have-a-doubt-in-roman-numericals.htmlThe below question, I am looking at the 80, They translated it as LXXX, but I can also write it as XXC right?...Tue, 18 Aug 2026 07:20:21 -0400https://liberty.io/how-do-inverse-functions-exist-for-exponential-functions.htmlhttps://liberty.io/how-do-inverse-functions-exist-for-exponential-functions.htmlI know that they exist for exponential functions (we currently have them in class), but to me it doesn't seem "reasonable" when I look at the definition of what an inverse function is. The inverse is ...Tue, 18 Aug 2026 07:14:30 -0400https://liberty.io/deriving-the-variance-of-the-bernoulli-distribution.htmlhttps://liberty.io/deriving-the-variance-of-the-bernoulli-distribution.htmlFor a Bernoulli distribution, $\mu_X = p$. I can easily derive this from the general equation for mean of a discrete random variable: $$ \mu_X=\sum_{i=1}^kx_iPr(X=x) $$ $$ \mu_X=1(p)+0(1-p)=p $$ I ......Tue, 18 Aug 2026 07:09:06 -0400https://liberty.io/why-do-hyperbolic-trig-functions-seem-to-be-encountered-rarely.htmlhttps://liberty.io/why-do-hyperbolic-trig-functions-seem-to-be-encountered-rarely.htmlHyperbolic "trig" functions such as $\sinh$, $\cosh$, have close analogies with regular trig functions such as $\sin$ and $\cos$. Yet the hyperbolic versions seem to be encountered relatively rarel......Tue, 18 Aug 2026 07:07:26 -0400https://liberty.io/mathematical-symbol-for-and.htmlhttps://liberty.io/mathematical-symbol-for-and.htmlI have found some pretty complete lists (I think) of mathematical symbols here and here, but I don't see a symbol for the word "and" on either list. A person could easily just write the word "and"......Tue, 18 Aug 2026 06:58:17 -0400https://liberty.io/geometrical-interpretation-of-trigonometric-antiderivative.htmlhttps://liberty.io/geometrical-interpretation-of-trigonometric-antiderivative.htmlI know about geometrical explanation of [definite] integral as an area under the curve, and I wonder if there are some ideas, which may give similar insight in taking antiderivatives [indefinite ...Tue, 18 Aug 2026 06:56:00 -0400https://liberty.io/are-the-graphs-of-inverse-functions-always-reflections-of-the-function.htmlhttps://liberty.io/are-the-graphs-of-inverse-functions-always-reflections-of-the-function.htmlFor an invertible function, is the graph of its inverse always the mirror image of the function's graph in the line $y=x$? And, if yes, then why is it so?...Tue, 18 Aug 2026 06:53:27 -0400https://liberty.io/what-books-are-prerequisites-for-spivak-039-s-calculus.htmlhttps://liberty.io/what-books-are-prerequisites-for-spivak-039-s-calculus.htmlFor financial reasons, I dropped out my senior year of college as a piano performance major. I will be returning to college to dual major in mathematics and computer science. I've taught myself to ...Tue, 18 Aug 2026 06:48:21 -0400https://liberty.io/what-calculation-shortcuts-exist-to-help-or-speed-up-mental-or-paper-c.htmlhttps://liberty.io/what-calculation-shortcuts-exist-to-help-or-speed-up-mental-or-paper-c.htmlAnything to speed up or simplify calculations. A simple example might be to get a multiple of $19$, for instance, $38 \cdot 19 = 38 \cdot 20 - 38$. (This is hard to tag with so few tags in play!) ...Tue, 18 Aug 2026 06:44:32 -0400https://liberty.io/using-operators-and-4-4-4-4-digits-find-all-formulas-that-would-resolv.htmlhttps://liberty.io/using-operators-and-4-4-4-4-digits-find-all-formulas-that-would-resolv.htmlI had a conversation with a colleague of mine and he brought up an interesting problem. Using the + - * / operators and four 4 4 4 4 digits, create an algorithm that will output all the formulas that ...Tue, 18 Aug 2026 06:44:31 -0400https://liberty.io/help-to-clarify-proof-of-euler-039-s-theorem-on-homogenous-equations.htmlhttps://liberty.io/help-to-clarify-proof-of-euler-039-s-theorem-on-homogenous-equations.htmlWhy is the last step (setting $\lambda = 1$ allowed? I have trouble accepting this because if I set $\lambda =1 $ at the very start, then: $f(\lambda x , \lambda y)=\lambda^r f(x,y)$ becomes $f......Tue, 18 Aug 2026 06:34:24 -0400https://liberty.io/how-do-i-find-the-composition-of-two-functions.htmlhttps://liberty.io/how-do-i-find-the-composition-of-two-functions.htmlIn a question sheet for my university course I am given the sets: $A = \{1, 2, 3, 4\}, B = \{x, y, z, w\}$ and $C = \{a, b, c\}$. I am then told to work out the functions: $f: A → B$ whi......Tue, 18 Aug 2026 06:33:22 -0400https://liberty.io/how-to-approximate-partial-derivatives-from-contour-map.htmlhttps://liberty.io/how-to-approximate-partial-derivatives-from-contour-map.htmlA problem of approximating partial derivatives from a given contour map Hi, I would like to know the connection between a contour map and partial derivatives. I understand that partial derivatives ......Tue, 18 Aug 2026 06:26:58 -0400https://liberty.io/creating-link-matrix.htmlhttps://liberty.io/creating-link-matrix.htmlI have a matrix called $ A $ with elements equal to $ 0 $ and $ 1 $. if element $ A _ { i j } $ is equal to $ 1 $, it means that member $ i $ is connected to member $ j $ directly. if elements $ A ......Tue, 18 Aug 2026 06:21:01 -0400https://liberty.io/i-can-039-t-find-a-way-to-simplify-this-boolean-algebra.htmlhttps://liberty.io/i-can-039-t-find-a-way-to-simplify-this-boolean-algebra.htmlxyz' + x'y + x'z the answer from boolean expression calculator on the internet is x'z + yz' I know the end x'z same like the answer. but I can't find a way to simplify xyz' + x'y = yz'...Tue, 18 Aug 2026 06:18:10 -0400https://liberty.io/how-many-integers-between-1-and-1000-use-exactly-three-digits.htmlhttps://liberty.io/how-many-integers-between-1-and-1000-use-exactly-three-digits.htmlThere is a problem in which I am obtaining a different answer than my professor. The problem is as follows: How many integers between $1$ and $1000$ use exactly three digits? The professor shows......Tue, 18 Aug 2026 06:15:36 -0400https://liberty.io/what-should-i-do-if-no-one-answers-my-question.htmlhttps://liberty.io/what-should-i-do-if-no-one-answers-my-question.htmlQ&A for people studying math at any level and professionals in related fields...Tue, 18 Aug 2026 06:14:34 -0400https://liberty.io/sum-of-total-scores-odds-will-end-even-or-odd.htmlhttps://liberty.io/sum-of-total-scores-odds-will-end-even-or-odd.htmlFriend and i are betting on the ending total scores of basketball game, he says odds are greater it will end with an even sum, i say the odds are equal. Who is right?...Tue, 18 Aug 2026 06:09:46 -0400https://liberty.io/this-projection-is-a-covering-map.htmlhttps://liberty.io/this-projection-is-a-covering-map.htmlIf $\tilde X$ is the product of $X$ with a discrete space, the projection $\tilde X \to X$ is a covering map. This question seems really easy, but as I'm a beginner there are some things a little ......Tue, 18 Aug 2026 06:05:03 -0400https://liberty.io/what-is-so-special-about-klein-4-group.htmlhttps://liberty.io/what-is-so-special-about-klein-4-group.htmlThis is my first course in abstract algebra and so far I am only learning about groups. So is there anyone who can explain to me why Klein 4-group is so special that it warrants a category of its o......Tue, 18 Aug 2026 05:59:44 -0400https://liberty.io/wrong-wolfram-alpha-result.htmlhttps://liberty.io/wrong-wolfram-alpha-result.htmlI have this function $$ f(x,y) = \left\{ \begin{array}{ll} \frac{x^3}{x^2 + y^2} & \mbox{if } (x,y) \neq (0,0) \\ 0 & \mbox{if } (x,y) = (0,0) \end{array} \right. $$ And I want to ......Tue, 18 Aug 2026 05:44:07 -0400https://liberty.io/find-a-vector-function-that-represents-the-curve-of-intersection-of-th.htmlhttps://liberty.io/find-a-vector-function-that-represents-the-curve-of-intersection-of-th.htmlThe cone $z=\sqrt{x^2+y^2}$ and the plane $z=1+y$ I parameterized the plane and put it into vector form: $t=1+y \to y=t-1$ $z= 1+t-1 \to z = t$ $y= t-1$, $z=t$ Since I'm finding the intersecti......Tue, 18 Aug 2026 05:32:38 -0400https://liberty.io/what-is-the-length-of-ef.htmlhttps://liberty.io/what-is-the-length-of-ef.htmlIn triangle $ABC$ , $AD$ is perpendicular to $BC$. It is given that $BC$=8 cm and $AD$=6 cm. If $E$ and $F$ are the mid points of $BD$ and $AC$ respectively, find $EF$. I tried using the mid point ...Tue, 18 Aug 2026 05:21:55 -0400https://liberty.io/mexican-hat-function-that-converges-to-1.htmlhttps://liberty.io/mexican-hat-function-that-converges-to-1.htmlThe Ricker wavelet, also known as the Mexican Hat function, converges to 0 for negative and positive infinity. I'm looking for a function with the same shape in the center, but that converges back......Tue, 18 Aug 2026 05:10:18 -0400https://liberty.io/reduction-of-order-method-for-3rd-order-de.htmlhttps://liberty.io/reduction-of-order-method-for-3rd-order-de.htmlUse the Reduction of Order method to solve $$(2-x)y'''+(2x-3)y''-xy'+y=0$$ (such that $x<2$) using $u(x)=e^x$. How do you use the method for $3$rd order... I have seen it been used on $2$nd but......Tue, 18 Aug 2026 04:58:54 -0400https://liberty.io/a-problem-with-progressive-percentage-incrementation.htmlhttps://liberty.io/a-problem-with-progressive-percentage-incrementation.htmlI have absolutely no clue if any of the terms I used in the title actually exist or make sense. I'm usually good at math (relatively) but I am facing this problem today that I just cannot solve. J......Tue, 18 Aug 2026 04:56:01 -0400https://liberty.io/is-zero-an-element-of-the-empty-set.htmlhttps://liberty.io/is-zero-an-element-of-the-empty-set.htmlI have a question about the empty set. Is $0$ (zero) an element of $\varnothing$? And what is the cardinality of {0}?...Tue, 18 Aug 2026 04:48:07 -0400https://liberty.io/rotate-a-point-in-circle-about-an-angle.htmlhttps://liberty.io/rotate-a-point-in-circle-about-an-angle.htmlHow should I rotate a point $(x,y)$ to location $(a,b)$ on the co-ordinate by any angle?...Tue, 18 Aug 2026 04:44:56 -0400https://liberty.io/equivalent-of-solutions-of-ivp.htmlhttps://liberty.io/equivalent-of-solutions-of-ivp.htmlConsider the IVP $y''-2y'+26y=0$, $y(0)=1$, $y'(0)=2$. From the characteristic equation $m^2-2m+26=0$, i found the roots as $m_1=1-5i$ and $m_2=1+5i$. Then when i use the basis solutions $y_1(x......Tue, 18 Aug 2026 04:37:38 -0400https://liberty.io/finding-new-standard-deviation-and-mean-after-adding-an-element.htmlhttps://liberty.io/finding-new-standard-deviation-and-mean-after-adding-an-element.htmlSay I have a mean and standard deviation for a dataset of 5 elements. I now add a sixth element. Is there a way to calculate the new mean and standard deviation using the information we had prior ......Tue, 18 Aug 2026 04:26:41 -0400https://liberty.io/two-holes-in-the-graph-when-dividing-polynomials.htmlhttps://liberty.io/two-holes-in-the-graph-when-dividing-polynomials.htmlSorry if this seems like a dumb question, or if it has been answered already. I did a quick search and didn't turn up anything, so here goes. I'm teaching a high school algebra class, and the book......Tue, 18 Aug 2026 04:21:36 -0400https://liberty.io/archimedes-trisection-of-equilateral-triangle.htmlhttps://liberty.io/archimedes-trisection-of-equilateral-triangle.htmlEquilateral triangle paradox Consider triangle ABC: Each side is trisected - AE, FC, GB each = 1/3 of their sides. First we established that triangle HIJ was equilateral. This was accomplished ......Tue, 18 Aug 2026 04:15:37 -0400https://liberty.io/tanget-to-the-curve-but-point-not-on-curve.htmlhttps://liberty.io/tanget-to-the-curve-but-point-not-on-curve.htmlQuestion is Find the $x$-coordinate of all points on the curve $$y=22x\sin(5x)+55\sqrt{3}x^2+68,\quad \frac\pi{10}<x<\frac{3\pi}{10}$$ where the tangent line passes through the point $P(0,68)......Tue, 18 Aug 2026 04:07:48 -0400https://liberty.io/minimum-number-of-steps-needed-to-solve-a-rubik-cube.htmlhttps://liberty.io/minimum-number-of-steps-needed-to-solve-a-rubik-cube.htmlLong time ago I've seen a book on group theory and there was an appendix about rubik cube. I remember there were only three steps that enabled me to solve my cube (three strings with letters encoding ...Tue, 18 Aug 2026 03:56:39 -0400https://liberty.io/how-to-graph-x-sin-x.htmlhttps://liberty.io/how-to-graph-x-sin-x.htmlGraph $x\sin(x)$ I am having problem with the period section of the graph. I can see that the function is even, and that as $x \rightarrow \pm\infty$, the amplitude $\rightarrow \infty$. I am use......Tue, 18 Aug 2026 03:56:24 -0400https://liberty.io/non-circular-ellipse-nomenclature.htmlhttps://liberty.io/non-circular-ellipse-nomenclature.htmlA circle a special case of an ellipse. Is there a general word to describe an ellipse that is specifically not a circle?...Tue, 18 Aug 2026 03:43:15 -0400https://liberty.io/why-does-the-integral-of-cot-x-have-absolute-value.htmlhttps://liberty.io/why-does-the-integral-of-cot-x-have-absolute-value.htmlThere's a proof of that integral on this website: http://math2.org/math/integrals/more/cot.htm I just don't understand why we need the absolute value....Tue, 18 Aug 2026 03:34:28 -0400https://liberty.io/formula-for-the-nth-partial-sum-of-a-telescoping-series.htmlhttps://liberty.io/formula-for-the-nth-partial-sum-of-a-telescoping-series.htmlFind the $n$th term for the sequence of partial sums for the series $$\sum_{n=1}^{\infty} \frac{5}{n(n+3)} =\sum_{n=1}^{\infty} \left(\frac{5}{3n}-\frac{5}{3(n+3)} \right)$$ and find $\lim\limits_{n\...Tue, 18 Aug 2026 03:32:02 -0400https://liberty.io/what-is-a-slower-decay-rate-than-hyperbolic.htmlhttps://liberty.io/what-is-a-slower-decay-rate-than-hyperbolic.htmlHyperbolic decay $$f_\alpha(x)=\frac{1}{\alpha x + 1} $$ is slower than exponential decay $$f(x) = e^{-x}$$ where $\alpha > 0$ is a scaling factor. The larger that $\alpha$ is, the steeper the d......Tue, 18 Aug 2026 03:29:57 -0400https://liberty.io/get-the-numbers-from-0-30-by-using-the-number-2-four-times.htmlhttps://liberty.io/get-the-numbers-from-0-30-by-using-the-number-2-four-times.htmlHow can I get the numbers from (0-30) by using the number $2$ four times.Use any common mathematical function and the (+,-,*,/,^) I tried to solve this puzzle, but I couldn't solve it completely. S......Tue, 18 Aug 2026 03:28:44 -0400https://liberty.io/expectation-on-1-x.htmlhttps://liberty.io/expectation-on-1-x.htmlIn general can one say that for a random variable X: $E[\frac{1}{X}] = \frac{1}{E[X]}$ ? I've worked out a few examples where this works but I'm not sure how widely this is useful......Tue, 18 Aug 2026 03:23:51 -0400https://liberty.io/too-simple-to-be-true.htmlhttps://liberty.io/too-simple-to-be-true.htmlAs indicates the title, this question is about "proofs" of true statements which are short and/or look elegant but are wrong. I mean example like Cayley-Hamilton's theorem, which states that for ......Tue, 18 Aug 2026 03:21:51 -0400https://liberty.io/why-are-box-topology-and-product-topology-different-on-infinite-produc.htmlhttps://liberty.io/why-are-box-topology-and-product-topology-different-on-infinite-produc.htmlWhy are box topology and product topology different on infinite products of topological spaces ? I'm reading Munkres's topology. He mentioned that fact but I can't see why it's true that they are ...Tue, 18 Aug 2026 03:13:33 -0400https://liberty.io/is-hubbard-and-hubbard-039-s-vector-calculus-linear-algebra-and-differ.htmlhttps://liberty.io/is-hubbard-and-hubbard-039-s-vector-calculus-linear-algebra-and-differ.htmlSince I have taken single variable calculus, proof based linear algebra and proof courses before multivariable calculus, I am used to some proof writing. Since I have taken about a year off due to ...Tue, 18 Aug 2026 03:09:41 -0400https://liberty.io/how-can-i-solve-this-hard-system-of-equations.htmlhttps://liberty.io/how-can-i-solve-this-hard-system-of-equations.htmlSolve the system below \begin{align} &\sqrt {3x} \left( 1+\frac {1}{x+y} \right) =2\\ &\sqrt {7y} \left( 1-\frac{1}{x+y} \right) =4\sqrt{2} \end{align} Frankly I am disappointed, beca......Tue, 18 Aug 2026 03:02:44 -0400https://liberty.io/express-this-curve-in-the-rectangular-form.htmlhttps://liberty.io/express-this-curve-in-the-rectangular-form.htmlExpress the curve $r = \dfrac{9}{4+\sin \theta}$ in rectangular form. And what is the rectangular form? If I get the expression in rectangular form, how am I able to convert it back to polar ...Tue, 18 Aug 2026 02:58:56 -0400https://liberty.io/how-to-prove-that-lim-limits-x-to0-frac-sin-x-x-1.htmlhttps://liberty.io/how-to-prove-that-lim-limits-x-to0-frac-sin-x-x-1.htmlHow can one prove the statement $$\lim_{x\to 0}\frac{\sin x}x=1$$ without using the Taylor series of $\sin$, $\cos$ and $\tan$? Best would be a geometrical solution. This is homework. In my math c......Tue, 18 Aug 2026 02:47:50 -0400https://liberty.io/prove-1-tan-2-theta-sec-2-theta.htmlhttps://liberty.io/prove-1-tan-2-theta-sec-2-theta.htmlProve the following trigonometric identity: $$1 + \tan^2\theta = \sec^2\theta$$ I'm curious to know of the different ways of proving this depending on different characterizations of tangent and se......Tue, 18 Aug 2026 02:39:17 -0400https://liberty.io/find-the-values-of-sin-69-circ-sin-18-circ-tan-23-circ.htmlhttps://liberty.io/find-the-values-of-sin-69-circ-sin-18-circ-tan-23-circ.htmlCalculate $\sin 69^{\circ},\sin 18^{\circ} , \tan 23^{\circ}$. accurate upto two decimal places or in surds . $\begin{align}\sin 69^{\circ}&=\sin (60+9)^{\circ}\\~\\ &=\sin (60^{\circ})\co......Tue, 18 Aug 2026 02:31:05 -0400https://liberty.io/how-to-find-the-boundary-conditions-of-the-differential-equation.htmlhttps://liberty.io/how-to-find-the-boundary-conditions-of-the-differential-equation.htmlI have a very complicated differential equation that can not be solved analytically and I show a on the simple example: Example 1: $$y''(x)-y(x)=0\tag{1}$$ with boundary conditions: $$y(1)=1,y'(......Tue, 18 Aug 2026 02:13:51 -0400https://liberty.io/what-are-the-units-of-cyclotomic-integers.htmlhttps://liberty.io/what-are-the-units-of-cyclotomic-integers.htmlThis question made me realize I had a misconception about the cyclotomic integers: I thought the units were exactly the roots of unity. There are only finitely many units but infinitely many intege......Tue, 18 Aug 2026 02:12:01 -0400https://liberty.io/why-is-039-catastrophic-cancellation-039-called-so.htmlhttps://liberty.io/why-is-039-catastrophic-cancellation-039-called-so.htmlI was studying Numerical Analysis by K. Mukherjee; there he discussed Loss of Significant Figures by Subtraction, as followed: In the subtraction of two approximate numbers, a serious type of er......Tue, 18 Aug 2026 02:04:55 -0400https://liberty.io/negative-mixed-fractions.htmlhttps://liberty.io/negative-mixed-fractions.htmlI'm comfortable with fractions like $\frac{-3}{8}$ being the same as $\frac{3}{-8}$ (though I'd think the latter anachronistic and would in any case probably prefer to write either of those two as $-\...Tue, 18 Aug 2026 01:59:17 -0400https://liberty.io/how-many-ways-can-you-make-change-for-a-dollar.htmlhttps://liberty.io/how-many-ways-can-you-make-change-for-a-dollar.htmlThis is a generic generating functions problem, and you are left with 1/(x-1)(x^5-1)....(x^50-1) however now I want to find the coefficient of the x^100 term of this expression... how can I do this......Tue, 18 Aug 2026 01:58:55 -0400https://liberty.io/chance-of-overlapping-events-in-milliseconds-per-day.htmlhttps://liberty.io/chance-of-overlapping-events-in-milliseconds-per-day.htmlI've got a server in which about 100 times per day a specific event happens. We now wanted to number those events using the unix time in milliseconds (the number of milliseconds that have passed si......Tue, 18 Aug 2026 01:52:06 -0400https://liberty.io/lax-wendroff-and-crank-nicolson-matlab-simulation-of-linear-advection.htmlhttps://liberty.io/lax-wendroff-and-crank-nicolson-matlab-simulation-of-linear-advection.htmlI am implementing two schemes: Lax-Wendroff and Crank Nicolson to solve the PDE linear advection in Matlab. My simulation seems to run; however, between the transition in time space the numerical ...Tue, 18 Aug 2026 01:47:21 -0400https://liberty.io/find-the-extreme-values-of-the-given-function-subject-to-constraints-f.htmlhttps://liberty.io/find-the-extreme-values-of-the-given-function-subject-to-constraints-f.htmlFind the extreme values of the function, subject to constraints $$f(x,y) = y^2 - x^2 ; x^2 + y^2 = 36$$ Here is what I have done: Using the substitution method we can say: $$y = \sqrt{-x^2 + 36}$$......Tue, 18 Aug 2026 01:42:40 -0400https://liberty.io/solving-ode-problem-with-calculator.htmlhttps://liberty.io/solving-ode-problem-with-calculator.htmlI'm trying to solve a below ODE problem with TI-nspire cas: $$ y''-4y'+3y=cos^2x $$ If I solve the above problem by hand (or using the wolfram alpha), I get the $$ y=c_1e^x+c_2e^{3x}+\frac16-\f......Tue, 18 Aug 2026 01:41:56 -0400https://liberty.io/multivariable-limit-of-a-piecewise-function.htmlhttps://liberty.io/multivariable-limit-of-a-piecewise-function.htmlConsider the following expression: $$\lim_{(x,y) \to (0,0)} g(x,y)= \begin{cases} \frac{\sin x}{x} y \text{ $\hspace{0.5in}$if $x \neq 0$}\\ y \text{ $\hspace{0.81in}$if $x=0$}\end{cases}$$ I am se......Tue, 18 Aug 2026 01:36:57 -0400https://liberty.io/why-is-the-graph-of-2logx-different-from-log-x-2-duplicate.htmlhttps://liberty.io/why-is-the-graph-of-2logx-different-from-log-x-2-duplicate.htmlAccording to Logarithmic exponent rules: log(x^y) = y ∙ log(x) For example: 2 log x = log x^2 However take the following graphs: y=2logx y=logx^2 The graphs seems to display something that sho......Tue, 18 Aug 2026 01:30:07 -0400