https://liberty.ioCurated star highlights with fresh energy.en-usTue, 01 Sep 2026 08:32:10 -0400https://liberty.io/x-axis-of-a-linearized-polynomial.htmlhttps://liberty.io/x-axis-of-a-linearized-polynomial.htmlThe other day in my Physics class we had some sample data that we wanted to linearize. The graph resembled a root curve. So to linearize it, we took the square root of all the x data and replotte......Tue, 01 Sep 2026 12:28:58 -0400https://liberty.io/how-to-expand-cos-nx-with-cos-x.htmlhttps://liberty.io/how-to-expand-cos-nx-with-cos-x.htmlMultiple Angle Identities: How to expand $\cos nx$ with $\cos x$, such as $$\cos10x=512(\cos x)^{10}-1280(\cos x)^8+1120(\cos x)^6-400(\cos x)^4+50(\cos x)^2-1$$ See a list of trigonometric ...Tue, 01 Sep 2026 12:18:06 -0400https://liberty.io/similar-matrices-how-do-you-find-them.htmlhttps://liberty.io/similar-matrices-how-do-you-find-them.htmlMy lecture notes write that two matrices are said to be similar if there exists an $n \times n$ invertible matrix $P$ such that $B = P^{-1} AP$. But suppose I am given two matrices, $A$ and $B$, ......Tue, 01 Sep 2026 12:03:29 -0400https://liberty.io/taking-partial-derivative-via-fourier-transform-using-matlab.htmlhttps://liberty.io/taking-partial-derivative-via-fourier-transform-using-matlab.htmlFor a task for university, I need to calculate in matlab a partial derivative using the 2D Fourier transform. Using equation (13) on pg 3 in http://www2.ph.ed.ac.uk/~wjh/teaching/Fourier/documents/...Tue, 01 Sep 2026 12:02:25 -0400https://liberty.io/why-can-natural-logarithm-receive-negative-x-in-the-context-of-derivat.htmlhttps://liberty.io/why-can-natural-logarithm-receive-negative-x-in-the-context-of-derivat.htmlIn our class we had this formula for $x<0$: $$ \ln(-x)' = \frac{1}{-x} \cdot (-x)' = \frac{1}{x} $$ The conclusion is that: $$ \int\frac{1}{x}dx = \ln|x| + c, x \neq 0 $$ I don't understand how......Tue, 01 Sep 2026 11:47:20 -0400https://liberty.io/formula-and-percentages.htmlhttps://liberty.io/formula-and-percentages.htmlI need help making a formula. Here is the basic concept: I have 1 twenty-sided die (from now on 1d20) and another 16-sided die (1d16). I roll each, and the die which displays the lowest number "win......Tue, 01 Sep 2026 11:32:29 -0400https://liberty.io/finding-non-closed-paths-so-that-a-line-integral-equals-certain-values.htmlhttps://liberty.io/finding-non-closed-paths-so-that-a-line-integral-equals-certain-values.htmlSo I'm required to find a non-closed path $C$ so that the line integral $$\displaystyle \int_C \mathbf{F} \cdot \mathrm d \mathbf{r}$$ equals $0$, and another non-closed path $C$ so that the line ...Tue, 01 Sep 2026 11:24:12 -0400https://liberty.io/what-are-the-length-of-the-longest-element-in-a-coexter-group-for-ever.htmlhttps://liberty.io/what-are-the-length-of-the-longest-element-in-a-coexter-group-for-ever.htmlWhat are the length of the longest element in a Coxeter group for every type? Thank you very much....Tue, 01 Sep 2026 11:14:02 -0400https://liberty.io/has-anyone-ever-attempted-to-find-all-splits-of-a-rectangle-into-small.htmlhttps://liberty.io/has-anyone-ever-attempted-to-find-all-splits-of-a-rectangle-into-small.htmlI'm enamored by the games Unproportional and Unproportional2. The only problem is that they have way too few levels. :) But, I'm a computer programmer, so I could make my own! With blackjack and ho......Tue, 01 Sep 2026 11:13:37 -0400https://liberty.io/composition-of-even-and-odd-functions-and-their-outcome.htmlhttps://liberty.io/composition-of-even-and-odd-functions-and-their-outcome.htmlGive an example of an even function. Give an example of an odd function. If f(x) is odd and g(x) is even, must f(g(x)) be even? Must g(f(x)) be even? I've tried generic functions like f(x) = x^3......Tue, 01 Sep 2026 11:06:20 -0400https://liberty.io/determining-whether-matrix-is-self-adjoint.htmlhttps://liberty.io/determining-whether-matrix-is-self-adjoint.htmlI have three matrices and I am trying to determine if they represent a self-adjoint linear transformation in some basis on an inner product space. The matrices are: $$A_1 = \left( \begin{array}{c......Tue, 01 Sep 2026 11:04:02 -0400https://liberty.io/half-full-cone-of-water.htmlhttps://liberty.io/half-full-cone-of-water.htmlI have been battling with this for a little while now, but can't really get my head around it. The question goes: how high must you pour water into a cone before it is half-full (by volume)? No ...Tue, 01 Sep 2026 11:02:42 -0400https://liberty.io/how-to-compute-these-riemann-stieltjes-integrals.htmlhttps://liberty.io/how-to-compute-these-riemann-stieltjes-integrals.htmlI have some doubt on this exercise which I have to solve: Compute The Integral $\displaystyle \int_0^4 F(x) \ \mathrm{d}G(x)$ for 1.) $F(x) = x$ for $x<2$, $1$ otherwise; $G(x)=x^2$ 2.) $F(x)......Tue, 01 Sep 2026 10:57:46 -0400https://liberty.io/finding-prime-implicants-and-essential-prime-implicants-for-boolean-fu.htmlhttps://liberty.io/finding-prime-implicants-and-essential-prime-implicants-for-boolean-fu.htmlI am trying to solve a EE problem and am unsure whether I doing it correctly. The problem is: Find all the prime implicants for the following Boolean functions, and determine which are essential: ......Tue, 01 Sep 2026 10:55:20 -0400https://liberty.io/what-does-f-039-and-f-039-039-mean.htmlhttps://liberty.io/what-does-f-039-and-f-039-039-mean.htmlI'm trying to learn what a Taylor series is, This is the equation I'm looking at and I know 0 calculus. I have been told that $F'(x)$ is a derivative but what does $F''(x)$ mean?...Tue, 01 Sep 2026 10:38:18 -0400https://liberty.io/proving-aoc-axiom-of-completeness-using-nip-nested-interval-property.htmlhttps://liberty.io/proving-aoc-axiom-of-completeness-using-nip-nested-interval-property.htmlNIP $\implies$ AoC. $\newcommand{\absval}[1]{\left\lvert #1 \right\rvert}$ Exercise 2.5.4 from Understanding Analysis by Abott. The Bolzanno-Weierstrass theorem is extremely important, and so is the ...Tue, 01 Sep 2026 10:38:06 -0400https://liberty.io/why-do-2-log-x-and-log-x-2-look-different-when-graphing-duplicate.htmlhttps://liberty.io/why-do-2-log-x-and-log-x-2-look-different-when-graphing-duplicate.htmlI understand how both graphs are drawn, but I do not understand why you cannot just convert one into another. It feels natural to me to just convert $2\log x$ to $\log x^2$ and not have to worry ab......Tue, 01 Sep 2026 10:37:02 -0400https://liberty.io/second-order-sde-solutions.htmlhttps://liberty.io/second-order-sde-solutions.htmlI have the following stochastic process: $$\ddot x(t)=a(t)+\nu(t)$$ where $a(t)$ is a deterministic function of the time and $\nu(t)$ is a random variable: $$\nu(t)\in\mathcal{N}(0,\sigma^2)$$ Obvi......Tue, 01 Sep 2026 10:31:35 -0400https://liberty.io/barycentric-coordinates-of-the-center-of-mass-of-a-triangle.htmlhttps://liberty.io/barycentric-coordinates-of-the-center-of-mass-of-a-triangle.htmlHow can I compute the barycentric coordinates of the center of mass of a triangle if its $3$ vertices are defined by $P_1, P_2$ and $P_3.$ This is what I tried: $$ \left\{ \begin{array}{cc} (m_1 ......Tue, 01 Sep 2026 10:12:28 -0400https://liberty.io/help-understanding-the-range-and-kernel-of-a-linear-transformation.htmlhttps://liberty.io/help-understanding-the-range-and-kernel-of-a-linear-transformation.htmlI'm having some trouble understanding the Range and Kernel of a linear transformation. The definition goes as follows: Let $T:V \longrightarrow W$ be a linear transformation. Define the sets $ke......Tue, 01 Sep 2026 10:09:00 -0400https://liberty.io/determining-ambiguity-in-context-free-grammars.htmlhttps://liberty.io/determining-ambiguity-in-context-free-grammars.htmlWhat are some common ways to determine if a grammar is ambiguous or not? What are some common attributes that ambiguous grammars have? For example, consider the following Grammar G: $S \rightarro......Tue, 01 Sep 2026 10:08:36 -0400https://liberty.io/homology-of-the-klein-bottle.htmlhttps://liberty.io/homology-of-the-klein-bottle.htmlI know that in general, $H_{n}(X)$ counts the number of $n$-cycles that are not $n$-boundaries of a simplicial complex $X$. So for the sphere, $H_{0}(X) \cong \mathbb{Z}$ since it is connected. Als......Tue, 01 Sep 2026 10:00:07 -0400https://liberty.io/what-are-the-real-world-applications-of-real-analysis.htmlhttps://liberty.io/what-are-the-real-world-applications-of-real-analysis.htmlI've read the wikipedia article on mathematical analysis and this, but I can't exactly find an answer. Is real analysis just some pure math, or does it really have something to with physical applic......Tue, 01 Sep 2026 09:50:58 -0400https://liberty.io/find-the-asymptotic-upper-bound.htmlhttps://liberty.io/find-the-asymptotic-upper-bound.htmlI need to find the asymptotic upper bounds in $O$ notation for $T(N)$ in two recurrences. Assuming that $T(N)$ is constant for sufficiently small $N$, I need to make the bounds as tight as possible......Tue, 01 Sep 2026 09:40:44 -0400https://liberty.io/matrix-calculator-tools.htmlhttps://liberty.io/matrix-calculator-tools.htmlDoes there exist any such software/online calculator where I can enter a matrix and am able to obtain the formula for an arbitrary power of the matrix? It would be an added benefit if the entries ......Tue, 01 Sep 2026 09:39:52 -0400https://liberty.io/curl-and-product-rule.htmlhttps://liberty.io/curl-and-product-rule.htmlI know the famous identity $\nabla \times (\vec{A} \times \vec{B}) = A(\nabla \cdot \vec{B}) -B(\nabla \cdot \vec{A}) + (B\cdot \nabla) \vec{A}- (A\cdot \nabla) \vec{B}$ My question: Can I distri......Tue, 01 Sep 2026 09:34:43 -0400https://liberty.io/stirling-039-s-formula-proof.htmlhttps://liberty.io/stirling-039-s-formula-proof.htmlSuppose we want to show that $$ n! \sim \sqrt{2 \pi} n^{n+(1/2)}e^{-n}$$ Instead we could show that $$\lim_{n \to \infty} \frac{n!}{n^{n+(1/2)}e^{-n}} = C$$ where $C$ is a constant. Maybe $C = \sqr......Tue, 01 Sep 2026 09:34:42 -0400https://liberty.io/in-which-sense-does-cauchy-riemann-equations-link-complex-and-real-ana.htmlhttps://liberty.io/in-which-sense-does-cauchy-riemann-equations-link-complex-and-real-ana.htmlOn page 12 of Stein, Shakarchi textbook 'Complex analysis', the authors state that the Cauchy-Riemann equations link complex and real analysis. I have completed courses on real and complex analysis......Tue, 01 Sep 2026 09:30:39 -0400https://liberty.io/how-to-integrate-a-three-products.htmlhttps://liberty.io/how-to-integrate-a-three-products.htmlI tried to integrate $x e^x \sin x$, using integration by parts, and setting $\frac{\, \mathrm dv}{\, \mathrm dx} = e^x \sin x$. Even though I got really close, I kept getting it wrong. Can some......Tue, 01 Sep 2026 09:26:57 -0400https://liberty.io/odds-of-a-specific-dice-distribution-occuring-when-playing-catan-2d6.htmlhttps://liberty.io/odds-of-a-specific-dice-distribution-occuring-when-playing-catan-2d6.htmlSo this is messing with me, because I have no idea how I would even go about starting calculating this. I have played a game of Catan. For those that don't know, it's a board game that involves rol......Tue, 01 Sep 2026 09:22:45 -0400https://liberty.io/change-of-basis-matrix-exercise-find-the-basis-given-the-matrix.htmlhttps://liberty.io/change-of-basis-matrix-exercise-find-the-basis-given-the-matrix.htmlThe matrix of basis change from basis $B = \{b_1,b_2\}$ to the basis $C = \{(1,1),(0,2)\}$ is $$ M = \begin{bmatrix} 1 & 0 \\2&3\end{bmatrix}$$ Find basis $B$. Well, here's what I did: ...Tue, 01 Sep 2026 09:22:32 -0400https://liberty.io/how-to-find-indicial-equation.htmlhttps://liberty.io/how-to-find-indicial-equation.htmlHow can I find the indicial equation of $x(x-1)y''+3y'-2y=0$? I tried the method of Frobenius but I keep getting lost in the algebra. Is there any other way to get the indicial equation?...Tue, 01 Sep 2026 09:04:19 -0400https://liberty.io/how-to-recognise-a-problem-requiring-the-use-of-calculus.htmlhttps://liberty.io/how-to-recognise-a-problem-requiring-the-use-of-calculus.htmlI've taken a calculus class for a semester at university as a mandatory part of my curriculum. In this class, we were taught formal definitions of what are derivatives and integrals, how to calculate ...Tue, 01 Sep 2026 08:57:51 -0400https://liberty.io/is-the-hilbert-schmidt-norm-of-a-matrix-induced-from-a-vector-norm-dup.htmlhttps://liberty.io/is-the-hilbert-schmidt-norm-of-a-matrix-induced-from-a-vector-norm-dup.htmlA general way to get a matrix norm is inducing it from a vector norm. The question is, can the Hilbert-Schmidt norm obtained this way? As a norm general question, is every matrix norm obtained th......Tue, 01 Sep 2026 08:47:54 -0400https://liberty.io/what-textbook-is-being-used-in-these-lectures-linear-algebra.htmlhttps://liberty.io/what-textbook-is-being-used-in-these-lectures-linear-algebra.htmlI am learning Linear Algebra from these lectures by Prof. Adrian Banner (Princeton University) Does anyone know what textbook they are using? This is a link to the playlist on YouTube: https://www....Tue, 01 Sep 2026 08:39:20 -0400https://liberty.io/what-exactly-is-significand.htmlhttps://liberty.io/what-exactly-is-significand.htmlper wiki, The significand is part of a number in scientific notation or a floating-point number, consisting of its significant digits. in this example $$123.45 = 0.12345 × 10^3$$ which part is ......Tue, 01 Sep 2026 08:31:00 -0400https://liberty.io/is-it-propositional-function-or-simply-proposition.htmlhttps://liberty.io/is-it-propositional-function-or-simply-proposition.htmlI was going through the text "Discrete Mathematics and Its Application" by Kenneth H Rosen (5th Edition) where I came across the use of $P(n)$ in the mathematical induction chapter and felt ...Tue, 01 Sep 2026 08:30:59 -0400https://liberty.io/how-to-calculate-x-with-percent-of-x-y.htmlhttps://liberty.io/how-to-calculate-x-with-percent-of-x-y.htmlI am sorry if this is impossible to compute in one equatation. I am trying to compute the following: for example y = 574 x - ((16% of x) + y) = 0 the goal is to find out what is x so as to make it ...Tue, 01 Sep 2026 08:16:29 -0400https://liberty.io/why-aren-039-t-logarithms-defined-for-negative-x.htmlhttps://liberty.io/why-aren-039-t-logarithms-defined-for-negative-x.htmlGiven a logarithm is true, if and only if, $y = \log_b{x}$ and $b^y = x$ (and $x$ and $b$ are positive, and $b$ is not equal to $1$)[1], are true, why aren't logarithms defined for negative ...Tue, 01 Sep 2026 08:08:30 -0400https://liberty.io/what-is-the-double-bracket-notation-used-here.htmlhttps://liberty.io/what-is-the-double-bracket-notation-used-here.htmlIts kind of a bracket but I'm not sure what it means. I have two ideas about it: It means the number of times the expression in satisfied or it changes for $1$ or $0$ depending on the result every......Tue, 01 Sep 2026 07:56:57 -0400https://liberty.io/why-is-pi-2-so-close-to-10.htmlhttps://liberty.io/why-is-pi-2-so-close-to-10.htmlNoam Elkies explained why $\pi^2=9.8696...$ is so close to $10$ using an inequality on Euler's solution to Basel problem $$\frac{\pi^2}{6}=\sum_{k=0}^{\infty} \frac{1}{\left(k+1\right)^2}$$ to f......Tue, 01 Sep 2026 07:55:23 -0400https://liberty.io/how-can-i-find-the-maximum-velocity-if-i-039-ve-already-found-when-it-.htmlhttps://liberty.io/how-can-i-find-the-maximum-velocity-if-i-039-ve-already-found-when-it-.htmlI've been working on this problem for awhile and got the second half figured out, but I can't seem to get what the maximum actually is. Here's the question: A bullet is fired in the air verticall......Tue, 01 Sep 2026 07:53:22 -0400https://liberty.io/deriving-the-derivative-formula-for-arcsecant-correctly.htmlhttps://liberty.io/deriving-the-derivative-formula-for-arcsecant-correctly.htmlI have been trying to derive the derivative of the arcsecant function, but I can't quite get the right answer (the correct answer is the absolute value of what I get). I first get $\frac{d}{dy}\sec......Tue, 01 Sep 2026 07:50:44 -0400https://liberty.io/interpreting-divergence-of-velocity-field.htmlhttps://liberty.io/interpreting-divergence-of-velocity-field.htmlThe wikipedia article on divergence describes one interpretation of divergence: "The velocity of the air at each point defines a vector field. While air is heated in a region, it expands in all ...Tue, 01 Sep 2026 07:48:48 -0400https://liberty.io/martingale-and-submartingale.htmlhttps://liberty.io/martingale-and-submartingale.htmlWhen $X_n$ is a martingale we know $(X_n)^+$, i.e. the positive part of $X_n$, is also a submartingale. Is this true? It is easy to show that this is SUB-m.g. but I couldn't find a counter-example ......Tue, 01 Sep 2026 07:47:56 -0400https://liberty.io/can-the-fibonacci-sequence-be-written-as-an-explicit-rule.htmlhttps://liberty.io/can-the-fibonacci-sequence-be-written-as-an-explicit-rule.htmlWhen I learned sums and sequences in algebra II with trig I learned about recursive rules and explicit rules. A recursive rule written with the formula of: $$a_n = r * a_{n-1}$$ Or as: $$a_n = a_{......Tue, 01 Sep 2026 07:46:45 -0400https://liberty.io/proving-the-identity-sin3x-sin-x-2-sin2x-cos-x.htmlhttps://liberty.io/proving-the-identity-sin3x-sin-x-2-sin2x-cos-x.htmlI need some help proving this identity: $$\sin3x + \sin x = 2\sin2x\cos x$$ I don't know where to start. I thought about expanding $\sin 3x$ into $\sin (2x + x)$ but I don't think that does me ......Tue, 01 Sep 2026 07:45:02 -0400https://liberty.io/integration-of-frac-1-x-ln-x-by-parts.htmlhttps://liberty.io/integration-of-frac-1-x-ln-x-by-parts.htmlI tried to solve $\int \frac{dx}{x\ln(x)}$ by parts. I have done it as usual: $$\int \frac{dx}{x\ln(x)}=\begin{bmatrix} f(x)=\frac{1}{\ln x} & g'(x)=\frac{1}{x}\\ f'(x)=-\frac{1}{x\ln^2x} &amp......Tue, 01 Sep 2026 07:34:19 -0400https://liberty.io/expressing-the-trace-of-a-2-by-trace-of-a.htmlhttps://liberty.io/expressing-the-trace-of-a-2-by-trace-of-a.htmlLet $A$ be a a square matrix. Is it possible to express $\operatorname{trace}(A^2)$ by means of $\operatorname{trace}(A)$ ? or at least something close?...Tue, 01 Sep 2026 07:22:56 -0400https://liberty.io/what-is-mutually-disjoint-sets.htmlhttps://liberty.io/what-is-mutually-disjoint-sets.htmlWhat is mutually disjoint sets? I know it has something to do with subsets but I don't know for sure....Tue, 01 Sep 2026 07:00:58 -0400https://liberty.io/a-small-doubt-about-the-dominated-convergence-theorem.htmlhttps://liberty.io/a-small-doubt-about-the-dominated-convergence-theorem.htmlTheorem $\mathbf{A.2.11}$ (Dominated convergence). Let $f_n : X \to \mathbb R$ be a sequence of measurable functions and assume that there exists some integrable function $g : X \to \mathbb R$ such......Tue, 01 Sep 2026 07:00:53 -0400https://liberty.io/find-first-term-and-common-difference-of-arithmetic-sequence-given-two.htmlhttps://liberty.io/find-first-term-and-common-difference-of-arithmetic-sequence-given-two.htmlThe 7th and 11th terms of an arithmetic sequence are 7b + 5c and 11b + 9c respectively. Given these i want to find the first term (term when n = 0) and the common difference. I tried a lot of ...Tue, 01 Sep 2026 06:58:33 -0400https://liberty.io/difference-between-matrix-mapping-and-matrix-multiplication.htmlhttps://liberty.io/difference-between-matrix-mapping-and-matrix-multiplication.htmlMy textbook says that a matrix mapping is a function f: $\mathbb{R}^n\to\mathbb{R}^m$ such that $f(\vec{x}) = A\vec{x}$ where $A$ is an $m \times n$ matrix So how is a result of a matrix mapping ...Tue, 01 Sep 2026 06:54:56 -0400https://liberty.io/how-to-solve-a-triangle-knowing-just-two-legs.htmlhttps://liberty.io/how-to-solve-a-triangle-knowing-just-two-legs.htmlGiven a right triangle, if I know two of its legs are $a = 3,5$ and $b = 5,5$, is it possible to solve the triangle with this info? This is, can I determine all of its sides and all of its angles w......Tue, 01 Sep 2026 06:54:49 -0400https://liberty.io/what-is-a-negative-number.htmlhttps://liberty.io/what-is-a-negative-number.htmlI'm trying to get to an abstract definition of a negative number that would fit in with the basic concept of addition/subtraction. There are questions here about multiplying and dividing negative ...Tue, 01 Sep 2026 06:52:35 -0400https://liberty.io/a-cube-abcdefgh-is-given-determine-the-locus-of-all-midpoints-of-segme.htmlhttps://liberty.io/a-cube-abcdefgh-is-given-determine-the-locus-of-all-midpoints-of-segme.htmlQuestion is:A cube $ABCDEFGH$ is given. Determine the locus of all midpoints of segments $MN$, where $M$ is any point on segment $AC$ and $N$ any point on segment $FH$. My understanding : Here $M$ ......Tue, 01 Sep 2026 06:44:04 -0400https://liberty.io/proof-by-induction-of-bernoulli-039-s-inequality-1-x-n-geq-1-nx.htmlhttps://liberty.io/proof-by-induction-of-bernoulli-039-s-inequality-1-x-n-geq-1-nx.htmlI'm asked to used induction to prove Bernoulli's Inequality: If $1+x>0$, then $(1+x)^n\geq 1+nx$ for all $n\in\mathbb{N}$. This what I have so far: Let $n=1$. Then $1+x\geq 1+x$. This is true. ......Tue, 01 Sep 2026 06:42:57 -0400https://liberty.io/calculate-angles-and-absolute-value-of-complex-number-matlab.htmlhttps://liberty.io/calculate-angles-and-absolute-value-of-complex-number-matlab.htmlhow to Calculate the angles and absolute value of complex number by using matlab commands: the complex number is:$$e^{3+4j}+e^{3-4j}$$ tyy!!...Tue, 01 Sep 2026 06:40:23 -0400https://liberty.io/moment-of-inertia-of-ellipse-around-x-axis.htmlhttps://liberty.io/moment-of-inertia-of-ellipse-around-x-axis.htmlHow can I calculate the moment of inertia of an ellipse ($\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$) using polar coordinates and the formula: $$I_x=\int_\Omega y^2dx dy$$ I know that the result is suppos......Tue, 01 Sep 2026 06:31:55 -0400https://liberty.io/find-the-exact-value-for-cos-alpha-beta-given-the-following-informatio.htmlhttps://liberty.io/find-the-exact-value-for-cos-alpha-beta-given-the-following-informatio.htmlSuppose $\alpha$ is interval $$\pi/2 \leq \alpha \leq \pi$$ and $$ \cos(\alpha) = - 1/3 $$ and $\beta$ is in the interval $$0 \leq \beta \leq \pi/2$$ and $$ \sin\beta = 2/5. $$ Use the facts abov......Tue, 01 Sep 2026 06:27:13 -0400https://liberty.io/coding-the-power-method-in-python.htmlhttps://liberty.io/coding-the-power-method-in-python.htmlI am a very beginner to coding and am trying to write a code for the power method/von Mises iteration for finding the largest eigenvalue (in magnitude) of a given matrix in Python. I'm referencing ...Tue, 01 Sep 2026 06:18:58 -0400https://liberty.io/volume-of-ellipsoid-using-triple-integrals.htmlhttps://liberty.io/volume-of-ellipsoid-using-triple-integrals.htmlGiven the general equation of the ellipsoid $\frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} =1$, I am supposed to use a 3D Jacobian to prove that the volume of the ellipsoid is $\frac{4}{3}\pi......Tue, 01 Sep 2026 06:17:59 -0400https://liberty.io/integral-of-tan-x.htmlhttps://liberty.io/integral-of-tan-x.htmlSo I wanted to find a way to work out any trigonometric integral by simply knowing the integral of the two main trigonometric functions; $\sin(x)$ and $\cos(x)$. This is since any trigonometric fun......Tue, 01 Sep 2026 06:12:29 -0400https://liberty.io/why-is-a-circle-not-a-convex-set.htmlhttps://liberty.io/why-is-a-circle-not-a-convex-set.htmlWhy is a circle not a convex set? My attempt: I was googling it, but I didn't find the correct answer, as I didn't understand the explanation why a circle is not a convex set as defined here: As ......Tue, 01 Sep 2026 06:10:27 -0400https://liberty.io/how-to-tell-the-difference-between-a-parabola-and-a-hyperbola-by-looki.htmlhttps://liberty.io/how-to-tell-the-difference-between-a-parabola-and-a-hyperbola-by-looki.htmlIf you are given a curve and you are told that it is either a parabola or a hyperbola, how can you tell definitively which one it is? No coordinates provided. Geometrical method preferred, i.e. v......Tue, 01 Sep 2026 06:07:23 -0400https://liberty.io/calculating-centre-of-rotation-given-point-coordinates-at-different-po.htmlhttps://liberty.io/calculating-centre-of-rotation-given-point-coordinates-at-different-po.htmlI am trying to work out if the centre of rotation of a measured sphere is actually at 0,0 or slightly offset from the centre. The situation is as follows: I have a machine tool with a table that r......Tue, 01 Sep 2026 05:49:00 -0400https://liberty.io/why-is-gradient-noise-better-quality-than-value-noise.htmlhttps://liberty.io/why-is-gradient-noise-better-quality-than-value-noise.htmlI have been reading about the mathematics behind Perlin noise, a gradient noise function often used in computer graphics, from Ken Perlin's presentation and Matt Zucker's FAQ. I understand that each ...Tue, 01 Sep 2026 05:43:29 -0400https://liberty.io/how-to-solve-int-tan-2-x-sec-x-dx-duplicate.htmlhttps://liberty.io/how-to-solve-int-tan-2-x-sec-x-dx-duplicate.htmlHow should one solve the following integral? $$\int \tan^2(x) \sec (x) \ dx$$ I can't think of any substitutions to be made involving $\tan^2(x)=\sec^2 (x)-1$ or $\sec^2(x)=\tan^2(x)+1$, which is ......Tue, 01 Sep 2026 05:27:50 -0400https://liberty.io/how-do-i-find-the-original-function-given-the-derivative-closed.htmlhttps://liberty.io/how-do-i-find-the-original-function-given-the-derivative-closed.htmlI have no idea how to start these problems. How does one get the original function given the derivative of the function? I would appreciate it if someone would guide me in doing problem 7 at least ......Tue, 01 Sep 2026 05:24:26 -0400https://liberty.io/what-is-the-brachistochrone-curve-with-an-initial-velocity.htmlhttps://liberty.io/what-is-the-brachistochrone-curve-with-an-initial-velocity.htmlI'm basically trying to solve the opposite of the brachistochrone problem. You know how the brachistochrone is the shortest path between two points A and B, falling under the influence of gravity?......Tue, 01 Sep 2026 05:16:38 -0400https://liberty.io/right-group-action.htmlhttps://liberty.io/right-group-action.htmlWikipedia says 'The difference between left and right actions is in the order in which a product like $gh$ acts on $x$. For a left action $h$ acts first and is followed by $g$, while for a right a......Tue, 01 Sep 2026 05:12:15 -0400https://liberty.io/matrix-is-conjugate-to-its-own-transpose.htmlhttps://liberty.io/matrix-is-conjugate-to-its-own-transpose.htmlMariano mentioned somewhere that everyone should prove once in their life that every matrix is conjugate to its transpose. I spent quite a bit of time on it now, and still could not prove it. At the ...Tue, 01 Sep 2026 05:11:57 -0400https://liberty.io/let-f-x-x-3-x-2-3x-1-and-h-x-frac-f-x-g-x-where-h-x-is-a-rational-func.htmlhttps://liberty.io/let-f-x-x-3-x-2-3x-1-and-h-x-frac-f-x-g-x-where-h-x-is-a-rational-func.htmlLet $f(x) = x^3-x^2–3x–1$ and $h(x)=\dfrac{f(x)}{g(x)}$ where $h(x)$ is a rational function such that (a) it is continuous every where except when $x=– 1$ (b) $\lim_{x\to\infty}=\infty$ (c) $\li......Tue, 01 Sep 2026 05:03:33 -0400https://liberty.io/irreducible-dual-representation.htmlhttps://liberty.io/irreducible-dual-representation.htmlFor a semisimple Lie Algebra $\mathfrak{g}$ with Cartan Subalgebra $\mathfrak{t}$, let $V(\lambda)$ be the unique irreducible highest weight module with highest weight $\lambda$. I am asked to sho......Tue, 01 Sep 2026 05:02:25 -0400https://liberty.io/find-the-angle-of-a-tangent-line-between-two-circles.htmlhttps://liberty.io/find-the-angle-of-a-tangent-line-between-two-circles.htmlI have two circles noted as $A$ and $B$. Each of these circles has known positions $\vec{P_A}$ and $\vec{P_B}$ with radii $r_A$ and $r_B$. I need to find the angle ($\theta_A$ or $\theta_B$) shown ......Tue, 01 Sep 2026 04:49:03 -0400https://liberty.io/rectangular-parallelepiped-of-greatest-volume-for-a-given-surface-area.htmlhttps://liberty.io/rectangular-parallelepiped-of-greatest-volume-for-a-given-surface-area.htmlI am trying to find the rectangular parallelepiped of greatest volume for a given surface area S using Lagrange's method. I tried solving by myself but at x=y=z = a, I am not getting maximum volu......Tue, 01 Sep 2026 04:46:10 -0400https://liberty.io/why-is-this-not-a-space-filling-curve.htmlhttps://liberty.io/why-is-this-not-a-space-filling-curve.htmlFrom Wikipedia, a space-filling curve is a curve (i.e. a continuous function whose domain is the unit interval $[0,1]$) whose range contains the entire 2-dimensional unit square. Many examples of ......Tue, 01 Sep 2026 04:42:27 -0400https://liberty.io/series-expansion-of-square-root-function.htmlhttps://liberty.io/series-expansion-of-square-root-function.htmlCould I please know how to expand: $$\sqrt{4-3x^2}$$ I simplified it to $\sqrt{1-\frac34x^2}$ but the question is if I let $x = x^2$, what will the coefficient of, say $x^5$ or $x^{10}$ be in the ...Tue, 01 Sep 2026 04:38:04 -0400https://liberty.io/roots-of-unity.htmlhttps://liberty.io/roots-of-unity.htmlThe $n$th roots of unity are the complex numbers: $1,w,w^2,...,w^{n-1}$, where $w = e^{\frac{2\pi i} {n}}$. If $n$ is even: The $n$th roots are plus-minus paired, $w^{\frac{n}{2}+j} = -w^j$. Squa......Tue, 01 Sep 2026 04:18:13 -0400https://liberty.io/how-to-prove-that-this-simple-graph-does-not-exist.htmlhttps://liberty.io/how-to-prove-that-this-simple-graph-does-not-exist.htmlShow that there is no simple graph with 6 vertices in which there are 5 vertices having following degrees 5,5,3,2,1 So, we have 5 vertices (=odd number of vertices) with an even number of degree......Tue, 01 Sep 2026 04:10:36 -0400https://liberty.io/finding-a-general-term-for-the-sequence.htmlhttps://liberty.io/finding-a-general-term-for-the-sequence.htmlFind a general term in simplest form for the sequence: $$2, 1, -4, 7, -10, 13, -16$$ This is what I tried: $a_n = a_1 + (n - 1)\cdot d$ $a_n = 2 + (n-1)\cdot 3$ $a_n = (-1)^n (2+(n-1)\cdot 3)$ which ...Tue, 01 Sep 2026 04:02:03 -0400https://liberty.io/need-help-in-understanding-how-to-find-an-elementary-matrix.htmlhttps://liberty.io/need-help-in-understanding-how-to-find-an-elementary-matrix.htmlI read this chapter in my book and thought I understood it, but I don't. I tried working a problem to test my understanding and I just don't know how to get started. Given the following matrices:......Tue, 01 Sep 2026 03:57:37 -0400https://liberty.io/saw-horses-with-legs-26-in-height.htmlhttps://liberty.io/saw-horses-with-legs-26-in-height.htmlI am building sawhorses with each end cut at a $14^{\circ}$ angle, where one end attaches to the underside of the sawhorse and the other $14^{\circ}$ angle hits the flat floor. My question: if I n......Tue, 01 Sep 2026 03:50:35 -0400https://liberty.io/user-ikshu-neithalath-mathematics-stack-exchange.htmlhttps://liberty.io/user-ikshu-neithalath-mathematics-stack-exchange.htmlQ&A for people studying math at any level and professionals in related fields...Tue, 01 Sep 2026 03:26:27 -0400https://liberty.io/range-of-sin-x-6-cos-x-6.htmlhttps://liberty.io/range-of-sin-x-6-cos-x-6.htmlI need to find the range of the function $y = (\sin x)^6+ (\cos x)^6$ I did find the answer but working in a crude way rather than a methodical step by step approach. I give below the steps I used , ...Tue, 01 Sep 2026 03:07:14 -0400https://liberty.io/on-the-horizontal-and-slanted-asymptotes-of-rational-functions.htmlhttps://liberty.io/on-the-horizontal-and-slanted-asymptotes-of-rational-functions.htmlI am really confused about the horizontal and slant asymptotes of a function. My textbook says that given a rational function: \begin{equation} y=f(x)=\frac{a_{n}x^{n}+a_{n-1}x^{n-1}+\cdot\cdot\cd......Tue, 01 Sep 2026 03:07:06 -0400https://liberty.io/what-does-the-derivative-of-a-function-at-a-point-describe-duplicate.htmlhttps://liberty.io/what-does-the-derivative-of-a-function-at-a-point-describe-duplicate.htmlI understand that the derivative of a function $f$ at a point $x=x_{0}$ is defined as the limit $$f'(x_{0})=\lim_{\Delta x\rightarrow 0}\frac{f(x_{0}+\Delta x)-f(x_{0})}{\Delta x}$$ where $\Delta x......Tue, 01 Sep 2026 03:06:45 -0400https://liberty.io/how-does-one-derive-the-centroid-formula-for-multiple-shapes.htmlhttps://liberty.io/how-does-one-derive-the-centroid-formula-for-multiple-shapes.htmlI know how to derive the formula for the centroid of n sets of finite points, each with $m_k$ points and centroid $C_k$. The formula is: $$C=\frac{\sum_{k=1}^{n}C_km_k}{\sum_{k=1}^{n}m_k}$$ However......Tue, 01 Sep 2026 02:57:38 -0400https://liberty.io/find-the-sum-or-value-of-following-expression-closed.htmlhttps://liberty.io/find-the-sum-or-value-of-following-expression-closed.htmlIn the problem I am not able to derive difference term. I know for solving summation we have to make difference term. How to proceed with this problem? $$\frac{{\displaystyle \sum_{n=1}^{99}} \s......Tue, 01 Sep 2026 02:48:57 -0400https://liberty.io/how-to-obtain-more-than-two-abcissas-for-one-ordinate.htmlhttps://liberty.io/how-to-obtain-more-than-two-abcissas-for-one-ordinate.htmlI'm reading a book on trigonometry and the author shown that some of the examples contain one abcissa for one ordinate and he showed that a parabola has two abscissas for every positive ordinates, ......Tue, 01 Sep 2026 02:48:20 -0400https://liberty.io/inverse-function-of-tanh-x.htmlhttps://liberty.io/inverse-function-of-tanh-x.htmlI have a problem while calculating inverse function of tanh(x). I know it is y = sinh(x)/cosh(x) and then I should express x, but I am stuck with that. Will you help me with this? thx a lot...Tue, 01 Sep 2026 02:34:16 -0400https://liberty.io/how-to-factor-4x-2-2x-1.htmlhttps://liberty.io/how-to-factor-4x-2-2x-1.htmlI want to know how to factor $4x^2 + 2x + 1$? I found the roots using quadratic equation and got $-1 + \sqrt{-3}$ and $-1 - \sqrt{-3}$, so I thought the factors would be $(x - (-1 + \sqrt{-3}))$ an......Tue, 01 Sep 2026 02:28:43 -0400https://liberty.io/why-does-the-function-r-theta-graph-a-spiral.htmlhttps://liberty.io/why-does-the-function-r-theta-graph-a-spiral.htmlIf $\theta$ denotes an angle (in radians I assume), and $r$ means the distance from the origin, then why does $r = \theta$ make a perfect spiral? I'm not that advanced in math (only in geometry and ...Tue, 01 Sep 2026 02:23:55 -0400https://liberty.io/calculus-finding-the-minimum-vertical-distance-between-graphs.htmlhttps://liberty.io/calculus-finding-the-minimum-vertical-distance-between-graphs.htmlQuestion:Find the minimum vertical distance between the graphs of $2+\sin x$ and $\cos x$? In order to find out the required distance, what should I do? It seems that there is a problem if I ...Tue, 01 Sep 2026 02:21:06 -0400https://liberty.io/how-to-calculate-the-number-of-pieces-in-the-border-of-a-puzzle.htmlhttps://liberty.io/how-to-calculate-the-number-of-pieces-in-the-border-of-a-puzzle.htmlIs there any way to calculate how many border-pieces a puzzle has, without knowing its width-height ratio? I guess it's not even possible, but I am trying to be sure about it. Thanks for your help......Tue, 01 Sep 2026 02:14:33 -0400https://liberty.io/linear-function.htmlhttps://liberty.io/linear-function.htmlI don't get this, need some help, examples and information The linear function $f$ is given by $$f(x) = 3x - 2 ,\quad -2 \leq x \leq 4.$$ Enter the independent variable and the dependent ...Tue, 01 Sep 2026 02:09:19 -0400https://liberty.io/lagrange-remainder-is-t-contained-within-an-open-or-closed-interval.htmlhttps://liberty.io/lagrange-remainder-is-t-contained-within-an-open-or-closed-interval.htmlAbove is the taylor series centred at a including the lagrange remainder. Can t be equal to a or b?...Tue, 01 Sep 2026 01:45:39 -0400https://liberty.io/expected-value-of-an-expected-value.htmlhttps://liberty.io/expected-value-of-an-expected-value.htmlI am looking at a proof that $\text{Var}(X)= E((X - EX)^2) = E(X^2) - (E(X))^2$ $E((X - EX)^2) =$ $E(X^2 - 2XE(X) + (E(X))^2) =$ $E(X^2) - 2E(X)E(X) + (E(X))^2)$ I can't see how the second line......Tue, 01 Sep 2026 01:41:55 -0400https://liberty.io/how-can-i-get-the-fourier-transform-of-this-generalized-function.htmlhttps://liberty.io/how-can-i-get-the-fourier-transform-of-this-generalized-function.htmlwhen function f(x) of the generalized function is $f(x) = x+2$, the fourier transform of the generalized function will be like ($F$ is the Fourier transform) $$\int_{-∞}^{∞}(x+2)(F[k(x)])dx$$ whe......Tue, 01 Sep 2026 01:40:15 -0400https://liberty.io/understanding-all-sub-sigma-algebras-of-a-sigma-algebra.htmlhttps://liberty.io/understanding-all-sub-sigma-algebras-of-a-sigma-algebra.htmlSuppose I have a set $\Omega$, and I partition $\Omega$ into $4$ sets $A_{1}, A_{2}, A_{3}, A_{4}$. Clearly, the $A_{i}$'s are pairwise disjoint and their union equals $\Omega$. Now, it's also cl......Tue, 01 Sep 2026 01:10:30 -0400