https://liberty.ioCurated star highlights with fresh energy.en-usMon, 10 Aug 2026 00:55:29 -0400https://liberty.io/midpoint-approximation-over-under-estimation.htmlhttps://liberty.io/midpoint-approximation-over-under-estimation.htmlSo left handed approximation underestimates the area under a increasing curve and over estimates for decreasing curves. And right handed approximation overestimates for increasing curves and ...Mon, 10 Aug 2026 04:55:04 -0400https://liberty.io/symmetry-group-of-equilateral-triangle.htmlhttps://liberty.io/symmetry-group-of-equilateral-triangle.htmlI have read at some places that the symmetry of equilateral triangle is C3v as well as some places mention it to be D3. The group tables for these two groups differ, hence they are not isomorphic......Mon, 10 Aug 2026 04:41:06 -0400https://liberty.io/linear-equation-to-matrix-form.htmlhttps://liberty.io/linear-equation-to-matrix-form.htmlHey everyone I'm practicing some linear equation matrix questions, so far they're easy to construct, but I am completely lost on this one since I cannot use the method of putting them in separate c......Mon, 10 Aug 2026 04:37:34 -0400https://liberty.io/is-there-a-rule-of-integration-that-corresponds-to-the-quotient-rule.htmlhttps://liberty.io/is-there-a-rule-of-integration-that-corresponds-to-the-quotient-rule.htmlWhen teaching the integration method of u-substitution, I like to emphasize its connection with the chain rule of integration. Likewise, the intimate connection between the product rule of derivati......Mon, 10 Aug 2026 04:17:27 -0400https://liberty.io/when-an-equation-has-no-solutions-denote-it-with-x-in-varnothing.htmlhttps://liberty.io/when-an-equation-has-no-solutions-denote-it-with-x-in-varnothing.htmlMy teacher claims that when an equation in variables $x_1,x_2,\ldots,x_n$ has no solutions, you should denote this fact with $(x_1,x_2,\ldots,x_n)\in\varnothing$. An empty set can't have an elemen......Mon, 10 Aug 2026 04:10:00 -0400https://liberty.io/probability-of-head-in-coin-flip-when-coin-is-flipped-two-times.htmlhttps://liberty.io/probability-of-head-in-coin-flip-when-coin-is-flipped-two-times.htmlProbability of getting a head in coin flip is $1/2$. If the coin is flipped two times what is the probability of getting a head in either of those attempts? I think both the coin flips are mutually ...Mon, 10 Aug 2026 04:08:07 -0400https://liberty.io/what-is-a-particular-and-a-homogenous-solution-of-a-differential-equat.htmlhttps://liberty.io/what-is-a-particular-and-a-homogenous-solution-of-a-differential-equat.htmlWhen solving linear nonhomogeneous equations, we deal with two types of solutions: particular homogeneous Why do we have these two types of solutions for differential equations? What does each of......Mon, 10 Aug 2026 04:03:19 -0400https://liberty.io/sum-of-fibonacci-numbers.htmlhttps://liberty.io/sum-of-fibonacci-numbers.htmlWhile trying to find find a formula to calculate the length of the golden spiral I came across the sum of the Fibonacci numbers. I noticed that $$\text{Fibonacci numbers: }1,1,2,3,5,8,13,21,34...$......Mon, 10 Aug 2026 03:57:25 -0400https://liberty.io/linear-combination-of-x.htmlhttps://liberty.io/linear-combination-of-x.htmlIs $3x + 4x^2 + y$ a valid linear combination of $x$? I'm not sure if "a linear combination of $x$" allows for other terms than $x$ which would just be treated as constants, or it should strictly c......Mon, 10 Aug 2026 03:55:01 -0400https://liberty.io/how-many-different-possible-wordle-tweets-can-exist.htmlhttps://liberty.io/how-many-different-possible-wordle-tweets-can-exist.htmlWordle is a viral web game where you have up to six attempts to guess a secret five-letter English word* using the letters A-Z. After each $5$-letter guess (which must itself be a real English word......Mon, 10 Aug 2026 03:40:38 -0400https://liberty.io/is-there-a-valid-shortcut-method-to-convert-a-quadratic-from-vertex-fo.htmlhttps://liberty.io/is-there-a-valid-shortcut-method-to-convert-a-quadratic-from-vertex-fo.htmlRecently, I had a couple of students give me an interesting way of converting from vertex form to factored form that I've never seen before. One told me he got it from his science teacher, and anot......Mon, 10 Aug 2026 03:29:38 -0400https://liberty.io/how-to-find-derivative-of-x-2-sin-x-using-only-the-limit-definition-of.htmlhttps://liberty.io/how-to-find-derivative-of-x-2-sin-x-using-only-the-limit-definition-of.htmlI'm trying to find the derivative of $x^2\sin x$ using only the limit definition of a derivative. I've tried two approaches, one using the difference quotient and another with the regular $x-a$ fo......Mon, 10 Aug 2026 03:16:24 -0400https://liberty.io/minimal-axioms-for-a-group.htmlhttps://liberty.io/minimal-axioms-for-a-group.htmlMy group theory is very rusty. If I want to just start with left inverses and left identities, must I link the axioms, or can I leave them independent? e.g., is it enough to say "there exists at l......Mon, 10 Aug 2026 03:13:09 -0400https://liberty.io/what-exactly-is-a-random-variable.htmlhttps://liberty.io/what-exactly-is-a-random-variable.htmlI don't really understand the definition of a random variable. I also find the Wikipedia entry on random variables kind of confusing. Can someone give me a clear explanation of the random variable?...Mon, 10 Aug 2026 03:05:18 -0400https://liberty.io/why-zero-divided-by-zero-is-undefined-and-not-infinity-duplicate.htmlhttps://liberty.io/why-zero-divided-by-zero-is-undefined-and-not-infinity-duplicate.htmlapologize in advance if this is a duplicate, but I found a lot questions related to this but none answering this specific question. My logic is: let's consider division the opposite of multiplicat......Mon, 10 Aug 2026 02:57:52 -0400https://liberty.io/how-to-denote-a-set-of-sequences.htmlhttps://liberty.io/how-to-denote-a-set-of-sequences.htmlHow do you denote a set of sequences of a certain length, consisting of real numbers? For example, is this correct? $$S=\{A:A=\{a_k\}_{k=1}^{10}\wedge (\forall k\in\Bbb{N},k\le 10:a_k\in\Bbb{R})\}$......Mon, 10 Aug 2026 02:54:56 -0400https://liberty.io/how-do-i-express-108-as-the-product-of-powers-of-its-prime-factors.htmlhttps://liberty.io/how-do-i-express-108-as-the-product-of-powers-of-its-prime-factors.htmlHow do I express $108$ as the product of powers of its prime factors? I've got $2^2 \times 3^3$ is this correct?...Mon, 10 Aug 2026 02:42:17 -0400https://liberty.io/what-means-a-setminus-logic-symbol.htmlhttps://liberty.io/what-means-a-setminus-logic-symbol.htmlI have a task to calculate a double integral on a region $D=A\setminus B$, where $A=\{(x,y): \text{some system of inequalities}\}, B= \{(x,y): \text{another system of inequalities}\}$. What does ...Mon, 10 Aug 2026 02:31:55 -0400https://liberty.io/what-is-the-volume-of-an-ellipsoid-closed.htmlhttps://liberty.io/what-is-the-volume-of-an-ellipsoid-closed.htmlHow to use integrals to find the volume of the following ellipsoid? What are the associated parametric equations for this sort of ellipsoid?...Mon, 10 Aug 2026 02:11:01 -0400https://liberty.io/lipschitz-continuous-rightarrow-uniformly-continuous.htmlhttps://liberty.io/lipschitz-continuous-rightarrow-uniformly-continuous.htmlThe Question: Prove that if a function $f$ defined on $S \subseteq \mathbb R$ is Lipschitz continuous then $f$ is uniformly continuous on $S$. Definition. A function $f$ defined on a set $S \su......Mon, 10 Aug 2026 01:59:20 -0400https://liberty.io/does-the-perimeter-of-a-semicircle-include-the-diameter-closed.htmlhttps://liberty.io/does-the-perimeter-of-a-semicircle-include-the-diameter-closed.htmlIs the perimeter of a semicircle $(\pi)(\text{radius})$ or $(\pi)(\text{radius})+\text{diameter}$?...Mon, 10 Aug 2026 01:55:38 -0400https://liberty.io/the-matrix-notation-of-signum.htmlhttps://liberty.io/the-matrix-notation-of-signum.htmlThe following question on a notation might look trivial but I am really not sure how to deal with it. If I have a variable $x$, I could write out: $$x=|x|\;\text {sgn} (x)$$ a notation that help......Mon, 10 Aug 2026 01:53:28 -0400https://liberty.io/how-do-i-change-the-output-of-matlab-when-using-inverse-laplace.htmlhttps://liberty.io/how-do-i-change-the-output-of-matlab-when-using-inverse-laplace.htmlHere is my input: syms s F = (2+2*s*(exp(-s))+4*(exp(-2*s)))/(s^2)*(s^2+2*s+10); ilaplace(F) An the output is : 20*t + 4*dirac(t - 1) + 4*dirac(t - 2) + 20*heaviside(t - 1) + 8*heaviside(t - 2) ......Mon, 10 Aug 2026 01:53:11 -0400https://liberty.io/write-in-terms-of-sine-and-cosine-and-simplify.htmlhttps://liberty.io/write-in-terms-of-sine-and-cosine-and-simplify.htmlI am having problems understanding how to solve this equation, or what it is asking, any clarification would be much apprreciated! Write the trigonometric expression in terms of sine and cosine, ......Mon, 10 Aug 2026 01:48:29 -0400https://liberty.io/how-to-get-the-perimeter-of-the-circular-sector-in-the-figure.htmlhttps://liberty.io/how-to-get-the-perimeter-of-the-circular-sector-in-the-figure.htmlI have a perimeter exercise of a circular sector, but my result is different. Where have I been wrong? The exercise is: With the formula: $\frac{\angle{O}}{360}2\pi r + 2r$ I have the circular s......Mon, 10 Aug 2026 01:45:14 -0400https://liberty.io/derivative-of-function-with-respect-to-multivariable-function.htmlhttps://liberty.io/derivative-of-function-with-respect-to-multivariable-function.htmlLet \begin{align*} V\colon \mathbb R & \to \mathbb R \\ x & \longmapsto V(x). \end{align*} and \begin{align*} w\colon \mathbb R^3 & \to \mathbb R \\ (a,b,c) & \longmapst......Mon, 10 Aug 2026 01:42:26 -0400https://liberty.io/bernoulli-markov-chain.htmlhttps://liberty.io/bernoulli-markov-chain.htmlLet $Y_{n}$, $n≥0$ be an i.i.d. sequence of Bernoulli $(p)$ random variables for some fixed $p ∈ (0, 1).$ Let $X_{n} = Y_{n−1} + Y_{n}$ for $n ≥ 1$. Is $X_{n}$, $n≥1$ a Markov chain? If no explain......Mon, 10 Aug 2026 01:35:41 -0400https://liberty.io/dot-product-of-a-row-vector-and-a-column-vector.htmlhttps://liberty.io/dot-product-of-a-row-vector-and-a-column-vector.htmlAs I understand it, the dot product of two vectors $\bf a$ and $\bf b$, both represented as $1 \times n$ matrices, is equal to $\bf a \bf b^T$. Is it possible to take the dot product of $\bf a$, a......Mon, 10 Aug 2026 01:35:07 -0400https://liberty.io/how-can-something-be-greater-than-100.htmlhttps://liberty.io/how-can-something-be-greater-than-100.htmlApologies, my mathematics is poor but I saw an advert recently which said: Up to $350\%$ more likely to quit How can something be greater than $100\%$? It can't be can it?...Mon, 10 Aug 2026 01:32:27 -0400https://liberty.io/each-natural-number-is-equal-to-the-set-of-all-natural-numbers-less-th.htmlhttps://liberty.io/each-natural-number-is-equal-to-the-set-of-all-natural-numbers-less-th.htmlPlease check my proof that each natural number is equal to the set of all natural numbers less than it. Let $\mathbb N$ is the set of natural numbers by Von Neumann construction (https://www.wikiw......Mon, 10 Aug 2026 01:26:59 -0400https://liberty.io/if-x-2-x-1-0-then-find-the-value-of-x-1999-x-2000.htmlhttps://liberty.io/if-x-2-x-1-0-then-find-the-value-of-x-1999-x-2000.htmlIf $x^2+x+1 = 0$ then find the value of $x^{1999}+x^{2000}$. I first tried finding the solution of the given equation and then substituting it in the expression whose value we have to find but I w......Mon, 10 Aug 2026 01:21:28 -0400https://liberty.io/find-the-derivative-of-this-function-at-a-specific-point-closed.htmlhttps://liberty.io/find-the-derivative-of-this-function-at-a-specific-point-closed.htmlWhat is the derivative of $$f(t)= \frac{t^3 +2} t$$ at point $(-2,3)$....Mon, 10 Aug 2026 01:06:07 -0400https://liberty.io/eigenfunctions-of-1d-gaussian-kernel.htmlhttps://liberty.io/eigenfunctions-of-1d-gaussian-kernel.htmlHeads up Im a physicist! I want to know explicitly the eigenfunctions of the 1D gaussian kernel $$ K(x,y) = e^{-(x-y)^2/\sigma^2} $$ when it is integrated, that is $$ (Kf)(n,x)=\int_{-\infty}^{\inf......Mon, 10 Aug 2026 00:58:40 -0400https://liberty.io/if-m-and-n-are-positive-integers-then-prove-that-m-2-n-2-does-not-equa.htmlhttps://liberty.io/if-m-and-n-are-positive-integers-then-prove-that-m-2-n-2-does-not-equa.htmlI have to use proof by contradiction to solve the problem and I believe that proving $m^2 - n^2 = 1$ is the way to go with this, but where do I start?...Mon, 10 Aug 2026 00:54:24 -0400https://liberty.io/what-is-the-difference-between-a-taylor-series-taylor-polynomial-analy.htmlhttps://liberty.io/what-is-the-difference-between-a-taylor-series-taylor-polynomial-analy.htmlI am using Wikipedia to brush up on these key concepts. I will write what I believe I read and maybe an expert can put me back on track. First , they are all functions. I get the idea that a Tay......Mon, 10 Aug 2026 00:51:09 -0400https://liberty.io/why-are-we-allowed-to-multiply-a-1x1-matrix-by-any-matrix-duplicate.htmlhttps://liberty.io/why-are-we-allowed-to-multiply-a-1x1-matrix-by-any-matrix-duplicate.htmlSo, in order to multiply 2 matrices, there must be the same number of columns in the left matrix as there are rows in the right matrix. So if $A$ is an $m \times n$ matrix and $B$ is a $p \times q$ ...Mon, 10 Aug 2026 00:50:44 -0400https://liberty.io/what-is-an-example-of-a-field-that-is-not-a-sigma-field.htmlhttps://liberty.io/what-is-an-example-of-a-field-that-is-not-a-sigma-field.htmlWhat is an example of a field, that is not a $\sigma$-field? I read a $\sigma$-field requires closure of it's elements (which are sets) under countable unions, countable intersection, and complemen......Mon, 10 Aug 2026 00:50:27 -0400https://liberty.io/what-does-it-mean-for-two-functions-to-be-equivalent-closed.htmlhttps://liberty.io/what-does-it-mean-for-two-functions-to-be-equivalent-closed.htmlWhat does it mean rigorously for two functions $f: \mathbb{R} \to \mathbb{R}$ and $g: \mathbb{R} \to \mathbb{R}$ to be equivalent? Does $f = g$ if and only if $\forall x \in \mathbb{R} \ \ f(x) = g......Mon, 10 Aug 2026 00:48:56 -0400https://liberty.io/how-do-you-calculate-the-dimensions-of-the-null-space-and-column-space.htmlhttps://liberty.io/how-do-you-calculate-the-dimensions-of-the-null-space-and-column-space.htmlI understand you are supposed to get the reduced row echelon form, which I did, and this is what I came up with: 1 -2 0 19 -6 0 -37 0 0 1 -6 2 0 6 0 0 0 0 0 1 3 0 ......Mon, 10 Aug 2026 00:46:49 -0400https://liberty.io/standard-notation-for-diagonal-matrices.htmlhttps://liberty.io/standard-notation-for-diagonal-matrices.htmlIs there standard notation for the set of diagonal matrices? Specifically if the elements must be nonnegative, i.e. the matrix is positive semi-definite?...Mon, 10 Aug 2026 00:40:21 -0400https://liberty.io/how-to-find-the-general-form-ax-by-c-0-of-a-line-with-an-undefined-slo.htmlhttps://liberty.io/how-to-find-the-general-form-ax-by-c-0-of-a-line-with-an-undefined-slo.htmlThis is how the question reads: "The equation of the line that goes through the points (3, -6) and (3, 10) can be written in general form Ax + By + C = 0 where A = _ B = _ and C = ____" I know the ......Mon, 10 Aug 2026 00:39:07 -0400https://liberty.io/how-do-i-convert-this-equation-to-the-standard-form-of-a-circle.htmlhttps://liberty.io/how-do-i-convert-this-equation-to-the-standard-form-of-a-circle.htmlI'm looking for the radius of the sphere of this: $4x^2 + 4y^2 +4z^2 -16x - 24y + 8z= 44$. I have to get it into standard form in order to find the radius. So I factored out a 4 and simplified it t......Mon, 10 Aug 2026 00:34:34 -0400https://liberty.io/how-to-prove-this-gradient-theorem-in-vector-calculus.htmlhttps://liberty.io/how-to-prove-this-gradient-theorem-in-vector-calculus.htmlI encountered a theorem in some online notes labeled the 'gradient theorem': $$\int_\Omega \nabla u = \int_\Gamma u \mathbf n$$ I've never seen this theorem before, and searches for 'gradient the......Mon, 10 Aug 2026 00:34:16 -0400https://liberty.io/finding-the-dimension-of-the-symplectic-group.htmlhttps://liberty.io/finding-the-dimension-of-the-symplectic-group.htmlHow do you find the dimension of the symplectic group $\operatorname{Sp}(2n,\mathbb{R})$? $\operatorname{Sp}(2n,\mathbb{R})\subset \operatorname{Gl}(2n,\mathbb{R})$ is the group of invertible matr......Mon, 10 Aug 2026 00:27:13 -0400https://liberty.io/definition-of-f-invariant-in-linear-algebra.htmlhttps://liberty.io/definition-of-f-invariant-in-linear-algebra.htmlI'm studying linear algebra for math undergrad students, yet I have no background in maths, so I was wondering what the highlighted part of the text means. I tried to "google" it yet I did not find......Mon, 10 Aug 2026 00:20:15 -0400https://liberty.io/why-is-the-catenoid-the-minimal-surface-of-revolution.htmlhttps://liberty.io/why-is-the-catenoid-the-minimal-surface-of-revolution.htmlThis is more of a soft question than anything and I'm asking for either a proof or intuitive explanation as to why this is. It seems that if one defines 2 points in the upper-half of the Cartesian......Mon, 10 Aug 2026 00:19:46 -0400https://liberty.io/what-is-the-right-symbol-for-the-max-like-pi-is-for-product.htmlhttps://liberty.io/what-is-the-right-symbol-for-the-max-like-pi-is-for-product.htmlI have a set of variables that have a value, and i want to find the max of those values. Here is the equivalent of what I want to do with "sum" $$\sum_{j:~N_j \in U_i}~ DA_j$$ For all j subjec......Mon, 10 Aug 2026 00:13:56 -0400https://liberty.io/intuition-for-a-basis-in-vector-spaces.htmlhttps://liberty.io/intuition-for-a-basis-in-vector-spaces.htmlSome confusion about the basis vectors. These questions came from the definition of a basis at the bottom. I'm wondering: What the relation is between basis and non-basis vectors. Or put another w......Mon, 10 Aug 2026 00:13:05 -0400https://liberty.io/find-2x2-matrix-such-that-its-inverse-equals-its-transpose.htmlhttps://liberty.io/find-2x2-matrix-such-that-its-inverse-equals-its-transpose.htmlFind some matrix $B\in GL_2 (\mathbb{R})$ such that $B^{-1} = B^T$ and $B \neq I$ What I tried: I tried to create a simultaneous equation i.e. if B = $\begin{bmatrix} a&b\\c & d\end{bmatri......Mon, 10 Aug 2026 00:08:19 -0400https://liberty.io/is-there-a-mathematical-symbol-for-evidence.htmlhttps://liberty.io/is-there-a-mathematical-symbol-for-evidence.htmlI'm currently writing a short paper on the evidence needed to hold a probability function(in the Bayesian sense) objectively and trying connecting it to entropy. Unfortunately, I couldn't find an......Mon, 10 Aug 2026 00:05:34 -0400https://liberty.io/power-series-method-to-solve-airy-s-differential-equation-duplicate.htmlhttps://liberty.io/power-series-method-to-solve-airy-s-differential-equation-duplicate.htmlUsing power series method, solve Airy’s equation $$y′′+ xy = 0.$$ How do I start solving this? Thanks in advance!...Mon, 10 Aug 2026 00:05:19 -0400https://liberty.io/correct-probability-calculation-for-minesweeper.htmlhttps://liberty.io/correct-probability-calculation-for-minesweeper.htmlWhen I play Minesweeper, every now and then I reach a point where I have to make a guess. I then try to calculate the probability for every option to be a mine in order to choose the safest move. But ...Mon, 10 Aug 2026 00:04:49 -0400https://liberty.io/what-is-the-probability-that-one-but-not-both-of-the-events-r-and-s-wi.htmlhttps://liberty.io/what-is-the-probability-that-one-but-not-both-of-the-events-r-and-s-wi.htmlSuppose that R and S are events in a certain probability space such that $P[R∣S]=0.52$, $P[S∣R]=0.65$ and $P[R\cap S]=0.26$. I have found that $P[R]=0.4$ and $P[S]=0.5$. I thought this was askin......Mon, 10 Aug 2026 00:04:32 -0400https://liberty.io/039-free-vector-space-039-and-039-vector-space-039.htmlhttps://liberty.io/039-free-vector-space-039-and-039-vector-space-039.htmlIn this very nice book, author has defined vector space as set of functions $f : S \rightarrow F$ where $S$ is a finite set and $F$ is a field. It turns out that this definition is closely resemble ...Sun, 09 Aug 2026 23:51:54 -0400https://liberty.io/what-is-the-exact-definition-of-a-spline.htmlhttps://liberty.io/what-is-the-exact-definition-of-a-spline.htmlI know that a spline basically involves joining several functions together. I am hoping to get a more precise definition so I know exactly when to use the word "spline". I have a few questions ...Sun, 09 Aug 2026 23:40:51 -0400https://liberty.io/the-degree-of-antipodal-map.htmlhttps://liberty.io/the-degree-of-antipodal-map.htmlI am trying to solve the problem A map without fixed points - two wrong approaches. But I am not certain about the degree of antipodal map. I my thought, since the preimage of a point $y \in S^k$ is ...Sun, 09 Aug 2026 23:40:41 -0400https://liberty.io/integrating-sin-2-x.htmlhttps://liberty.io/integrating-sin-2-x.htmlI was just wondering that when doing the integral of $\sin^2(x)$ why we can't have the answer as $$ \frac13\,\sin^3x\,\frac{1}{\cos x} $$ I think this problem comes from the fact that calculus in my ...Sun, 09 Aug 2026 23:36:56 -0400https://liberty.io/computing-a-limit-of-riemann-sum-to-evaluate-an-integral.htmlhttps://liberty.io/computing-a-limit-of-riemann-sum-to-evaluate-an-integral.htmlI am told to evaluate this integral by computing the limit of Riemann sums. From another post (https://math.stackexchange.com/a/890615/502497) I gathered this statement The right endpoint Riemann ......Sun, 09 Aug 2026 23:34:59 -0400https://liberty.io/calculating-the-lateral-area-of-a-cone.htmlhttps://liberty.io/calculating-the-lateral-area-of-a-cone.htmlOn my book, it says ...thinking of the lateral surface as swept out by revolving a generator (i.e. slant height) about the axis: the lateral area equals the length of this generator multiplied b......Sun, 09 Aug 2026 23:24:51 -0400https://liberty.io/finding-the-square-root-of-6-4-sqrt-2.htmlhttps://liberty.io/finding-the-square-root-of-6-4-sqrt-2.htmlI found this standupmaths video on YouTube about the A4 paper puzzle. I really liked the puzzle and managed to get the answer by using a calculator. However, the answer (which I won't spoil), led ......Sun, 09 Aug 2026 23:22:41 -0400https://liberty.io/how-would-the-intersection-of-two-uncountable-sets-be-finite.htmlhttps://liberty.io/how-would-the-intersection-of-two-uncountable-sets-be-finite.htmlThis is a problem from Discrete Mathematics and its Applications Here is my book's definition on countable and definition of having the same cardinality The only example that my book gave of ...Sun, 09 Aug 2026 23:18:35 -0400https://liberty.io/prove-that-a-triangle-is-right-triangle.htmlhttps://liberty.io/prove-that-a-triangle-is-right-triangle.htmlI'd like to know if there's any theorem to prove that the triangle ACB' is a right triangle and that the angle ACB' is 90°. We know that ACB and A'B'C' are right triangles, so in my opinion ACB' is ...Sun, 09 Aug 2026 23:12:10 -0400https://liberty.io/how-to-find-a-f-such-that-f-grad-f.htmlhttps://liberty.io/how-to-find-a-f-such-that-f-grad-f.htmlI need help with this problem: For the conservative field $F$ find a function $f$ such that $F=$ grad$f$. $$F(x,y,z)=\left(\frac{x}{r},\frac{y}{r},\frac{z}{r}\right)$$, where $r=\sqrt{x^2+y^2+......Sun, 09 Aug 2026 23:05:42 -0400https://liberty.io/inner-product-and-norm-of-a-function.htmlhttps://liberty.io/inner-product-and-norm-of-a-function.htmlI have recently started a undergraduate linear algebra course in which these definitions came up: Let $V$ be the vector space $C[a, b]$ of all continuous functions on $[a, b]$. Then the inner prod......Sun, 09 Aug 2026 22:58:29 -0400https://liberty.io/is-frac-1-x-a-monotonic-decreasing-function.htmlhttps://liberty.io/is-frac-1-x-a-monotonic-decreasing-function.htmlThe derivative of $f(x)=\frac{1}{x}$ is always negative for all $x$ except $x=0$ (where it is not defined). But $f(-1)<f(1)$ , so the statement that $f(x)$ either decreases or remains constant a......Sun, 09 Aug 2026 22:57:48 -0400https://liberty.io/why-absolute-value-function-is-not-a-continuous-function.htmlhttps://liberty.io/why-absolute-value-function-is-not-a-continuous-function.htmlI am doing a pre-calculus course from: https://www.edx.org/course/pre-university-calculus It tells me that a continuous function has continuous graphs while a piece-wise function has kind of step......Sun, 09 Aug 2026 22:57:33 -0400https://liberty.io/does-lambert-w-product-log-count-as-an-explicit-solution.htmlhttps://liberty.io/does-lambert-w-product-log-count-as-an-explicit-solution.htmlSay I have an equation that I can solve in $x$ as follows: $$ x = LambertW_{-1}(y)$$ Where LambertW is the product-log function. Can I say I have an explicit solution for $x$? It looks like that......Sun, 09 Aug 2026 22:50:48 -0400https://liberty.io/solving-the-system-sqrt-x-y-7-x-sqrt-y-11.htmlhttps://liberty.io/solving-the-system-sqrt-x-y-7-x-sqrt-y-11.htmlI want to solve the following nonlinear system of algebraic equations. Indeed, I am curious about a step by step solution for pedagogical purposes. I am wondering if you can come up with anything. I ...Sun, 09 Aug 2026 22:48:35 -0400https://liberty.io/is-there-anything-deep-about-fuzzy-sets-fuzzy-logic.htmlhttps://liberty.io/is-there-anything-deep-about-fuzzy-sets-fuzzy-logic.htmlI keep coming across the term "fuzzy". I browsed around a bit and read a few articles.. It seems to me that there's absolutely nothing deep or foundational about fuzzy set theory or fuzzy logic. I ......Sun, 09 Aug 2026 22:48:02 -0400https://liberty.io/why-does-newton-039-s-method-work.htmlhttps://liberty.io/why-does-newton-039-s-method-work.htmlI find many sites explaining how to use Newton's method, but none explaining why it works. Could someone give me the intuition behind it? Thanks....Sun, 09 Aug 2026 22:44:21 -0400https://liberty.io/what-is-the-last-digit-of-2003-2003.htmlhttps://liberty.io/what-is-the-last-digit-of-2003-2003.htmlWhat is the last digit of this number? $$2003^{2003}$$ Thanks in advance. I don't have any kind of idea how to solve this. Except that $3^3$ is $27$....Sun, 09 Aug 2026 22:29:38 -0400https://liberty.io/understanding-how-to-check-subspaces.htmlhttps://liberty.io/understanding-how-to-check-subspaces.htmlSo here's my problem. I'm trying to do some practice on Subspace of vectors, and this was a problem in the book. Determine if the given set is a subspace of $P_n$, the set of polynomials of degr......Sun, 09 Aug 2026 22:29:34 -0400https://liberty.io/find-the-average-value-of-f-over-region-d-f-x-y-6xy-d-is-the-triangle-.htmlhttps://liberty.io/find-the-average-value-of-f-over-region-d-f-x-y-6xy-d-is-the-triangle-.htmlI have no idea how to find the limits for this. I've tried plotting it over x but cant figure it out....Sun, 09 Aug 2026 22:16:35 -0400https://liberty.io/expected-value-of-function-of-exponential-random-variable.htmlhttps://liberty.io/expected-value-of-function-of-exponential-random-variable.htmlLet $X \sim Exp(\lambda)$ (exponential distribution) and $Y=X^a$. For what values of $a\in R$ is $E[Y]< \infty$? I computed $E[Y] = \lambda \int x^a e^{-\lambda x}dx$. How to proceed? Does the ...Sun, 09 Aug 2026 22:08:51 -0400https://liberty.io/equations-of-lines-tangent-to-an-ellipse.htmlhttps://liberty.io/equations-of-lines-tangent-to-an-ellipse.htmlDetermine the equations of the lines that are tangent to the ellipse $\displaystyle{\frac{1}{16}x^2 + \frac{1}{4}y^2 = 1}$ and pass through $(4,6)$. I know one tangent should be $x = 4$ because it......Sun, 09 Aug 2026 22:05:07 -0400https://liberty.io/variance-of-ax-b.htmlhttps://liberty.io/variance-of-ax-b.htmlHow do I show that $$\text{Var}(aX+b)=a^2\text{Var}(X)$$. Since, I am reading statistic first time, I don't have any idea how to start. Thanks for helping me....Sun, 09 Aug 2026 22:01:25 -0400https://liberty.io/graph-f-x-y-ln-x-y.htmlhttps://liberty.io/graph-f-x-y-ln-x-y.htmlI'm trying to graph $f(x,y)=\ln(x)-y$, however, I am not sure how as all of my tools are refusing to graph it. Can you please help me? Thanks...Sun, 09 Aug 2026 21:46:57 -0400https://liberty.io/what-is-the-name-of-the-solid-created-by-joining-two-tetrahedra-face-t.htmlhttps://liberty.io/what-is-the-name-of-the-solid-created-by-joining-two-tetrahedra-face-t.htmlWhen I connect the bases of two pyramids, I will receive an octahedron. I have a question about an analogical situation with two tetrahedra. What is the name of a geometrical figure, which is a ...Sun, 09 Aug 2026 21:29:29 -0400https://liberty.io/are-there-any-k-n-that-have-euler-trails-but-not-euler-cycles.htmlhttps://liberty.io/are-there-any-k-n-that-have-euler-trails-but-not-euler-cycles.htmlI know that in $K_n$, a complete graph. ALL the vertices must be of even degree for a euler cycle to exist. So n should be odd in this case if we want euler cycles. And if n is even, then $K_n$ has......Sun, 09 Aug 2026 21:22:58 -0400https://liberty.io/converse-of-fermat-039-s-little-theorem.htmlhttps://liberty.io/converse-of-fermat-039-s-little-theorem.htmlIf $a^n\equiv a \pmod n$ for all integers $a$, does this imply that $n$ is prime? I believe this is the converse of Fermat's little theorem....Sun, 09 Aug 2026 21:20:08 -0400https://liberty.io/is-2n-the-same-as-2-n.htmlhttps://liberty.io/is-2n-the-same-as-2-n.htmlI am trying to determine the convergence of a series $$\sum_{n=17}^{\infty} \frac{(n!)}{(2n)!}.$$ Using the ratio test, I have simplified $a_{n+1}/a_n$ to $$\frac{(2n!)}{2(n)!}.$$ If $(2n)!$ is th......Sun, 09 Aug 2026 21:19:23 -0400https://liberty.io/can-you-self-teach-yourself-mathematics-closed.htmlhttps://liberty.io/can-you-self-teach-yourself-mathematics-closed.htmlIf you are not a natural genius, can you teach yourself Mathematics (through textbooks) enough to make a successful mathematical career?...Sun, 09 Aug 2026 21:16:58 -0400https://liberty.io/independence-vs-marginal-independence.htmlhttps://liberty.io/independence-vs-marginal-independence.htmlI known that for give $x,y$, if we have $$p(x,y)=p(x)p(y)$$ Then we call $x,y$ are independent. For marginal independence, I found the definition here. Random variable $x$ is marginal independ......Sun, 09 Aug 2026 21:08:02 -0400https://liberty.io/differential-on-total-complex.htmlhttps://liberty.io/differential-on-total-complex.htmlGiven a bicomplex $C_{\ast,\ast}$, i.e. objects $C_{p,q}$ in an abelian category with horizontal $$d_{p,q}^h:C_{p,q}\to C_{p-1,q}$$ and vertical $$d_{p,q}^v:C_{p,q}\to C_{p,q-1}$$ differentials that ...Sun, 09 Aug 2026 21:02:32 -0400https://liberty.io/cis-graphs-finding-values.htmlhttps://liberty.io/cis-graphs-finding-values.htmlFor this question: If you guys are not familiar with the $\operatorname{cis}$ notation $$ \operatorname{cis}(x) = (\cos(x) + i\sin(x)) $$ I've tried doing this: By comparing arguments $$......Sun, 09 Aug 2026 20:52:43 -0400https://liberty.io/express-in-terms-of-single-vector.htmlhttps://liberty.io/express-in-terms-of-single-vector.htmlCan you guys give me a hint for this problem. ABCDEF is a regular hexagon. Express in terms of a single vector the sum of the vectors, $4\overrightarrow{AB}$, $2\overrightarrow{AC}$, $\overrightar......Sun, 09 Aug 2026 20:50:53 -0400https://liberty.io/calculate-sin-1-sin-45-circ-without-a-calculator.htmlhttps://liberty.io/calculate-sin-1-sin-45-circ-without-a-calculator.htmlWe know that in this triangle, $h^2=a^2+a^2$. Therefore, $$ h=\sqrt{a^2+a^2}=\sqrt{2a^2}=\sqrt{2}\:a\:. $$ As $h$ can be written as $\sqrt{2}\:a$, we now know how to write an expression for $\si......Sun, 09 Aug 2026 20:44:56 -0400https://liberty.io/definitions-of-baire-first-and-second-category-sets.htmlhttps://liberty.io/definitions-of-baire-first-and-second-category-sets.htmlFrom Planetmath A meager or Baire first category set in a topological space is one which is a countable union of nowhere dense sets. A Baire second category set is one which contains a countable u......Sun, 09 Aug 2026 20:42:18 -0400https://liberty.io/how-to-design-shape-a-polyhedron-to-be-nearly-spherically-symmetrical-.htmlhttps://liberty.io/how-to-design-shape-a-polyhedron-to-be-nearly-spherically-symmetrical-.htmlThere are only 5 platonic solids, but take a look at these images: How are these things designed? How are they shaped? It looks to me like those hexagons are all the same size and shape, and evenly ...Sun, 09 Aug 2026 20:29:13 -0400https://liberty.io/how-to-factorise-x-4-equations.htmlhttps://liberty.io/how-to-factorise-x-4-equations.htmlThis is my previous question I'm facing a problem to factorise this $64x^4+64x^3-88x^2-51x+39=0$. How to factorise $x^4$ equations?...Sun, 09 Aug 2026 20:20:07 -0400https://liberty.io/write-the-trigonometric-expression-as-an-algebraic-expression.htmlhttps://liberty.io/write-the-trigonometric-expression-as-an-algebraic-expression.htmlWrite the trigonometric expression as an algebraic expression. $6 \cos(2 \cos^{-1} x)$ Can someone explain to me how to do this? I tried it on my own after watching a youtube video from patrickjm......Sun, 09 Aug 2026 20:18:35 -0400https://liberty.io/find-distance-as-function-of-time.htmlhttps://liberty.io/find-distance-as-function-of-time.htmlA car starts from rest and accelerates in $a = \frac{2\cdot m}{3\cdot s^3}t$, After $3$ seconds, The car will be $27$ metres from beginning. Find distance as function of time. I know i have to ...Sun, 09 Aug 2026 20:13:09 -0400https://liberty.io/evaluate-definite-integral-using-limit-of-summations.htmlhttps://liberty.io/evaluate-definite-integral-using-limit-of-summations.htmlI need to evaluate a definite integral using sums. $$ \int_{-5}^{5} (x-\sqrt{25-x^2})dx $$ I start by getting de value of $\Delta x$: $$ \Delta x = \frac{5-(-5)}{n}=\frac{10}{n} $$ Then for $x......Sun, 09 Aug 2026 20:09:00 -0400https://liberty.io/finding-a-derivative-using-multiple-chain-rules.htmlhttps://liberty.io/finding-a-derivative-using-multiple-chain-rules.htmlFind the derivative of the function. $y = [x + (x + \sin^2 x)^3]^4$ I know how to use the chain rule and I found the derivative to be: $$4[x+(x+\sin^2(x))^3]^3 \cdot (1 + 3(x + \sin^2(x))^2) \cd......Sun, 09 Aug 2026 20:04:27 -0400https://liberty.io/some-confusion-about-what-a-function-really-is.htmlhttps://liberty.io/some-confusion-about-what-a-function-really-is.htmlDespite my username, my background is mostly in functional analysis where (at least to my understanding), a function $f$ is considered as a mathematical object in its own right distinctly different......Sun, 09 Aug 2026 20:00:39 -0400https://liberty.io/what-fraction-of-a-sphere-can-an-external-observer-see.htmlhttps://liberty.io/what-fraction-of-a-sphere-can-an-external-observer-see.htmlHere is a geometry problem. Let there be a ball of radius R and let's call it the Moon. Let there be an external observer: A. A is at a distance d to (the surface of) the Moon. [Edit] A is a Cy......Sun, 09 Aug 2026 19:59:33 -0400https://liberty.io/standard-symbol-for-to-prove-or-need-to-show-in-proofs.htmlhttps://liberty.io/standard-symbol-for-to-prove-or-need-to-show-in-proofs.htmlIs there a standard symbol used as shorthand for "to prove" or "need to show" in a proof? I've seen "N.T.S." but was wondering if there is anything more abstract — not bound to English....Sun, 09 Aug 2026 19:57:44 -0400https://liberty.io/spherical-harmonics-for-dummies.htmlhttps://liberty.io/spherical-harmonics-for-dummies.htmlAdding to the for dummies. The real spherical harmonics are orthonormal basis functions on the surface of a sphere. I'd like to fully understand that sentence and what it means. Still grappling ......Sun, 09 Aug 2026 19:52:43 -0400https://liberty.io/proof-that-a-number-can-039-t-be-both-odd-and-even.htmlhttps://liberty.io/proof-that-a-number-can-039-t-be-both-odd-and-even.htmlI'm trying to proof that a number can't be both odd and even at the same time. The definitions are the follows: A number is called even if there's $m\in\mathbb{Z}$ that satisfies $k = 2m$. A numbe......Sun, 09 Aug 2026 19:51:43 -0400https://liberty.io/where-exactly-are-complex-numbers-used-in-the-real-world.htmlhttps://liberty.io/where-exactly-are-complex-numbers-used-in-the-real-world.htmlI've always enjoyed solving problems in the complex numbers during my undergrad. However, I've always wondered where are they used and for what? In my domain (computer science) I've rarely seen it be ...Sun, 09 Aug 2026 19:50:52 -0400