https://liberty.ioCurated star highlights with fresh energy.en-usSun, 30 Aug 2026 23:23:23 -0400https://liberty.io/optimization-of-the-surface-area-of-a-open-rectangular-box-to-find-the.htmlhttps://liberty.io/optimization-of-the-surface-area-of-a-open-rectangular-box-to-find-the.htmlA rectangular storage container with an open top is to have a volume of 10 cubic meters. The length of the box is twice its width. Material for the base costs ten dollars per square meter and for the ...Mon, 31 Aug 2026 03:21:54 -0400https://liberty.io/total-distance-travelled-using-integrals.htmlhttps://liberty.io/total-distance-travelled-using-integrals.htmlFor time $t \geq 0$, the velocity of a particle moving along the $x$-axis is given by $\mathrm{v}\left(t\right) = \sin\left(t\right)$, where $t$ is measured in seconds and $\mathrm{v}\left(t\right)......Mon, 31 Aug 2026 03:14:46 -0400https://liberty.io/is-a-triangle-with-two-equal-angles-always-isosceles.htmlhttps://liberty.io/is-a-triangle-with-two-equal-angles-always-isosceles.htmlAn isosceles triangle is a triangle with two sides that are equal in length. This means that two angle will also be equal to each other. Is there any way that a triangle could have two equal angles......Mon, 31 Aug 2026 03:13:53 -0400https://liberty.io/p-norm-with-p-to-infinity-duplicate.htmlhttps://liberty.io/p-norm-with-p-to-infinity-duplicate.htmlI have to show that: for all vectors $v\in \Bbb R^n$: $\lim_{p\to \infty}||v||_p = \max_{1\le i \le n}|v_i|$ with the $||\cdot ||_p$ norm defined as $$ ||\cdot ||_p: (v_1, \dots ,v_n) \to (\sum^n_{......Mon, 31 Aug 2026 03:08:52 -0400https://liberty.io/how-to-integrate-sec-3-x-dx-duplicate.htmlhttps://liberty.io/how-to-integrate-sec-3-x-dx-duplicate.htmlPossible Duplicate: Indefinite integral of secant cubed How to integrate $\sec^3 x \, dx$? Can someone please give a method, I tried separating $\sec^3 x$ as $\sec x(\sec^2 x)$ then applying by-...Mon, 31 Aug 2026 02:59:22 -0400https://liberty.io/change-from-product-to-sum.htmlhttps://liberty.io/change-from-product-to-sum.htmlWe know that : $$a \times b = \underbrace{a + a + a + ... + a}_{\text{b times}}$$ That's how we convert from a product to a sum. So what happens if we go a little further? That is : $$\prod\limits_......Mon, 31 Aug 2026 02:57:51 -0400https://liberty.io/which-of-these-sentences-are-propositions-what-are-the-truth-values-of.htmlhttps://liberty.io/which-of-these-sentences-are-propositions-what-are-the-truth-values-of.htmlWhich of these sentences are propositions? What are the truth values of those that are propositions? a) Boston is the capital of Massachusetts. b) Miami is the capital of Florida. c) 2 + 3 = 5. d) ......Mon, 31 Aug 2026 02:53:07 -0400https://liberty.io/utilization-of-m-m-c-queue.htmlhttps://liberty.io/utilization-of-m-m-c-queue.htmlWhat is the utilization of m/m/c queue ? in some texts such as http://www.amazon.com/Queueing-Networks-Markov-Chains-Applications/dp/0471565253 said: individual server utilization $\rho =(\lambda)......Mon, 31 Aug 2026 02:49:20 -0400https://liberty.io/some-commutator-identities.htmlhttps://liberty.io/some-commutator-identities.htmlOn the way to study Lang's algebra, I cannot solve this problem. See page 69. Let G be a group and denote the commutator of x and y by & $[x,y]=xyx^{-1}y^{-1}$. I wanna prove that if $[x,y]=......Mon, 31 Aug 2026 02:45:19 -0400https://liberty.io/why-is-it-valid-to-multiply-both-sides-of-an-equation-by-its-complex-c.htmlhttps://liberty.io/why-is-it-valid-to-multiply-both-sides-of-an-equation-by-its-complex-c.htmlThis is the closest explanation I can find, although I still don't fully understand. We know what’s happening: division is subtracting an angle and shrinking the magnitude. By multiplying top ......Mon, 31 Aug 2026 02:44:13 -0400https://liberty.io/how-often-in-years-do-calendars-repeat-with-the-same-day-date-combinat.htmlhttps://liberty.io/how-often-in-years-do-calendars-repeat-with-the-same-day-date-combinat.htmlE.g. I'm using this formulas for calculating day of week (Julian calendar): \begin{align} a & = \left\lfloor\frac{14 - \text{month}}{12}\right\rfloor\\ y & = \text{year} + 4800 - a \\ m &a......Mon, 31 Aug 2026 02:25:18 -0400https://liberty.io/clarification-of-the-definition-of-tempered-growth.htmlhttps://liberty.io/clarification-of-the-definition-of-tempered-growth.htmlThis is a rather simple question, but I don't have many resources I've been able to consult in this regard (meaning that a Google search hasn't yielded any meaningful results). I just want to clari......Mon, 31 Aug 2026 02:16:58 -0400https://liberty.io/what-is-different-between-g-n-and-gn-n.htmlhttps://liberty.io/what-is-different-between-g-n-and-gn-n.htmlWhat is different between $G/N$ and $GN/N$? ($G$ denotes a group, and N does a normal subgroup of G) If you look at the element of each group above, I think $G/N$ contains element of form $gN$, ......Mon, 31 Aug 2026 02:10:56 -0400https://liberty.io/in-a-math-paper-what-is-a-remark.htmlhttps://liberty.io/in-a-math-paper-what-is-a-remark.htmlI sometimes see paragraphs labeled 'Remark.' However, papers that include remarks also include unlabeled explanatory paragraphs (i.e. all the other writing in the article) that seem to be remarks. ......Mon, 31 Aug 2026 02:06:33 -0400https://liberty.io/the-definition-of-distance-and-how-to-prove-the-ruler-postulate-in-euc.htmlhttps://liberty.io/the-definition-of-distance-and-how-to-prove-the-ruler-postulate-in-euc.htmlI have started to read some books about geometry. At the moment I've just started to read Hilbert's axioms and also some elementary books for highschool. From the basic perspective of an axiomatic ......Mon, 31 Aug 2026 02:04:52 -0400https://liberty.io/proof-for-symmetry-property-of-congruent-segments.htmlhttps://liberty.io/proof-for-symmetry-property-of-congruent-segments.htmlI am starting to learn geometrical proofs, and I have come across the Symmetry property of segment congruence (if $AB$ is congruent to $CD$, then $CD$ is congruent to $AB$). One of the exercises i......Mon, 31 Aug 2026 01:55:40 -0400https://liberty.io/what-is-the-difference-between-o-2-n-and-2-o-n.htmlhttps://liberty.io/what-is-the-difference-between-o-2-n-and-2-o-n.htmlIn the context of complexity, I have seen both $O(2^{n})$ and $2^{O(n)}$ used. Whats the difference between the two?...Mon, 31 Aug 2026 01:44:39 -0400https://liberty.io/what-is-the-norm-of-a-complex-number.htmlhttps://liberty.io/what-is-the-norm-of-a-complex-number.htmlMany articles on the web state that the norm of $z = a +bi$ is \begin{align} \sqrt{z*z^*} &= \sqrt{(a + bi)(a - bi)}\\ &= \sqrt{a^2 - abi + abi - b^2i^2}\\ &= \sqrt{a^2 - b^2\sqrt{-1......Mon, 31 Aug 2026 01:41:47 -0400https://liberty.io/recurrence-tree-method-t-n-3-2n-3-1-why-is-the-solution-in-log-3-2-and.htmlhttps://liberty.io/recurrence-tree-method-t-n-3-2n-3-1-why-is-the-solution-in-log-3-2-and.htmlI feel this is more of a question in the logarithms, but I am confused why this tree reoccurrence $$T(n) = 3T(2n/3)+1$$ solution is derived from $\log_{3/2}$ rather than $\log_{2/3}$? My thinking......Mon, 31 Aug 2026 01:30:50 -0400https://liberty.io/how-does-an-exponent-work-when-it-039-s-less-than-one.htmlhttps://liberty.io/how-does-an-exponent-work-when-it-039-s-less-than-one.htmlI'm rather familiar with exponents, I know that $y^x = y_1 \cdot y_2 \cdot y_3 .... y_x$, but what if the exponent is less than one, how would that work? I put in my computer $25^{1/2}$ anyway, ...Mon, 31 Aug 2026 01:30:08 -0400https://liberty.io/sweepstakes-probability.htmlhttps://liberty.io/sweepstakes-probability.htmlIs there a way to determine the probability of winning a particular sweepstakes with only the following information: the estimated odds of winning are $1$ in $13,000$ contestants must be owners (...Mon, 31 Aug 2026 01:16:56 -0400https://liberty.io/can-a-circle-have-a-negative-radius.htmlhttps://liberty.io/can-a-circle-have-a-negative-radius.htmlI am working on a problem that asks if a given sphere intersects the zx-plane. The equation of the sphere is $(x-2)^2+(y+6)^2+(z-4)^2=5^2$ Can someone please explain to me how the sphere does not ...Mon, 31 Aug 2026 01:12:57 -0400https://liberty.io/infinity-divided-by-infinity-and-dirac-delta.htmlhttps://liberty.io/infinity-divided-by-infinity-and-dirac-delta.htmlA dirac delta produces something that's infinitely long, and it could also be seen as infinitely thin. Why do we define the surface of a dirac delta to be 1. If length$\times$width $= \infty \cdot......Mon, 31 Aug 2026 01:07:22 -0400https://liberty.io/necessary-condition-for-the-mean-value-theorem.htmlhttps://liberty.io/necessary-condition-for-the-mean-value-theorem.htmlGive an example which demonstrates that continuity is a necessary condition for the mean value theorem: I thought in this function: $$g(x) = \begin{cases} x + 1 & x < 1 \\[4pt] x - 1 &......Mon, 31 Aug 2026 00:54:49 -0400https://liberty.io/best-math-books-or-apps-for-adults-to-learn-math-from-the-beginning.htmlhttps://liberty.io/best-math-books-or-apps-for-adults-to-learn-math-from-the-beginning.htmlI lost a possible job because I didn't know how to multiply and subtract negative valued integers. I also don't know how fraction manipulation works. What reference books can I read that can help f......Mon, 31 Aug 2026 00:43:48 -0400https://liberty.io/how-to-prove-that-frac-d-dx-sin-x-cos-x.htmlhttps://liberty.io/how-to-prove-that-frac-d-dx-sin-x-cos-x.htmlI have to prove that $\dfrac{d}{dx}\sin(x)=\cos(x)$. I used the definition of a derivative: $$\dfrac{d}{dx}f(x)=\lim\limits_{h\to 0} \dfrac{f(x+h)-f(x)}{h}$$ $$\dfrac{d}{dx}\sin(x)=\lim\limits_{h\t......Mon, 31 Aug 2026 00:41:05 -0400https://liberty.io/why-does-the-base-b-not-matter-in-the-change-of-base-logarithm-equatio.htmlhttps://liberty.io/why-does-the-base-b-not-matter-in-the-change-of-base-logarithm-equatio.htmlI've recently started learning about logarithms and was just curious as to why the base "b" in the "change the base" formula doesn't matter, as in it doesn't matter what value y......Mon, 31 Aug 2026 00:39:26 -0400https://liberty.io/solving-2-sin-theta-cos-theta-sin-theta-0.htmlhttps://liberty.io/solving-2-sin-theta-cos-theta-sin-theta-0.htmlThe question is to solve the following question in the range $-\pi \le \theta \le \pi$ $$2\sin\theta\cos\theta + \sin\theta = 0$$ I missed the obvious sin factorisation so proceeded as below. I s......Mon, 31 Aug 2026 00:37:27 -0400https://liberty.io/how-to-calculate-q-expansion-coefficients-of-a-modular-form.htmlhttps://liberty.io/how-to-calculate-q-expansion-coefficients-of-a-modular-form.htmlCan you please explain for me how we can calculate the coefficients of $q$-expansion series of a modular form function $f$? I am really confused....Mon, 31 Aug 2026 00:35:24 -0400https://liberty.io/find-number-of-solutions-of-2x-2-5x-3-x-1-0.htmlhttps://liberty.io/find-number-of-solutions-of-2x-2-5x-3-x-1-0.htmlProblem: Find number of solutions of $|2x^2-5x+3|+x-1=0$ Solution: Case 1: When $2x^2-5x+3 \geq 0$ Then we get, $2x^2-5x+3+x-1=0$ x=1,1 Case 2: When $2x^2-5x+3 < 0$ Then we get, $-2x^2+5x-3+x-1=......Mon, 31 Aug 2026 00:33:10 -0400https://liberty.io/difference-between-mathbb-r-linear-and-mathbb-c-linear-maps.htmlhttps://liberty.io/difference-between-mathbb-r-linear-and-mathbb-c-linear-maps.htmlSuppose we have a $f\colon \mathbb{C} \to \mathbb{C} $. I am confused when people say $f$ is $\mathbb{R}$-linear map or $\mathbb{C}$-linear map. What is the difference between these two ?...Mon, 31 Aug 2026 00:21:18 -0400https://liberty.io/what-039-s-the-difference-between-a-logic-an-internal-logic-language-o.htmlhttps://liberty.io/what-039-s-the-difference-between-a-logic-an-internal-logic-language-o.htmlmaybe this question doesn't make sense at all. I don't know exactly the meaning of all these concepts, except the internal language of a topos (and searching on the literature is not helping at all......Mon, 31 Aug 2026 00:15:30 -0400https://liberty.io/euler-angles-and-gimbal-lock.htmlhttps://liberty.io/euler-angles-and-gimbal-lock.htmlCan someone show mathematically how gimbal lock happens when doing matrix rotation with Euler angles for yaw, pitch, roll? I'm having a hard time understanding what is going on even after reading ...Mon, 31 Aug 2026 00:12:46 -0400https://liberty.io/when-is-the-intersection-of-a-cone-and-a-cylinder-is-planar.htmlhttps://liberty.io/when-is-the-intersection-of-a-cone-and-a-cylinder-is-planar.htmlI'm trying to understand better the link between the algebraic and geometric interpretation of the ellipse, and I've wondered about the intersection of a right circular cylinder and a right circula......Mon, 31 Aug 2026 00:04:31 -0400https://liberty.io/trace-of-the-cissoid-of-diocles.htmlhttps://liberty.io/trace-of-the-cissoid-of-diocles.htmlI want to show that you can parameterized the cissoid of Diocles using $$ \alpha(t) = \left(\frac{2at^2}{1+t^2}, \frac{2at^3}{1+t^2}\right),\quad\text{where }t=\tan\theta. $$ Here is an image of the ...Mon, 31 Aug 2026 00:04:07 -0400https://liberty.io/algorithms-for-symbolic-manipulation.htmlhttps://liberty.io/algorithms-for-symbolic-manipulation.htmlIf you take a look at WolframAlpha, or other computer algebraic system, you will find that it is able to do symbolic manipulation like real humans. For example, if you type in an integral, it can ......Sun, 30 Aug 2026 23:59:46 -0400https://liberty.io/what-does-the-integral-of-position-with-respect-to-time-mean.htmlhttps://liberty.io/what-does-the-integral-of-position-with-respect-to-time-mean.htmlThe integral of acceleration with respect to time is velocity. The integral of velocity with respect to time is position. What is the integral of position with respect to time, and what does it me......Sun, 30 Aug 2026 23:58:45 -0400https://liberty.io/equivalence-of-repeating-decimal.htmlhttps://liberty.io/equivalence-of-repeating-decimal.htmlGRE exam question asks what is greater $.\overline{717}$ or $.\overline{71}$ I believe both are equal, but GRE says that $.\overline{717}$ is greater. But why? If they repeat for infinity, is......Sun, 30 Aug 2026 23:47:52 -0400https://liberty.io/existance-of-multiple-of-n-with-only-0-and-1-as-it-039-s-digits-duplic.htmlhttps://liberty.io/existance-of-multiple-of-n-with-only-0-and-1-as-it-039-s-digits-duplic.htmlPossible Duplicate: Proof that a natural number multiplied by some integer results in a number with only one and zero as digits I read this somewhere recently: For any natural number $n$, there ex......Sun, 30 Aug 2026 23:46:21 -0400https://liberty.io/find-the-tangential-and-normal-components-of-the-acceleration-vector-f.htmlhttps://liberty.io/find-the-tangential-and-normal-components-of-the-acceleration-vector-f.htmlFind the tangential and normal components of the acceleration vector for the curve $$ \vec r (t) = 〈 2t,t^2 ,\ln t 〉$$ I found this problem in a notebook and I did a part, but I was stuck and I ......Sun, 30 Aug 2026 23:45:17 -0400https://liberty.io/find-x-intercept-of-a-logarithmic-function.htmlhttps://liberty.io/find-x-intercept-of-a-logarithmic-function.htmlFind the x-intercept of: $3-4\log _{10}\left(2x\right) $ My process: $4\log _{10}\left(2x\right)=3 $ $\log _{10}\left(2x\right)\:=\:\frac{3}{4}$ How would I find the x-intercept from here? $......Sun, 30 Aug 2026 23:38:00 -0400https://liberty.io/largest-subset-with-no-pair-summing-to-power-of-two.htmlhttps://liberty.io/largest-subset-with-no-pair-summing-to-power-of-two.htmlFor positive integer $n$, define the set $A_n=\{0,1,\ldots,n\}$. What is the size of the largest subset of $A_n$ such that the sum of any two (not necessarily distinct) elements in it is not a powe......Sun, 30 Aug 2026 23:35:33 -0400https://liberty.io/understanding-the-heine-cantor-theorem.htmlhttps://liberty.io/understanding-the-heine-cantor-theorem.htmlThe "Heine–Cantor theorem" states: If $f : M → N$ is a continuous function between two metric spaces, and $M$ is compact, then $f$ is uniformly continuous. I do not doubt its validity, of course, ......Sun, 30 Aug 2026 23:33:15 -0400https://liberty.io/boolean-algebra-finding-the-maxterm-expansion-of-f-a-b-c-d-a-b-d-a-039.htmlhttps://liberty.io/boolean-algebra-finding-the-maxterm-expansion-of-f-a-b-c-d-a-b-d-a-039.htmlI wanted to show the maxterm expansion of the following Boolean Function below $f(A,B,C,D) = (A + B + D)(A' + C)(C + D)$ What I know: From what I know, the domain is f(A, B, C, and D). No terms ar......Sun, 30 Aug 2026 23:30:07 -0400https://liberty.io/best-program-for-multiplying-many-multivariable-polynomials.htmlhttps://liberty.io/best-program-for-multiplying-many-multivariable-polynomials.htmlI want to make a table with two columns: The first column will consist of many(possibly hundreds) of polynomials in two variables. The second column will be a function applied to all of the polyn......Sun, 30 Aug 2026 23:10:47 -0400https://liberty.io/how-to-find-inverse-fourier-transform.htmlhttps://liberty.io/how-to-find-inverse-fourier-transform.htmlI have the function $$ \delta(f-2) $$ How can we inverse Fourier transform it? It's easy if $f$ is replaced with $w$. But based on my knowledge, $w = 2\pi f$. The correct answer is $$ e^{4\pi ......Sun, 30 Aug 2026 23:09:30 -0400https://liberty.io/derivative-on-endpoints.htmlhttps://liberty.io/derivative-on-endpoints.htmlwhat's the derivative of $f(x)= x^{2}$ ($x\geq 0$) when x=0? from my understand, it doesn't exist because even $\lim_{h \to 0^{+}}\frac{f(x+h)-f(x)}{h}$ is 0, but $\lim_{h \to 0^{-}}\frac{f(x+h)-f......Sun, 30 Aug 2026 23:06:45 -0400https://liberty.io/user-raymond-hettinger-mathematics-stack-exchange.htmlhttps://liberty.io/user-raymond-hettinger-mathematics-stack-exchange.htmlQ&A for people studying math at any level and professionals in related fields...Sun, 30 Aug 2026 23:04:43 -0400https://liberty.io/comparison-test-of-sum1-ln-n-n.htmlhttps://liberty.io/comparison-test-of-sum1-ln-n-n.htmlHere's what I came up with: For this problem, I'm required to use a comparison test to determine if $\Sigma1/ln(n)^n$ converges or diverges. By intuition, I am thinking that $\Sigma1/ln(n)^n$ conv......Sun, 30 Aug 2026 22:56:43 -0400https://liberty.io/is-x-not-actually-equal-to-e-ln-x.htmlhttps://liberty.io/is-x-not-actually-equal-to-e-ln-x.htmlYeah, this is a silly question, but I can't seem to convince myself that the graph $f(x)=x$ is really equal to the graph of $g(x)=e^{\ln(x)}$. Specifically, doesn't this fail on negative values of ......Sun, 30 Aug 2026 22:56:07 -0400https://liberty.io/why-is-frac-987654321-123456789-8-0000000729.htmlhttps://liberty.io/why-is-frac-987654321-123456789-8-0000000729.htmlMany years ago, I noticed that $987654321/123456789 = 8.0000000729\ldots$. I sent it in to Martin Gardner at Scientific American and he published it in his column!!! My life has gone downhill si......Sun, 30 Aug 2026 22:52:21 -0400https://liberty.io/units-in-exponent-e-g-solve-2-3-years.htmlhttps://liberty.io/units-in-exponent-e-g-solve-2-3-years.htmlWhat happens to units in an exponent? My math textbook just introduced the exponential equation: $$A_t = Pe^{rt}$$ I've always made it a point in solving math problems to include the units in ev......Sun, 30 Aug 2026 22:51:13 -0400https://liberty.io/formula-for-a-stadium-shape-2d-capsule.htmlhttps://liberty.io/formula-for-a-stadium-shape-2d-capsule.htmlThis shape (Known as a stadium, it's basically a 2D capsule.) I need a formula to draw this shape on a graph, preferably in that orientation (on its side). What I've tried: I've tried adapting......Sun, 30 Aug 2026 22:42:45 -0400https://liberty.io/checking-if-matrix-columns-are-linearly-independent.htmlhttps://liberty.io/checking-if-matrix-columns-are-linearly-independent.htmlaccording to the definition of linear dependency vectors $v_1,...,v_n$ are linearly independent $iff$ $c_1v_1$+$c_2v_2$+$...$+$c_nv_n≠0$. One can also do the gaussian elemination to get which colu......Sun, 30 Aug 2026 22:37:17 -0400https://liberty.io/area-in-an-equilateral-triangle-that-is-closer-to-the-centroid-than-to.htmlhttps://liberty.io/area-in-an-equilateral-triangle-that-is-closer-to-the-centroid-than-to.htmlWhat percent of the area of an equilateral triangle is closer to the centroid of the triangle than to any edge of the triangle?...Sun, 30 Aug 2026 22:36:54 -0400https://liberty.io/why-is-sqrt-12-2-sqrt-3.htmlhttps://liberty.io/why-is-sqrt-12-2-sqrt-3.htmlWhy $\sqrt {12} = 2 \sqrt 3$? It is obvious? If we considered the function $f(s) = s^2 $ it is injective on positive numbers so we obtain the conclusion. But in the same time it is an equality bet......Sun, 30 Aug 2026 22:32:59 -0400https://liberty.io/solution-of-a-given-legendre-equation.htmlhttps://liberty.io/solution-of-a-given-legendre-equation.htmlI was trying to solve following question : $$ (1-x^2)y''-2xy'+n(n+1)y=0 $$ has the $n^{th}$ degree polynomial solution $y_n(x)$ such that $y_n(1) =3$. We are given $\int_{-1}^{1} [y_n^2(x) + y_{......Sun, 30 Aug 2026 22:29:52 -0400https://liberty.io/how-to-be-good-at-proving-duplicate.htmlhttps://liberty.io/how-to-be-good-at-proving-duplicate.htmlI'm starting my Discrete Math class, and I was taught proving techniques such as proof by contradiction, contrapositive proof, proof by construction, direct proof, equivalence proof etc. I know ho......Sun, 30 Aug 2026 22:27:22 -0400https://liberty.io/partial-derivative-with-constant.htmlhttps://liberty.io/partial-derivative-with-constant.htmlGiven $$c = 0.03 + 0.08a$$ What is $\dfrac{\partial c}{\partial a}$? I'm guessing $\dfrac{\partial c}{\partial a} = 0.03 + 0.08 = 0.11$? The constant is confusing me. Thanks....Sun, 30 Aug 2026 22:15:31 -0400https://liberty.io/why-is-the-definition-of-the-absolute-value-x-1-the-way-it-is.htmlhttps://liberty.io/why-is-the-definition-of-the-absolute-value-x-1-the-way-it-is.htmlIn my notebook it is given that for the above function, we would have: $f(x) = {-(x+1), x<-1; (x+1), x\geq-1}$ What I don't get is why did we take $-1$ instead of $0$ as is the case for the fu......Sun, 30 Aug 2026 21:42:19 -0400https://liberty.io/equivalence-relation-showing-that-an-operation-is-well-defined.htmlhttps://liberty.io/equivalence-relation-showing-that-an-operation-is-well-defined.htmlLet $r \in \mathbb{N}$. Observe the equivalence relation $\sim \subseteq \mathbb{Z} \times \mathbb{Z}$ defined as $x \sim y :\Leftrightarrow (\exists k \in \mathbb{Z})(y=x+kr)$. Show that an operat......Sun, 30 Aug 2026 21:41:59 -0400https://liberty.io/the-derivative-of-the-electric-field-for-a-uniformly-charged-rod.htmlhttps://liberty.io/the-derivative-of-the-electric-field-for-a-uniformly-charged-rod.htmlThe formula for the electric field at a point due to a charge $Q$ (just considering the magnitude) at some distance $x$ away from the point is $E=\dfrac{k_eQ}{x^2}$ where $k_e$ is a constant equal to ...Sun, 30 Aug 2026 21:28:12 -0400https://liberty.io/showing-ha-h-if-and-only-if-a-belongs-to-h.htmlhttps://liberty.io/showing-ha-h-if-and-only-if-a-belongs-to-h.htmlLet be $H$ a subgroup. Show $Ha=H$ if and only if $a$ belongs to $H$. Here what I have understand of the proof: $a$ must belong to $Ha$ because $ea=a$ where $e$ is the identity in $H$. So the onl......Sun, 30 Aug 2026 21:24:55 -0400https://liberty.io/summing-the-digits-of-pi.htmlhttps://liberty.io/summing-the-digits-of-pi.htmlI find $\pi$ and other transcendental numbers quite fascinating. Lately I've been playing around with sequences and $\pi$ and found something relatively intriguing. Consider $\pi$ written in base......Sun, 30 Aug 2026 21:17:29 -0400https://liberty.io/what-is-2-32-1-in-decimal-notation.htmlhttps://liberty.io/what-is-2-32-1-in-decimal-notation.htmlI came across decimal notation, and surprisingly I have never heard of it. I have heard of scientific notation and just thought this was decimal notation. However, when I want to find a definition ......Sun, 30 Aug 2026 21:16:00 -0400https://liberty.io/definition-of-a-sheaf-making-sense-of.htmlhttps://liberty.io/definition-of-a-sheaf-making-sense-of.htmlI have started to read an english translation of Serre's FAC. Immediately a sheaf is defined. The more categorical definition given on wikipedia actually makes more sense to me, but I would like to ...Sun, 30 Aug 2026 21:15:52 -0400https://liberty.io/fourier-transform-of-a-radial-function.htmlhttps://liberty.io/fourier-transform-of-a-radial-function.htmlConsider a function $f \in L^2(\mathbb{R}^n)$ such that $f$ is radial. My question is, is the Fourier transform $\hat{f}(\xi)$ automatically radial (I can see it is even in each variable $x_i$), or......Sun, 30 Aug 2026 21:14:49 -0400https://liberty.io/proving-that-the-dot-product-is-distributive.htmlhttps://liberty.io/proving-that-the-dot-product-is-distributive.htmlI know that one can prove that the dot product, as defined "algebraically", is distributive. However, to show the algebraic formula for the dot product, one needs to use the distributive property i......Sun, 30 Aug 2026 21:03:51 -0400https://liberty.io/relation-between-curvature-and-radius-of-curvature-duplicate.htmlhttps://liberty.io/relation-between-curvature-and-radius-of-curvature-duplicate.htmlWhy is the radius of curvature the reciprocal of the curvature? How to see this intuitively as well show it rigorously?...Sun, 30 Aug 2026 21:01:52 -0400https://liberty.io/in-a-triangle-abc-the-real-part-of-the-equation-acosb-bcosa-i-asinb-bs.htmlhttps://liberty.io/in-a-triangle-abc-the-real-part-of-the-equation-acosb-bcosa-i-asinb-bs.htmlIf A B C are the angles of a triangle, and a, b, c are the corresponding sides, then the real part of $ (acosB+bcosA + i(asinB-bsinA))^n $is? My book says that the real part of this is $(acosB+bc......Sun, 30 Aug 2026 21:00:20 -0400https://liberty.io/open-loop-transfer-function.htmlhttps://liberty.io/open-loop-transfer-function.htmla block diagram system is given as per the attached image. what is the open loop transfer function? i say it is found by opening the loop and multiplying all of the terms. So that: U = KFYX. Then ...Sun, 30 Aug 2026 20:51:53 -0400https://liberty.io/what-is-the-logic-rationale-behind-the-vector-cross-product.htmlhttps://liberty.io/what-is-the-logic-rationale-behind-the-vector-cross-product.htmlI don't think I ever understood the rationale behind this. I get that the dot product $\mathbf{a} \cdot \mathbf{b} =\lVert \mathbf{a}\rVert \cdot\lVert \mathbf{b}\rVert \cos\theta$ is derived fro......Sun, 30 Aug 2026 20:48:44 -0400https://liberty.io/does-every-matrix-have-a-ref.htmlhttps://liberty.io/does-every-matrix-have-a-ref.htmlI'm wondering if anyone here has experience using Wolframalpha? I can't seem to get a row echelon form (REF) or echelon form, it always does reduced row echelon form (R-REF). Anybody know how to ge......Sun, 30 Aug 2026 20:43:41 -0400https://liberty.io/calculate-arbitrary-points-from-a-plane-equation.htmlhttps://liberty.io/calculate-arbitrary-points-from-a-plane-equation.htmlI understand how one can calculate a plane equation (ax+by+cz=d) from three points but how can you go in reverse? How can you calculate arbitrary points from a plane equation?...Sun, 30 Aug 2026 20:42:19 -0400https://liberty.io/understanding-the-derivative-as-a-linear-transformation.htmlhttps://liberty.io/understanding-the-derivative-as-a-linear-transformation.htmlIt's been a while now I am studying multivariable calculus and the concept of differentiation in space (or higher dimension). I saw relative posts but one question remains. I can't understand the c......Sun, 30 Aug 2026 20:18:58 -0400https://liberty.io/evaluate-the-integral-of-sec-2x-1-dx.htmlhttps://liberty.io/evaluate-the-integral-of-sec-2x-1-dx.htmlI got $\ln|\sec(2x +1) + \tan(2x+1)| + \text C$ as an answer. I saw that the integral of $\sec x$ is $\ln|\sec x + \tan x| + \text C$. But I feel I may have left something out because that was too ......Sun, 30 Aug 2026 20:17:50 -0400https://liberty.io/why-is-the-integral-of-dx-equal-to-x.htmlhttps://liberty.io/why-is-the-integral-of-dx-equal-to-x.htmlPlease give a thorough explanation, not just "$dx$ is the derivative of $x$, so the antiderivative of $dx$ is $x$, duh"....Sun, 30 Aug 2026 20:05:13 -0400https://liberty.io/using-exponential-shift-to-find-general-solution-of-an-ode.htmlhttps://liberty.io/using-exponential-shift-to-find-general-solution-of-an-ode.htmlWe have these 2 theorems/definitions. *For each natural number n, $(D-m)^n e^{mx} y = e^{mx}D^ny$ *If f(D) is a polynomial in D with constant coefficients then $e^{mx}f(D)y=f(D-m)e^{mx}y$$\,\,\,\......Sun, 30 Aug 2026 20:02:32 -0400https://liberty.io/is-there-a-name-for-sec-x-039-s-relationship-with-tan-x.htmlhttps://liberty.io/is-there-a-name-for-sec-x-039-s-relationship-with-tan-x.htmlIn a couple of trig identities, esp to do with integrals and derivatives, you see a relationship between tan(x) and sec(x). Similarly between csc(x) and cot(x). $ \frac{d}{dx}\tan(x) = \sec^2(x) $......Sun, 30 Aug 2026 20:00:36 -0400https://liberty.io/on-the-notion-of-set-equality.htmlhttps://liberty.io/on-the-notion-of-set-equality.htmlI should start by providing some context. I want to understand the foundations of mathematics and in doing so have been looking at different texts involving ZF set theory and category theory. I ...Sun, 30 Aug 2026 19:57:28 -0400https://liberty.io/isomorphism-between-h-h-cap-n-and-hn-n.htmlhttps://liberty.io/isomorphism-between-h-h-cap-n-and-hn-n.htmlSuppose we have a group G under operation +, and let H be a subgroup and N a normal subgroup. I want to prove that $H/H\cap N$ is isomorphic to $HN/N$. Where, if I am not mistaken: $H/H\cap N = \......Sun, 30 Aug 2026 19:56:23 -0400https://liberty.io/if-x-is-domain-and-y-is-range-what-is-z.htmlhttps://liberty.io/if-x-is-domain-and-y-is-range-what-is-z.htmlThis is a really small stupid question, but I'm currently self-teaching myself Multivariable Calculus, and I'm just curious if there is a term for this. I did a small google search and didn't find ...Sun, 30 Aug 2026 19:49:06 -0400https://liberty.io/prove-every-odd-number-is-of-the-form-2k-5-is-every-even-number-of-the.htmlhttps://liberty.io/prove-every-odd-number-is-of-the-form-2k-5-is-every-even-number-of-the.htmlI know even numbers are of the form $n=2k, k \in \mathbb Z$, and that odd numbers are of the form $n=2k+1, k \in \mathbb Z$. I'm being asked to prove that all odd numbers are of the form $n=2k+......Sun, 30 Aug 2026 19:45:06 -0400https://liberty.io/how-to-derive-the-closed-form-solution-of-geometric-series.htmlhttps://liberty.io/how-to-derive-the-closed-form-solution-of-geometric-series.htmlI have the following equation: $$g(n) = 1 + c^2 + c^3 + ... + c^n$$ The closed form solution of this series is $$g(n) = \frac{c^{n+1} -1}{c-1}$$ However, I am having a difficult time seeing the ...Sun, 30 Aug 2026 19:38:55 -0400https://liberty.io/sum-of-series-question-s-n-1-1-4-1-9-1-16-1-25-1-n-2-2.htmlhttps://liberty.io/sum-of-series-question-s-n-1-1-4-1-9-1-16-1-25-1-n-2-2.htmlProve that for any nonzero natural $n$ it is true that $$S_n = 1 + 1/4 + 1/9 + 1/16 + 1/25 + … + 1/n^2 < 2.$$ I'm sort of at a loss here. I'm not sure if there exists some formula or method to......Sun, 30 Aug 2026 19:24:50 -0400https://liberty.io/what-does-r-0-mean.htmlhttps://liberty.io/what-does-r-0-mean.htmlI'm reading this paper: http://www.aaai.org/Papers/KDD/1996/KDD96-027.pdf, and the authors used the symbol $R_{0}^{+}$ in the definition of Exact Exception Problem, such as $D: P(I) \rightarrow R_{......Sun, 30 Aug 2026 19:12:20 -0400https://liberty.io/why-do-we-essentially-need-complete-measure-space.htmlhttps://liberty.io/why-do-we-essentially-need-complete-measure-space.htmlWhile reading the motivation of complete measure space on Wikipedia, what I concluded was, completeness is not really necessary when we define on one measure space and it is necessary when we wan......Sun, 30 Aug 2026 19:06:57 -0400https://liberty.io/geodesic-sphere-using-only-regular-pentagons-and-hexagons.htmlhttps://liberty.io/geodesic-sphere-using-only-regular-pentagons-and-hexagons.htmlI know geodesic approximation to a construct a spherical dome shape needs 12 pentagons and these pentagons are regular pentagons. However when I look closely hexagons are slightly different in th......Sun, 30 Aug 2026 18:53:35 -0400https://liberty.io/continuously-changing-dimensions-of-a-rectangle.htmlhttps://liberty.io/continuously-changing-dimensions-of-a-rectangle.htmlThe dimension of a rectangle are continousl changing. The width increases at a rate of 3 in/s while the length decreases at the rate of 2 in/s. At one instant the rectangle is a 20-in square. How ......Sun, 30 Aug 2026 18:40:12 -0400https://liberty.io/show-that-b-x-n-1-p-1-b-n-x-1-n-p.htmlhttps://liberty.io/show-that-b-x-n-1-p-1-b-n-x-1-n-p.htmlIn my statistics book $B$ is defined as: $$B(x; \ n, \ p) = \sum_{y = 0}^{x}b(y; \ n, \ p) = \sum_{y = 0}^{x}{n \choose y}p^{y}(1 - p)^{n - y}$$ I want to show that: $$B(x; \ n, \ 1 - p) = 1 - B(n......Sun, 30 Aug 2026 18:33:28 -0400https://liberty.io/how-to-find-explicit-formula-for-a-sequence-seems-to-be-geometric-but-.htmlhttps://liberty.io/how-to-find-explicit-formula-for-a-sequence-seems-to-be-geometric-but-.htmlI recently came across this interesting problem, where the progression was neither arithmetic nor geometric. The problem asks for an explicit formula for the following recursive formula: $a_{1} = ......Sun, 30 Aug 2026 18:28:23 -0400https://liberty.io/how-does-full-row-rank-imply-column-space-is-r-m-for-a-m-times-n-matri.htmlhttps://liberty.io/how-does-full-row-rank-imply-column-space-is-r-m-for-a-m-times-n-matri.htmlFrom Gilbert Strang's textbook Introduction to Linear Algebra (p.159) Every matrix with full row rank has these properties $Ax=b$ has a solution for every right side $b$. The column space is the......Sun, 30 Aug 2026 18:25:42 -0400https://liberty.io/expected-number-of-steps-between-states-in-a-markov-chain.htmlhttps://liberty.io/expected-number-of-steps-between-states-in-a-markov-chain.htmlSuppose I am given a state space $S=\{0,1,2,3\}$ with transition probability matrix $\mathbf{P}= \begin{bmatrix} \frac{2}{3} & \frac{1}{3} & 0 & 0 \\[0.3em] \frac{2}{......Sun, 30 Aug 2026 18:22:19 -0400https://liberty.io/confusion-regarding-fermat-039-s-theorem-and-closed-interval-method.htmlhttps://liberty.io/confusion-regarding-fermat-039-s-theorem-and-closed-interval-method.htmlIn James Stewart's Calculus (8th edition) Fermat's Theorem is stated as If $f$ has a local maximum or minimum at $c$, and if $f'(c)$ exists, then $f'(c) = 0$. The author cautions that the convers......Sun, 30 Aug 2026 18:10:59 -0400https://liberty.io/why-divide-diameter-by-square-root-of-2-to-get-a-diameter-of-a-circle-.htmlhttps://liberty.io/why-divide-diameter-by-square-root-of-2-to-get-a-diameter-of-a-circle-.htmlI would be very grateful if you can help me with this problem. I am trying to explain in the simplest terms possible the sequence of f-stops in photography. The common f-stop rounded sequence is: f......Sun, 30 Aug 2026 18:06:05 -0400https://liberty.io/definition-for-covariant-derivative.htmlhttps://liberty.io/definition-for-covariant-derivative.htmlWhat is simple definition of the covariant derivative that looks like the definition of the derivative of a function in calculus? definition of the derivative of a function in calculus is: $$\fra......Sun, 30 Aug 2026 18:01:24 -0400https://liberty.io/has-there-ever-been-an-application-of-dividing-by-0.htmlhttps://liberty.io/has-there-ever-been-an-application-of-dividing-by-0.htmlRegarding the expression $a/0$, according to Wikipedia: In ordinary arithmetic, the expression has no meaning, as there is no number which, multiplied by $0$, gives $a$ (assuming $a\not= 0$), a......Sun, 30 Aug 2026 18:01:13 -0400https://liberty.io/proving-this-limit-lim-limits-n-to-infty-sqrt-n-n-1.htmlhttps://liberty.io/proving-this-limit-lim-limits-n-to-infty-sqrt-n-n-1.htmlThis is supposed to be proven: $$\lim\limits_{n\to\infty}\sqrt [n]n=1$$ The sequence is monotone decreasing and has a lower bound of $1$. So $\epsilon=0$ With the Archimedean Property we get: $......Sun, 30 Aug 2026 17:45:50 -0400https://liberty.io/finding-maclaurin-series-of-a-function.htmlhttps://liberty.io/finding-maclaurin-series-of-a-function.htmlThe prompt is to find the Maclaurin series of the function $$f(x) = xe^{x^2-1}$$ and evaluate the 100th derivative. At what value of x does this series converge? We know that $$e^x = \sum \frac{x......Sun, 30 Aug 2026 17:28:40 -0400https://liberty.io/how-do-derivatives-work-in-the-xyz-3d-plane.htmlhttps://liberty.io/how-do-derivatives-work-in-the-xyz-3d-plane.htmlIn just the XY plane, a derivative is the slope of the tangent line. How does it work when you have a 3D shape? I recall 3D math, and the Z-axis. Was this Calc 3? Let's say you have a "mou......Sun, 30 Aug 2026 17:22:04 -0400