https://liberty.ioCurated star highlights with fresh energy.en-usThu, 03 Sep 2026 18:23:30 -0400https://liberty.io/how-to-calculate-the-coordinates-of-a-triangle-039-s-orthocentre.htmlhttps://liberty.io/how-to-calculate-the-coordinates-of-a-triangle-039-s-orthocentre.htmlHow does one calculate the coordinates of a triangle's orthocentre? I was surfing through the net and got this formula: $$x-\rm{coordinate}= \frac{x_1\tan A+x_2\tan B+x_3\tan C}{\tan A+\tan B+\tan......Thu, 03 Sep 2026 22:22:41 -0400https://liberty.io/find-intersection-of-two-3d-lines.htmlhttps://liberty.io/find-intersection-of-two-3d-lines.htmlI have two lines $(5,5,4) (10,10,6)$ and $(5,5,5) (10,10,3)$ with same $x$, $y$ and difference in $z$ values. Please some body tell me how can I find the intersection of these lines. EDIT: By usi......Thu, 03 Sep 2026 22:21:30 -0400https://liberty.io/determine-if-sum-n-1-infty-frac-1-nn-2-n-n-3-1-converges-or-diverges.htmlhttps://liberty.io/determine-if-sum-n-1-infty-frac-1-nn-2-n-n-3-1-converges-or-diverges.htmlAnother series I found I'm struggling with. Determine if the following series converges or diverges.$$\sum_{n=1}^{\infty}\frac{(-1)^nn^2+n}{n^3+1}$$ Ratio test and n-th root test are both ...Thu, 03 Sep 2026 22:15:55 -0400https://liberty.io/what-does-039-card-039-stands-for.htmlhttps://liberty.io/what-does-039-card-039-stands-for.htmlI read some book where 'card' symbol is used to signify number of elements in the set. But where is it from? What the full word could be for 'card'?...Thu, 03 Sep 2026 22:07:40 -0400https://liberty.io/geometry-how-to-determine-if-two-lines-are-parallel-in-3d-based-on-coo.htmlhttps://liberty.io/geometry-how-to-determine-if-two-lines-are-parallel-in-3d-based-on-coo.htmlI am a Belgian engineer working on software in C# to provide smart bending solutions to a manufacturer of press brakes. In this context I am searching for the best way to determine if two lines are ...Thu, 03 Sep 2026 22:02:23 -0400https://liberty.io/show-that-measure-is-trivial.htmlhttps://liberty.io/show-that-measure-is-trivial.htmlLet $\mu(dx,dy)$ be a Borel measure with compact support on $\left\{(x,y) \mid x>0, y>0 \right\}$ and let $\mu(dx,dy)$ satisfy $$ \mu(dx,dy) = \lambda^{\alpha} \mu \left( d \frac{x}{\lambd......Thu, 03 Sep 2026 21:44:49 -0400https://liberty.io/how-do-you-distinguish-the-difference-between-a-negative-sign-and-a-mi.htmlhttps://liberty.io/how-do-you-distinguish-the-difference-between-a-negative-sign-and-a-mi.htmlBoth the negative sign symbol and subtraction's minus symbol look the same, so how does anyone tell them apart?...Thu, 03 Sep 2026 21:44:02 -0400https://liberty.io/proving-a-tautology-by-applying-a-chain-of-logical-identities.htmlhttps://liberty.io/proving-a-tautology-by-applying-a-chain-of-logical-identities.htmlI need help showing that $[ (p \land q) \Rightarrow (p \Rightarrow q) ]$ is a tautology by applying a chain of logical identities. The question also asks to identify each identity I use. I have no ......Thu, 03 Sep 2026 21:42:39 -0400https://liberty.io/can-someone-explain-geometric-multiplicity.htmlhttps://liberty.io/can-someone-explain-geometric-multiplicity.htmlI'm reading my textbook and I'm really confused about geometric multiplicity. I've read the definition and they have given an example but I'm still lost. I've tried looking it up on other websites.......Thu, 03 Sep 2026 21:41:37 -0400https://liberty.io/why-is-sqrt-2-sqrt-3-neq-sqrt-6.htmlhttps://liberty.io/why-is-sqrt-2-sqrt-3-neq-sqrt-6.htmlWhy is $\sqrt{-2} \cdot \sqrt{-3} \neq \sqrt{6}$? Are there other examples where regular arithmetic goes wrong for complex numbers?...Thu, 03 Sep 2026 21:40:50 -0400https://liberty.io/show-that-tan-pi-over-8-sqrt-2-1.htmlhttps://liberty.io/show-that-tan-pi-over-8-sqrt-2-1.htmlShow that $\tan {\pi \over 8} = \sqrt 2 - 1$, using the identity $\tan 2\theta = {{2\tan \theta } \over {1 - {{\tan }^2}\theta }}$ Using $\tan 2\theta = {{2\tan \theta } \over {1 - {{\tan }^2}\...Thu, 03 Sep 2026 21:29:32 -0400https://liberty.io/in-general-what-does-the-location-of-the-rightmost-1-in-a-binary-numbe.htmlhttps://liberty.io/in-general-what-does-the-location-of-the-rightmost-1-in-a-binary-numbe.htmlIn general, what does the location of the rightmost “1” in a binary number tell you? Is it different for positive and for negative numbers?...Thu, 03 Sep 2026 21:27:43 -0400https://liberty.io/linear-bound-on-angles-in-an-euclidean-triangle.htmlhttps://liberty.io/linear-bound-on-angles-in-an-euclidean-triangle.htmlI am trying to understand a proof in the book of Burago "A Course in metric geometry" (Lemma 10.8.13 page 383). I have difficulties with a certain inequality for the angle of euclidean triangles: ...Thu, 03 Sep 2026 21:27:23 -0400https://liberty.io/about-algebraic-integer-and-algebraic-number.htmlhttps://liberty.io/about-algebraic-integer-and-algebraic-number.htmlLet $u$ a algebraic number. Prove that exists a natural number $n\in \mathbb{Z}$ such that $nu$ is a algebraic integer If $u$ is algebraic integer and $n\in \mathbb{Z}$ then $u+n$ and $nu$ are ...Thu, 03 Sep 2026 21:25:46 -0400https://liberty.io/pre-multiplying-and-post-multiplying-matrices-give-the-same-diagonal-e.htmlhttps://liberty.io/pre-multiplying-and-post-multiplying-matrices-give-the-same-diagonal-e.htmlIf $$X = \left[ \begin{array}{ccc} 3 & 4 & 1\\ 4 & 1 & 3\\ 1 & 3 & 4\end{array} \right]$$ find the possible matrix $Y$ such that: $$XY - YX = I$$ The method my ...Thu, 03 Sep 2026 21:13:44 -0400https://liberty.io/question-about-heart-shaped-graph.htmlhttps://liberty.io/question-about-heart-shaped-graph.htmlFirstly, I am sorry for my poor english since it is only my second language. During my spare time, I tried to form a hearshaping function. Here's what I did. If I drew a ellipse that looks like......Thu, 03 Sep 2026 21:09:48 -0400https://liberty.io/lagrangian-mechanics-derivatives.htmlhttps://liberty.io/lagrangian-mechanics-derivatives.htmlI don't really know whether to put this in Physics forums since it is relating to Mechanics, or Math since the question is actually about the math being done. Don't criticize me over it. So for the ...Thu, 03 Sep 2026 21:09:44 -0400https://liberty.io/how-to-write-reciprocal-squared-function-shifted-right-by-3-and-down-b.htmlhttps://liberty.io/how-to-write-reciprocal-squared-function-shifted-right-by-3-and-down-b.htmlGiven a reciprocal squared function that is shifted right by $3$ and down by $4$, write this as a rational function. I am uncertain how to denote this. My attempt: $$\frac{1}{x^2-3}-4$$ But I need......Thu, 03 Sep 2026 21:05:57 -0400https://liberty.io/tail-sum-for-expectation.htmlhttps://liberty.io/tail-sum-for-expectation.htmlIn Pitman's Probability, the tail sum formula for expectation is introduced for a nonnegative (0,1,...) discrete random variable $X$: $$E(X) = \sum_{i=0}^\infty P(X > i).$$ I wonder if there i......Thu, 03 Sep 2026 20:57:59 -0400https://liberty.io/particular-solution-of-second-order-differential-equation.htmlhttps://liberty.io/particular-solution-of-second-order-differential-equation.htmlGiven the ode: $$ y''-2y'+y=e^t, $$ how can I find the form of the particular solution? At first, I tried the form $y=Ae^t$ but $$ \begin{split} &\frac{d^2y}{dt^2}Ae^t=\frac{dy}{dt}Ae^t=......Thu, 03 Sep 2026 20:52:33 -0400https://liberty.io/quadratic-equations-and-factorized-form.htmlhttps://liberty.io/quadratic-equations-and-factorized-form.htmlI have trouble understanding why: Given the equation $x^2+4x-21=0$, the solution given is $(x+7)(x-3)=0$ in its factorization form. Using quadratic equations, $$ x = \frac{-b\pm \sqrt{b^2-4ac}}{......Thu, 03 Sep 2026 20:46:57 -0400https://liberty.io/an-infinite-dictionary-countably-infinite-or-uncountably-infinite.htmlhttps://liberty.io/an-infinite-dictionary-countably-infinite-or-uncountably-infinite.htmlThis question concerns Ian Stewart's "Hyperwebster", an uncountable dictionary. Say a publishing company wants to publish every possible permutation (of any length) of the characters A-Z. The ...Thu, 03 Sep 2026 20:43:46 -0400https://liberty.io/linear-combinations-of-sine-and-cosine.htmlhttps://liberty.io/linear-combinations-of-sine-and-cosine.htmlIf you take a linear combination of the cosine and sine function, then the result is again a sinusoid, but with a new amplitude and phase shift. $$a \cos(\theta) + b \sin(\theta) = A \cos(\theta + \...Thu, 03 Sep 2026 20:42:55 -0400https://liberty.io/why-does-squaring-of-the-radius-of-a-circle-times-pi-give-us-the-area.htmlhttps://liberty.io/why-does-squaring-of-the-radius-of-a-circle-times-pi-give-us-the-area.htmlWhy does squaring the radius of a circle times pi equal the area? (I.e., why does the area of circle representable as $\pi r^2$ ?)What is the relationship between the radius of a circle and the area?...Thu, 03 Sep 2026 20:38:31 -0400https://liberty.io/how-does-one-find-the-common-area-shared-between-two-polar-curves.htmlhttps://liberty.io/how-does-one-find-the-common-area-shared-between-two-polar-curves.htmlThe problem: Consider the circle $r = (\sqrt{2} - 1)\cos(\theta)$ and the cardioid $r = 1 - \cos(\theta)$. Find the area $A$ of the region $R$ common to the inside of the circle and the inside of the ...Thu, 03 Sep 2026 20:24:22 -0400https://liberty.io/is-there-a-math-symbol-meaning-if-or-when.htmlhttps://liberty.io/is-there-a-math-symbol-meaning-if-or-when.htmlTo introduce a logic premise normally people use the word If, or Iff (if and only if). Is there a math symbol with the same meaning?...Thu, 03 Sep 2026 20:19:35 -0400https://liberty.io/what-does-the-dot-over-x-or-y-mean.htmlhttps://liberty.io/what-does-the-dot-over-x-or-y-mean.htmlI am just starting to read up on differential equations. The problem is (in my materials) nowhere is explained what do these dots mean. Can anyone shed some light? $$\begin{align}&\begin{cases......Thu, 03 Sep 2026 20:04:04 -0400https://liberty.io/prove-3-n-ge-n-3-by-induction.htmlhttps://liberty.io/prove-3-n-ge-n-3-by-induction.htmlYep, prove $3^n \ge n^3$, $n \in \mathbb{N}$. I can do this myself, but can't figure out any kind of "beautiful" way to do it. The way I do it is: Assume $3^n \ge n^3$ Now, $(n+1)^3 = n^3 + ......Thu, 03 Sep 2026 19:57:56 -0400https://liberty.io/given-the-equation-x-sqrt-y-does-the-equation-represent-y-as-a-functio.htmlhttps://liberty.io/given-the-equation-x-sqrt-y-does-the-equation-represent-y-as-a-functio.htmlI can justify why the answer might be no, but also why it might be yes, so which line of reasoning is correct? I. $x=\sqrt{y}$ does not represent y as a function of x because if $x$ is a negative ......Thu, 03 Sep 2026 19:46:15 -0400https://liberty.io/how-to-understand-force-of-interest-in-simple-interest.htmlhttps://liberty.io/how-to-understand-force-of-interest-in-simple-interest.htmlI am struggling to understand "force of interest" in simple interest. I understand that "force of interest" can be interpreted as compound interest with infinitely small time interval. That is, the......Thu, 03 Sep 2026 19:45:26 -0400https://liberty.io/help-solving-a-differential-equation.htmlhttps://liberty.io/help-solving-a-differential-equation.htmlmy Calculus II class is nearing the end of the quarter and we've just started differential equations to get ready for Calculus III. In my homework, I came upon these problems. One of the problems ......Thu, 03 Sep 2026 19:37:00 -0400https://liberty.io/equation-of-a-spherocylinder-capsule.htmlhttps://liberty.io/equation-of-a-spherocylinder-capsule.htmlI want to know if a 3D spherocylinder/capsule, that looks like this: Can be approximated by a known equation. I want a shape with volume. Edit: In my program I already have a cylinder with two ...Thu, 03 Sep 2026 19:36:44 -0400https://liberty.io/not-getting-what-it-means-v-ai-bj-ck-for-some-vector-v.htmlhttps://liberty.io/not-getting-what-it-means-v-ai-bj-ck-for-some-vector-v.htmlI am watching Khan Academy Linear Algebra series and I am really getting confused when Sal writes some sort of equation for the vector, such as $v = ai + bj + ck$. I understand that vector $v$ shou......Thu, 03 Sep 2026 19:34:29 -0400https://liberty.io/how-may-one-show-that-sin-x-sin-y-le-x-y-duplicate.htmlhttps://liberty.io/how-may-one-show-that-sin-x-sin-y-le-x-y-duplicate.htmlI think I saw the inequality $$|\sin x-\sin y|\le |x-y|$$ somewhere recently. I've been trying to see why it could be true but so far have been unable to come up with any success. Could someone s......Thu, 03 Sep 2026 19:33:36 -0400https://liberty.io/fractional-linear-transformations-and-their-matrix-form.htmlhttps://liberty.io/fractional-linear-transformations-and-their-matrix-form.htmlFor $z \in \mathbb{C}$, a fractional linear transformation of $z$, with an associated matrix $M$ is: $$M = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \\ T_{M}(z) = \frac{az + b}{cz+d}.$$ ...Thu, 03 Sep 2026 19:29:03 -0400https://liberty.io/proving-the-cauchy-schwarz-inequality-by-induction.htmlhttps://liberty.io/proving-the-cauchy-schwarz-inequality-by-induction.htmlI ran across this problem in some old notes, and I frustratingly can't figure out how to do it Let $a_i$ and $b_i$ be sequences of natural numbers, use induction to show $\sum_{i=1}^n (a_ib_i)^{......Thu, 03 Sep 2026 19:25:40 -0400https://liberty.io/how-can-i-find-the-equation-of-a-parabola-only-given-it-039-s-x-interc.htmlhttps://liberty.io/how-can-i-find-the-equation-of-a-parabola-only-given-it-039-s-x-interc.htmlI received a problem in my math class the other day that left me stumped. The problem went something like this. Mr. Lots-O-Cash would like to order a parabola that passes through the points $(-4......Thu, 03 Sep 2026 19:17:47 -0400https://liberty.io/a-jacobi-elliptic-integral-with-cosine.htmlhttps://liberty.io/a-jacobi-elliptic-integral-with-cosine.htmlAfter reduction of a problem, I find myself in front of these integrals, $$ \int_0^{2\pi} cn \left( \frac{2K(1/2)}{\pi} \theta, \frac{1}{2} \right) \cos(m \theta) \mathrm{d}\theta, $$ with $m$ posi......Thu, 03 Sep 2026 19:11:00 -0400https://liberty.io/using-shell-method-find-the-volume-of-the-solid-generated-when-y-3-x-2.htmlhttps://liberty.io/using-shell-method-find-the-volume-of-the-solid-generated-when-y-3-x-2.htmlQuestion Via the Shell Method, what is the volume of the solid of revolution formed by revolving a portion of $y=3-x^2$ about the x-axis bounded by $y=0$ and $y=3$. Context The answer from Wolfram......Thu, 03 Sep 2026 19:10:37 -0400https://liberty.io/intuitive-explanation-of-duhamel-039-s-principle.htmlhttps://liberty.io/intuitive-explanation-of-duhamel-039-s-principle.htmlThis is with regards to the first section of Wikipedia article Duhamel's principle (revision from July 2012). I want to see if I am understanding this. Basically the inhomogeneous equation says tha......Thu, 03 Sep 2026 18:58:39 -0400https://liberty.io/translate-to-first-order-logic.htmlhttps://liberty.io/translate-to-first-order-logic.htmlI have this two sentences: All men love the woman to whom they are married. No woman loves a man who does not respect all women. and in the exercise they are translated to First Order Logic as fo......Thu, 03 Sep 2026 18:53:30 -0400https://liberty.io/variance-of-x-y.htmlhttps://liberty.io/variance-of-x-y.htmlIf X and Y are random variables with correlation coefficient 0.7, each of which has variance 6, what is the variance of X−Y? Enter your answer as a decimal. Using the information given, I was abl......Thu, 03 Sep 2026 18:50:06 -0400https://liberty.io/ordinary-differential-equations-v-dx-dt.htmlhttps://liberty.io/ordinary-differential-equations-v-dx-dt.htmlThe question is: What ODE does v=dx/dt solve? And what is the limit as t approaches infinity of v(t). It sounds almost too easy, but I am confused about what to do exactly. Intuitively, as the ques......Thu, 03 Sep 2026 18:49:05 -0400https://liberty.io/how-the-product-of-two-integrals-is-iterated-integral-int-cdot-int-iin.htmlhttps://liberty.io/how-the-product-of-two-integrals-is-iterated-integral-int-cdot-int-iin.htmlLet $\large{I} = \Large \int\limits_{-\infty}^{\infty} \normalsize e^{\small -\frac{y^2}{2}}\ dy$. Then, my textbook says, $$\tag{1} I^2 = \left( \int\limits_{-\infty}^{\infty} \normalsize e^{\......Thu, 03 Sep 2026 18:47:24 -0400https://liberty.io/simplify-sqrt-sqrt-3-5-sqrt-3-4.htmlhttps://liberty.io/simplify-sqrt-sqrt-3-5-sqrt-3-4.htmlDenest $\sqrt {\sqrt[3]{5}-\sqrt[3]{4}}$. I have tried completing square by several method but all failed. Can anyone help me please? Thank you. p.s. I'm a poor question-tagger....Thu, 03 Sep 2026 18:46:11 -0400https://liberty.io/cauchy-schwarz-inequality-for-nonsymmetric-matrices.htmlhttps://liberty.io/cauchy-schwarz-inequality-for-nonsymmetric-matrices.htmlI am interested in Cauchy-Schwarz inequalities for inner products of the form $\langle Ax,y\rangle$ where $A$ is some matrix. It is rather easy to see that the Cauchy-Schwarz inequality continues t......Thu, 03 Sep 2026 18:45:59 -0400https://liberty.io/consistency-of-a-system-of-linear-equations-using-unknowns-in-the-matr.htmlhttps://liberty.io/consistency-of-a-system-of-linear-equations-using-unknowns-in-the-matr.htmlCan anyone tell me how would I answer these type of questions? I already know how to answer the normal way which doesn't include any variables such as Alpha in the question....Thu, 03 Sep 2026 18:40:20 -0400https://liberty.io/how-find-this-sum-sin-2-x-sin-2-2x-sin-2-3x-cdots-sin-2-nx.htmlhttps://liberty.io/how-find-this-sum-sin-2-x-sin-2-2x-sin-2-3x-cdots-sin-2-nx.htmlQuestion: Find the value $$f^{(2)}_{n}(x)=\sin^2{x}+\sin^2{(2x)}+\sin^2{(3x)}+\cdots+\sin^2{(nx)}$$ My solution: since $$\sin^2{x}=\dfrac{1}{2}(1-\cos{(2x)})$$ so $$f_{n}(x)=\dfrac{n}{2}-\dfra......Thu, 03 Sep 2026 18:37:29 -0400https://liberty.io/change-of-polynomial-basis-exercise.htmlhttps://liberty.io/change-of-polynomial-basis-exercise.htmlCan someone please check if my solution is correct? Thanks. Considering the following basis: $$ B= \left\{ x^2,1+x^2,x-1 \right\}\\ C= \left\{ x,1-x,x^2 \right\} $$ a) Find the change of basis mat......Thu, 03 Sep 2026 18:26:53 -0400https://liberty.io/derivative-of-matrix-exponential-w-r-t-to-each-element-of-the-matrix.htmlhttps://liberty.io/derivative-of-matrix-exponential-w-r-t-to-each-element-of-the-matrix.htmlI have $x= \exp(At)$ where $A$ is a matrix. I would like to find derivative of $x$ with respect to each element of $A$. Could anyone help with this problem?...Thu, 03 Sep 2026 18:20:43 -0400https://liberty.io/i-wonder-if-the-set-of-all-characteristic-function-on-a-set-a-is-simil.htmlhttps://liberty.io/i-wonder-if-the-set-of-all-characteristic-function-on-a-set-a-is-simil.htmlRecently I studied something known as characteristic function (Indicator function), after going through it deeply I want know if the set of all characteristic functions on A will be similar to its ......Thu, 03 Sep 2026 18:14:27 -0400https://liberty.io/riemann-sheets-for-combined-roots.htmlhttps://liberty.io/riemann-sheets-for-combined-roots.htmlSo, I am aware of the fact that if I do have a function like $$f(z) = z^{1/2} + z^{1/3}$$ being $2, 3$ relatively primes, I have $$\mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/3\mathbb{Z}$$ hence ......Thu, 03 Sep 2026 18:14:11 -0400https://liberty.io/proof-for-rational-roots.htmlhttps://liberty.io/proof-for-rational-roots.htmlLet $f(x) = a_0+a_1x+\cdots+a_nx^n$ be a polynomial of degree $n$ over $\Bbb Z$. A: If a rational number $\frac{p}{q}$ is a root of $f(X)$, show that $p\mid a_0$ and $q\mid a_n$. Assume $\gcd(p, q......Thu, 03 Sep 2026 18:03:50 -0400https://liberty.io/derivative-of-operatorname-arcsec-3x.htmlhttps://liberty.io/derivative-of-operatorname-arcsec-3x.htmlThis seems straightforward to me, but it was marked wrong on my test with no clarification and no ability to ask questions. $y=\operatorname{arcsec}(3x)$. Calculate the derivative. I take the se......Thu, 03 Sep 2026 17:57:58 -0400https://liberty.io/prove-that-in-any-triangle-abc-cos-2a-cos-2b-cos-2c-geq-frac-3-4.htmlhttps://liberty.io/prove-that-in-any-triangle-abc-cos-2a-cos-2b-cos-2c-geq-frac-3-4.htmlI have two similar looking questions. (1) Prove that in triangle $ABC$, $$\cos^2A+\cos^2B+\cos^2C\geq\frac{3}{4}.$$ (2) If $\Delta ABC$ is an acute angled, then prove that $$\cos^2A+\cos^2B+\cos^2......Thu, 03 Sep 2026 17:56:16 -0400https://liberty.io/can-i-subtract-infinity-from-infinity.htmlhttps://liberty.io/can-i-subtract-infinity-from-infinity.htmlI was stuck when solving a problem on limits. It was like----> $\lim_{x\to\infty} (x-x)$. What should I do now?...Thu, 03 Sep 2026 17:55:32 -0400https://liberty.io/aaa-similarity-theorem.htmlhttps://liberty.io/aaa-similarity-theorem.htmlDef: Two triangles are called similar if and only if one can be scaled into the other. AAA Theorem: If two triangles have the same angles if and only if they are similar. How does one prove the T......Thu, 03 Sep 2026 17:36:53 -0400https://liberty.io/how-to-find-the-nth-term-in-the-following-sequence-1-1-2-2-4-4-8-8-16-.htmlhttps://liberty.io/how-to-find-the-nth-term-in-the-following-sequence-1-1-2-2-4-4-8-8-16-.htmlI'm having difficulty in finding the formula for the sequence above, when I put this in WolframAlpha it gave me a rather complex formula which I'm not convinced even works properly but I'm sure the......Thu, 03 Sep 2026 17:34:17 -0400https://liberty.io/is-x-1-a-linear-function.htmlhttps://liberty.io/is-x-1-a-linear-function.htmlI want to know if $\frac1x$ is a linear function or not. I read that a linear function can also be defined as a function of the first degree. Since $\frac1x=x^{-1}$, the function is of the first d......Thu, 03 Sep 2026 17:28:59 -0400https://liberty.io/orthogonal-set-vs-orthogonal-basis.htmlhttps://liberty.io/orthogonal-set-vs-orthogonal-basis.htmlFor some reason my book distinguishes the two names. If a set is an orthogonal set, doesn't that make it immediately a basis for some subspace $W$ since all the vectors in the orthogonal set are ...Thu, 03 Sep 2026 17:26:11 -0400https://liberty.io/why-are-all-non-prime-numbers-divisible-by-a-prime-number.htmlhttps://liberty.io/why-are-all-non-prime-numbers-divisible-by-a-prime-number.htmlIn Euclid's infinite prime numbers proof, the logic is as follows: Assume a set $S$ of all prime numbers in existence is finite (there are a finite amount of primes) Then there must be a greatest......Thu, 03 Sep 2026 17:14:21 -0400https://liberty.io/double-pendulum-lagrangian.htmlhttps://liberty.io/double-pendulum-lagrangian.htmlcould someone help me with the following problem, for part $b$ i know that $$\Bbb L=T-U$$ but i've forgotten how i can calculate the kinetic energy and potential energies...Thu, 03 Sep 2026 17:14:07 -0400https://liberty.io/gram-schmidt-process-and-polynomial-basis-inner-product-exercise.htmlhttps://liberty.io/gram-schmidt-process-and-polynomial-basis-inner-product-exercise.htmlLet the vector space $P_2$ have the inner product: $\langle p,q\rangle=\int\limits_{-1}^{1}p(x)q(x)dx$ Apply the Gram-Schmidt process to transform the standard $S=\{1,x,x^2\}$ into an orthonormal b......Thu, 03 Sep 2026 17:09:20 -0400https://liberty.io/secant-line-and-diameter-of-a-circle.htmlhttps://liberty.io/secant-line-and-diameter-of-a-circle.htmlA secant line incident to a circle at points $A$ and $C$ intersects the circle's diameter at point $B$ with a $45^\circ$ angle. If the length of $AB$ is $1$ and the length of $BC$ is $7$, then what......Thu, 03 Sep 2026 17:04:40 -0400https://liberty.io/meaning-of-vector-derivative-of-velocity-with-respect-to-position.htmlhttps://liberty.io/meaning-of-vector-derivative-of-velocity-with-respect-to-position.htmlI would like to compute an expression for $\frac{\partial v}{\partial x}$, where $x$ is a position vector and $v = \dot{x}$ is a velocity vector. Using the Chain Rule, we should have: $$\frac{\par......Thu, 03 Sep 2026 16:45:53 -0400https://liberty.io/for-what-values-does-the-geothmetic-meandian-converge.htmlhttps://liberty.io/for-what-values-does-the-geothmetic-meandian-converge.htmlThe geothmetic meandian, $G_{MDN}$ is defined in this XKCD as $$F(x_1, x_2, ..., x_n) = \left(\frac{x_1 +x_2+\cdots+x_n}{n}, \sqrt[n]{x_1 x_2 \cdots x_n}, x_{\frac{n+1}{2}} \right)$$ $$G_{MDN}(x_1,......Thu, 03 Sep 2026 16:34:21 -0400https://liberty.io/how-do-you-find-the-turning-points-of-a-polynomial-without-using-calcu.htmlhttps://liberty.io/how-do-you-find-the-turning-points-of-a-polynomial-without-using-calcu.htmlI have a polynomial $P(x) = -x^3+12x+3$, and I am asked to find the turning points of it, and hence state how many zeroes it has. Since this chapter is separate from calculus, we are expected to so......Thu, 03 Sep 2026 16:29:20 -0400https://liberty.io/harvard-math-25-and-55-prerequisites-closed.htmlhttps://liberty.io/harvard-math-25-and-55-prerequisites-closed.htmli'll be attending harvard next year (deferred entry) and am planning to use this year to study calculus and linear algebra in preparation for either math 25 or 55. (See here for the various courses......Thu, 03 Sep 2026 16:28:05 -0400https://liberty.io/how-do-i-find-approximate-the-jacobian-of-this-ode-system.htmlhttps://liberty.io/how-do-i-find-approximate-the-jacobian-of-this-ode-system.html$$x_1'=(x_2-x_4)x_5-x_1+8$$ $$x_2'=(x_3-x_5)x_1-x_2+8$$ $$x_3'=(x_4-x_1)x_2-x_3+8$$ $$x_4'=(x_5-x_2)x_3-x_4+8$$ $$x_5'=(x_1-x_3)x_4-x_5+8$$ I am asked for either an exact or approximate solution. I ...Thu, 03 Sep 2026 16:27:13 -0400https://liberty.io/arma-1-2-model-auto-covariance-function.htmlhttps://liberty.io/arma-1-2-model-auto-covariance-function.htmlI am struggling with finding the Autocovariance function $\gamma(k)$, of the following ARMA(1,2) model: $x_t-0.9x_{t-1}=e_t+2e_{i-1}+0.5e_{t-2}$. I have already found this model to be stationary,......Thu, 03 Sep 2026 16:25:10 -0400https://liberty.io/complex-conjugate-clarification.htmlhttps://liberty.io/complex-conjugate-clarification.htmlSo I understand that something like $4-i$ is a complex conjugate of $4+i$, however are things like; (a) $-4+i = \overline{-4-i}$, (b) $-4+i = \overline{4+i}$, and (c) $-4-i = \overline{4+i}$. Gene......Thu, 03 Sep 2026 16:22:48 -0400https://liberty.io/3-sided-coin-odds-you-roll-two-heads-two-tails-and-two-feet.htmlhttps://liberty.io/3-sided-coin-odds-you-roll-two-heads-two-tails-and-two-feet.htmlAssume it's a fair coin. What are the odds you roll two heads, two tails, and two feet on six rolls? I got the following: $ 3 \times \binom{6}{2} \over 3^6 $ $3$ for the different sides, $6 \choo......Thu, 03 Sep 2026 16:16:42 -0400https://liberty.io/is-pi-equal-to-180-circ.htmlhttps://liberty.io/is-pi-equal-to-180-circ.html$$ \begin{array}{ccc} \sin{(\theta+180^{\circ})}=-\sin{\theta} & \cos{(\theta+180^{\circ})}=-\cos{\theta} & \tan{(\theta+180^{\circ})}=\tan{\theta} \\ \sin{(\theta+\pi)}=-\sin{\theta} &......Thu, 03 Sep 2026 16:15:48 -0400https://liberty.io/find-the-length-of-ad-to-the-nearest-whole-number.htmlhttps://liberty.io/find-the-length-of-ad-to-the-nearest-whole-number.htmlA sketch of a rectangular ground ABCD is given here. (i) Based on the information marked on the sketch, find the length of AD to the nearest whole number. Any Ideas on how to begin?...Thu, 03 Sep 2026 16:07:23 -0400https://liberty.io/logic-of-step-to-convert-rectangular-equation-of-circle-to-parametric.htmlhttps://liberty.io/logic-of-step-to-convert-rectangular-equation-of-circle-to-parametric.htmlI recently learned in math class how to convert the rectangular equation of a circle into a parametric equation: $x^2+y^2=1$ $\cos^2(t)+\sin^2(t)=1$ $x^2+y^2=\cos^2(t)+\sin^2(t)$ $x^2=\cos^2(t)......Thu, 03 Sep 2026 16:00:42 -0400https://liberty.io/how-to-find-the-anti-derivative-of-sin-2-x-closed.htmlhttps://liberty.io/how-to-find-the-anti-derivative-of-sin-2-x-closed.htmlI already know that this problem can be solved through a trigonometric identity. However, since this is a relatively simple expression, why can’t the antiderivative just be traced back? In such cas......Thu, 03 Sep 2026 15:48:01 -0400https://liberty.io/defining-the-l-2-norm-of-a-vector-valued-function.htmlhttps://liberty.io/defining-the-l-2-norm-of-a-vector-valued-function.htmlI am considering a collection of function of the type, $ f:[0,2\pi]\rightarrow \mathbb{R^2}$. I want to define the $L^2$ norm of the function in that space. I am defining the a norm of $f(=(f_1,f_......Thu, 03 Sep 2026 15:45:52 -0400https://liberty.io/finding-the-fourth-roots-of-a-complex-number.htmlhttps://liberty.io/finding-the-fourth-roots-of-a-complex-number.htmlHow would I find the fourth roots of $-81i$ in the complex numbers? Here is what I currently have: $w = -81i$ $r = 9$ $\theta = \arctan (-81)$? Although I am not sure it's correct or if ......Thu, 03 Sep 2026 15:43:14 -0400https://liberty.io/an-open-interval-is-an-open-set.htmlhttps://liberty.io/an-open-interval-is-an-open-set.htmlFirst, the question I have is very similar to this question, but I hope it doesn't get closed as duplicate since I'm stuck nevertheless. I'm trying to algebraically prove that an open interval is an ...Thu, 03 Sep 2026 15:35:15 -0400https://liberty.io/user-andrew-elkouri-mathematics-stack-exchange.htmlhttps://liberty.io/user-andrew-elkouri-mathematics-stack-exchange.htmlQ&A for people studying math at any level and professionals in related fields...Thu, 03 Sep 2026 15:29:38 -0400https://liberty.io/convert-from-dfa-to-nfa.htmlhttps://liberty.io/convert-from-dfa-to-nfa.htmlFor this language $\{ w \mid w \text{ contains at least three 1's} \}$, its DFA diagram is defined as follows: While trying to convert it to NFA, but I realized that its NFA would be identical to ......Thu, 03 Sep 2026 15:07:03 -0400https://liberty.io/what-do-you-call-a-number-that-represents-20-percent-written-as-0-2-an.htmlhttps://liberty.io/what-do-you-call-a-number-that-represents-20-percent-written-as-0-2-an.htmlLets say I want to represent 20 Percent What do you call/name the "0.2" and "20%" notations I am calling them Fractional and WholeValue notations for the time being, but wondering if there was ...Thu, 03 Sep 2026 15:00:28 -0400https://liberty.io/need-help-with-the-integral-int-frac-2-tan-x-3-5-sin-2-x-4-dx.htmlhttps://liberty.io/need-help-with-the-integral-int-frac-2-tan-x-3-5-sin-2-x-4-dx.htmlI'm having a problem resolving the following integral, spent almost all day trying. Any help would be appreciated. $$\int \frac{2\tan(x)+3}{5\sin^2(x)+4}\,dx$$...Thu, 03 Sep 2026 14:41:05 -0400https://liberty.io/why-cos-2-2x-frac-1-2-1-cos-4x.htmlhttps://liberty.io/why-cos-2-2x-frac-1-2-1-cos-4x.htmlWhy: $$\cos ^2(2x) = \frac{1}{2}(1+\cos (4x))$$ I don't understand this, how I must to multiply two trigonometric functions? Thanks a lot....Thu, 03 Sep 2026 14:16:12 -0400https://liberty.io/what-is-the-isometry-and-isometry-group.htmlhttps://liberty.io/what-is-the-isometry-and-isometry-group.htmlCan someone explain with some simple examples what is meant by the isometry and isometry group? I'm a student of physics and I often come across this term in texts of general relativity for various ...Thu, 03 Sep 2026 14:15:15 -0400https://liberty.io/volume-of-a-pencil-closed.htmlhttps://liberty.io/volume-of-a-pencil-closed.htmlA pencil sharpened at both ends is formed of a cylindrical barrel together with two conical points. The pencil measures 110mm from tip to tip, has radius $r$ mm, and the cylindrical barrel is $h$ m......Thu, 03 Sep 2026 14:05:20 -0400https://liberty.io/prove-that-the-limit-exists-and-then-find-its-limit.htmlhttps://liberty.io/prove-that-the-limit-exists-and-then-find-its-limit.htmlWe are given a function $h(x)$ which strictly positive. The function $h$ is defined on $\mathbb{R}$ and that satisfies the following:\ $$\lim_{x\to 0}(h(x)+\frac{1}{h(x)})=2$$ The question is to ......Thu, 03 Sep 2026 14:02:35 -0400https://liberty.io/cubic-equation-in-pocket-calculator.htmlhttps://liberty.io/cubic-equation-in-pocket-calculator.htmlIs it possible to input and automaticly get solutions for the cubic equation using Texas Instruments TI-30X IIS calculator?...Thu, 03 Sep 2026 13:56:10 -0400https://liberty.io/how-to-find-values-from-1-equation-with-3-variables.htmlhttps://liberty.io/how-to-find-values-from-1-equation-with-3-variables.htmlIs there any way to get something out of: $50x + 5y - 18z = 0$ ? I found this trying to solve this: \begin{equation} xyz+xyz+xyz=zzz \end{equation} in which xyz are not multiplying themselves. T......Thu, 03 Sep 2026 13:51:12 -0400https://liberty.io/how-can-i-find-the-hypotenuse-and-opposite-sides-of-triangle.htmlhttps://liberty.io/how-can-i-find-the-hypotenuse-and-opposite-sides-of-triangle.htmlBased on the given information how can I find the pixel values of the hypotenuse and opposite sides of the triangle?...Thu, 03 Sep 2026 13:49:53 -0400https://liberty.io/how-do-you-draw-a-direction-field-for-2x2-matrix.htmlhttps://liberty.io/how-do-you-draw-a-direction-field-for-2x2-matrix.htmlI understand $AX = X'$ and by doing so, you get both equation for derivative of $x_1 and $x_2$. When I make a $x_1$ and $x_2$ plot, I am confused regarding which derivative of $x_1$ or $x_2$ to ch......Thu, 03 Sep 2026 13:45:06 -0400https://liberty.io/how-do-we-decide-the-direction-of-vector-that-is-orthogonal-to-some-ot.htmlhttps://liberty.io/how-do-we-decide-the-direction-of-vector-that-is-orthogonal-to-some-ot.htmlassume that two vector given with relationship below: $$\vec{n}\cdot \vec{u} = 0 $$ Then $\vec{n}$ and $\vec{u}$ are orthogonal vectors. Assume that we now have the direction of $\vec{n}$ Then t......Thu, 03 Sep 2026 13:36:48 -0400https://liberty.io/what-is-a-formal-polynomial.htmlhttps://liberty.io/what-is-a-formal-polynomial.htmlI'm starting to study Field Theory by myself, the books don't say explicitly what a polynomial is, I mean, what the $x$ of $f(x)$ in $F[x]$ is? $x\in F$? When I take $f(\alpha)$ am I taking the ele......Thu, 03 Sep 2026 13:31:33 -0400https://liberty.io/is-dodecahedron-tesselation-somehow-possible.htmlhttps://liberty.io/is-dodecahedron-tesselation-somehow-possible.htmlIn this video (at 3:25) there is an animation of planets inside a dodecahedron matrix (or any data-structure that best fit this 3d mosaic). I tried reproducing it with 12 sided dices, or in Blender......Thu, 03 Sep 2026 13:28:44 -0400https://liberty.io/definite-integral-word-problem.htmlhttps://liberty.io/definite-integral-word-problem.htmlThe sales of a plastic widget were estimated to be: $P(t)  =  8000te ^{−0.5t}$ where $t$ is in weeks, and $P(t)$ is in units per week. How many widgets were sold in the first $9$ weeks? To sta......Thu, 03 Sep 2026 13:27:52 -0400https://liberty.io/what-is-the-difference-between-the-dot-product-and-the-scalar-projecti.htmlhttps://liberty.io/what-is-the-difference-between-the-dot-product-and-the-scalar-projecti.htmlI don't understand the difference between the dot product of two vectors and the scalar projection of a vector onto another one. To me it looks like they are both (geometrically) the length of the ...Thu, 03 Sep 2026 13:21:31 -0400https://liberty.io/multivariable-differential-equation.htmlhttps://liberty.io/multivariable-differential-equation.htmlGiven $u=f(2x-y)+g(x-2y)$, show that $$2 \frac{\mathrm{d}^2 u}{\mathrm{d}x^2} + 5 \frac{\mathrm{d}^2 u}{\mathrm{d}x\,\mathrm{d}y} + 2\frac{\mathrm{d}^2 u}{\mathrm{d}y^2} = 0.$$ I'm not even sure wh......Thu, 03 Sep 2026 13:18:42 -0400https://liberty.io/from-randn-to-bivariate-gaussian-distribution-image.htmlhttps://liberty.io/from-randn-to-bivariate-gaussian-distribution-image.htmlIn Matlab, I generated a bivariate Gaussian distribution with mean 50 and standard deviation 50: G=50*randn((100*100)/2,2)+50; histogram2(G(:,1),G(:,2)); The 3D histogram gives me the output I am ...Thu, 03 Sep 2026 13:17:00 -0400https://liberty.io/calculate-line-integral-with-respect-to-arc-length.htmlhttps://liberty.io/calculate-line-integral-with-respect-to-arc-length.htmlCalculate the line integral with respect to arc length of $\int_Czds$, where $C$ is parametrized by $γ(t) = (tcost,tsint,t), 0 ≤ t ≤ t_0.$ I calculate the norm of $γ'(t)$ and I get $\sqrt {2+t^2}$.......Thu, 03 Sep 2026 13:11:19 -0400https://liberty.io/how-do-i-simplify-frac-sin-2x-1-cos-2x.htmlhttps://liberty.io/how-do-i-simplify-frac-sin-2x-1-cos-2x.html$$\dfrac{\sin(2x)}{1-\cos (2x)}$$ How do I simplify given expression? My attempt: We know the double-angle identity for $\sin(2x)$ and $\cos(2x)$ as shown below $$\sin(2x) = 2\sin (x)\cos(x)$$......Thu, 03 Sep 2026 13:07:23 -0400