https://liberty.ioCurated star highlights with fresh energy.en-usWed, 12 Aug 2026 11:17:35 -0400https://liberty.io/what-is-the-derivative-of-log-y-with-respect-to-x.htmlhttps://liberty.io/what-is-the-derivative-of-log-y-with-respect-to-x.htmlI was solving the following sum: We started off with the first equation and then he took log on both sides to simplify. Everything was fine here but the next step's LHS confused me, he said he used......Wed, 12 Aug 2026 15:15:34 -0400https://liberty.io/cutting-a-cube-with-a-plane.htmlhttps://liberty.io/cutting-a-cube-with-a-plane.htmlMe and a couple of friends have recently started solving Math problems in our breaks at school. We came along the Georg Mohr tournament, which is Denmark's first qualifications tournament for the I......Wed, 12 Aug 2026 15:13:22 -0400https://liberty.io/british-flag-theorem-generalized-and-inspired-from-british-flag-theore.htmlhttps://liberty.io/british-flag-theorem-generalized-and-inspired-from-british-flag-theore.htmlBritish Flag theorem: Let $P$ be a point in the plane, let $ABCD$ be a rectangle in the plane then: $$PA^2+PC^2=PB^2+PD^2$$ Generalization: Let $ABCD$ be a rectangle in a plane, Let $P$ be a poin......Wed, 12 Aug 2026 14:57:17 -0400https://liberty.io/is-this-a-solution-for-the-problem-a-3-b-3-c-3-has-no-nonzero-integer-.htmlhttps://liberty.io/is-this-a-solution-for-the-problem-a-3-b-3-c-3-has-no-nonzero-integer-.htmlIs this a solution for the problem: $\ a^3 + b^3 = c^3\ $ has no nonzero integer solutions? Suppose $\ a^3 + b^3 = c^3,\ a,b,c \in \mathbb Z^*,\ $then: $c^3 - b ^ 3 = (c - b)((c - b) ^ 2 + 3cb......Wed, 12 Aug 2026 14:56:34 -0400https://liberty.io/backward-stability-vs-stability.htmlhttps://liberty.io/backward-stability-vs-stability.htmlI'm reading a numerical analysis book, which gave me very confusing descriptions about stability and backward stability. It says that an algorithm $f$ is stable, if for any $x$, $$ \frac{\|f_b(x)......Wed, 12 Aug 2026 14:53:56 -0400https://liberty.io/express-using-logic-symbols.htmlhttps://liberty.io/express-using-logic-symbols.htmlI have to also decide whether it is true or not after I express it in logic symbols. This is what I have so far..Am I correct? and I don't know how to do a).. a) There is a smallest positive numbe......Wed, 12 Aug 2026 14:53:23 -0400https://liberty.io/degree-of-rational-function.htmlhttps://liberty.io/degree-of-rational-function.htmlThis might sound like a very trivial question but I found different answers on the web. Assume one has a rational function $$\frac{f(x)}{g(x)} ,$$ where $f(x)$ and $g(x)$ are polynomials. What i......Wed, 12 Aug 2026 14:51:21 -0400https://liberty.io/do-all-straight-lines-have-an-inverse-function.htmlhttps://liberty.io/do-all-straight-lines-have-an-inverse-function.htmlDo all straight lines have an inverse function? It would seem to make sense. All linear lines would pass the horizontal line test and thus when reflected across $y=x$ it would still be a function. ...Wed, 12 Aug 2026 14:49:55 -0400https://liberty.io/quaternions-problem-with-understanding-essence-of.htmlhttps://liberty.io/quaternions-problem-with-understanding-essence-of.htmlI can't calm down when I can't realized of principles of working something. Quaternion is that place where I spend а few days! Unfortunately, not one lectures or books what I saw or read don't expl......Wed, 12 Aug 2026 14:36:30 -0400https://liberty.io/trees-on-six-vertices.htmlhttps://liberty.io/trees-on-six-vertices.htmlShow that there are exactly six non-isomorphic trees on six vertices. ("Introduction to Combinatorics" p.183) The solutions were provided in the book. But why can we be sure that there are ONLY si......Wed, 12 Aug 2026 14:27:43 -0400https://liberty.io/what-is-the-difference-between-equation-and-formula.htmlhttps://liberty.io/what-is-the-difference-between-equation-and-formula.htmlSometimes equation and formula are used interchangeably, but I was wondering if there is a difference. For example, suppose we can calculate a car's fuel efficiency as: mpg = distance traveled in......Wed, 12 Aug 2026 14:18:18 -0400https://liberty.io/are-the-endpoints-of-the-domain-of-a-function-counted-as-critical-poin.htmlhttps://liberty.io/are-the-endpoints-of-the-domain-of-a-function-counted-as-critical-poin.htmlDo the end points of a domain come under critical points? I know we say critical point is a point where the derivative is zero or the derivative doesn't exist. For example: $$ f:[0,\pi] \to [-1,1......Wed, 12 Aug 2026 14:06:38 -0400https://liberty.io/cdf-for-a-continuous-random-variable.htmlhttps://liberty.io/cdf-for-a-continuous-random-variable.htmlLet $Y$ be a continuous random variable with probability density function$$ f_Y(x)=\begin{cases} xe^{-\frac{x^2}2};&x>0\\ 0;&x\le0\end{cases}.$$ Compute $\mathbb{E}(Y)$ and determine the ...Wed, 12 Aug 2026 14:05:00 -0400https://liberty.io/finding-the-radius-of-the-smallest-circle-that-can-circumscribe-an-equ.htmlhttps://liberty.io/finding-the-radius-of-the-smallest-circle-that-can-circumscribe-an-equ.htmlQ:A puzzle board is in the form of an equilateral triangle that has an area of $7\sqrt{3}$ if the board is placed on a circular table, what should be the min area of the table so that the whole board ...Wed, 12 Aug 2026 14:00:38 -0400https://liberty.io/how-to-find-a-circle-given-a-segment.htmlhttps://liberty.io/how-to-find-a-circle-given-a-segment.htmlIn this page, we get mathematical formulas for calculating the area of a circle segment under a few different conditions, if you already know the radius of the circle. My question is, how do you d......Wed, 12 Aug 2026 13:58:16 -0400https://liberty.io/continuous-differentiability-implies-lipschitz-continuity.htmlhttps://liberty.io/continuous-differentiability-implies-lipschitz-continuity.htmlHere's a statement in Zygmund's Measure and Integral on page 17: If $f$ has a continuous derivative on $[a,b]$, then (by the mean-value theorem) $f$ satisfies a Lipschitz condition on $[a,b]$. ......Wed, 12 Aug 2026 13:49:34 -0400https://liberty.io/equipotent-sets.htmlhttps://liberty.io/equipotent-sets.htmlWe know by definition that if a bijection between two sets A and B exists,then A and B are equivalent.The book I was reading took the function f:Z→N f(x)= -2x,for x<0 2x+1,for x=>0 w......Wed, 12 Aug 2026 13:38:42 -0400https://liberty.io/difference-between-direction-cosine-matrix-dcm-and-rotation-matrix.htmlhttps://liberty.io/difference-between-direction-cosine-matrix-dcm-and-rotation-matrix.htmlI am a bit confused about the difference between direction cosine matrix(DCM) and rotation matrix. I have searched through the literature but found no explicit explanation if they are different or ......Wed, 12 Aug 2026 13:38:37 -0400https://liberty.io/definition-of-suspension-topology.htmlhttps://liberty.io/definition-of-suspension-topology.htmlI am a bit confused with the definition of a suspension. This the definition. For a space $X$, denote $SX$ the suspension of $X$ in which this is the quotient space $$\frac{X \times I}{\sim}$$ w......Wed, 12 Aug 2026 13:37:38 -0400https://liberty.io/what-is-f-prime.htmlhttps://liberty.io/what-is-f-prime.htmlcurrently taking Measure and Integration course, which seems to have a different definition of f'. traditionally, $$f'(x)=\lim_{h\to 0} \frac{f(x+h)-f(x)}{h}$$ but in folland's book, it se......Wed, 12 Aug 2026 13:30:14 -0400https://liberty.io/points-in-general-position.htmlhttps://liberty.io/points-in-general-position.htmlI'm really confused by the definition of general position at wikipedia. I understand that the set of points/vectors in $\mathbb R^d$ is in general position iff every $(d+1)$ points are not in any ...Wed, 12 Aug 2026 13:21:33 -0400https://liberty.io/domain-and-range-of-a-sequence.htmlhttps://liberty.io/domain-and-range-of-a-sequence.htmlA sequence X is define as numbers smaller than 10 that are divisible by 2 and a natural number. What are the domain and ranges? I assume that the domain and range here is: dom X = <1,2,3,4,5,......Wed, 12 Aug 2026 13:05:34 -0400https://liberty.io/chain-rule-using-tree-diagram-why-does-it-work.htmlhttps://liberty.io/chain-rule-using-tree-diagram-why-does-it-work.htmlIn multivariable calculus, I was taught to compute the chain rule by drawing a "tree diagram" (a directed acyclic graph) representing the dependence of one variable on the others. I now want to ...Wed, 12 Aug 2026 12:52:28 -0400https://liberty.io/help-with-proof-of-expected-value-of-gamma-distribution.htmlhttps://liberty.io/help-with-proof-of-expected-value-of-gamma-distribution.htmlI am struggling with this proof of the expected value for the gamma distribution. I need help with the step indicated by the red arrow. Could someone please break it down for me. Thanks....Wed, 12 Aug 2026 12:39:09 -0400https://liberty.io/essentially-bounded-function-on-mathbb-r.htmlhttps://liberty.io/essentially-bounded-function-on-mathbb-r.htmlI don't really know a lot about measure (just finishing my undergrad) so I'm not really on good terms with this. So, let $L^\infty[a,b]$ denote the space of all essentially bounded functions on $[a......Wed, 12 Aug 2026 12:24:33 -0400https://liberty.io/prove-transitivity-property-congruence-mod-m.htmlhttps://liberty.io/prove-transitivity-property-congruence-mod-m.htmlProve transitivity property of congruence mod m. Show that if $x\equiv y \pmod m$ and $y \equiv z\pmod m$ then $x\equiv z\pmod m$ . I didn't really get the tutors explanation of this, I get what ...Wed, 12 Aug 2026 12:23:04 -0400https://liberty.io/how-can-i-plot-a-2d-figure-in-matlab-by-connecting-the-dots.htmlhttps://liberty.io/how-can-i-plot-a-2d-figure-in-matlab-by-connecting-the-dots.htmlI'm an amateur at MATLAB and I know how to do the basic stuff such as plotting functions, writing M files, functions, for loops, etc but I've be tasked to do something I have never done before. I ......Wed, 12 Aug 2026 12:22:02 -0400https://liberty.io/transformation-matrix-for-matrix-indices-to-cartesian-coordinates.htmlhttps://liberty.io/transformation-matrix-for-matrix-indices-to-cartesian-coordinates.htmlIn MatLab matrices, the indices are as follows: (1,1) (1,2) (1,3) (2,1) (2,2) (2,3) (3,1) (3,2) (3,3) This is an example 3x3 matrix. In corresponding cartesian coordinate system, the representation ...Wed, 12 Aug 2026 12:14:06 -0400https://liberty.io/solving-a-stratonovich-sde.htmlhttps://liberty.io/solving-a-stratonovich-sde.htmlI am trying to solve the following Stratonovich SDE $$dN_t=rN_tdt+\gamma N_t\circ dB_t$$ In my notes, the Stratonovich integral is defined as $$\int^t_0 N_s\circ dB_s=\int^t_0 N_sdB_s+\frac{1}{2}\l......Wed, 12 Aug 2026 12:11:38 -0400https://liberty.io/why-special-limit-of-euler-039-s-number-fails-at-higher-powers.htmlhttps://liberty.io/why-special-limit-of-euler-039-s-number-fails-at-higher-powers.htmlIn the "Calculus 6th edition" by Edwards and Penney, in the chapter of Transcendental functions, there is an interesting question about special limit that leads to the famous Euler's number 2.71828......Wed, 12 Aug 2026 12:06:58 -0400https://liberty.io/conversion-of-a-vector-in-a-cartesian-coordinate-system-to-a-cylindric.htmlhttps://liberty.io/conversion-of-a-vector-in-a-cartesian-coordinate-system-to-a-cylindric.htmlI'm having trouble converting a vector from the Cartesian coordinate system to the cylindrical coordinate system (second year vector calculus) Represent the vector $\mathbf A(x,y,z) = z\ \hat i - ......Wed, 12 Aug 2026 12:02:37 -0400https://liberty.io/positivity-in-algebraic-geometry.htmlhttps://liberty.io/positivity-in-algebraic-geometry.htmlNow I am studying vanishing theorems in algebraic geometry, and I am so curious about why the word $\textbf{Positivity}$ is used. So if somebody can give any intuition about that word, it will be ...Wed, 12 Aug 2026 11:56:34 -0400https://liberty.io/understanding-the-proof-to-egorov-039-s-theorem.htmlhttps://liberty.io/understanding-the-proof-to-egorov-039-s-theorem.htmlTheorem (Egorov). Let $\{f_n\}$ be a sequence of measurable functions converging almost everywhere on a measurable set $E$ to a function $f$. Then, given any $\delta > 0$, there exists a ...Wed, 12 Aug 2026 11:55:14 -0400https://liberty.io/average-smallest-prime-factors.htmlhttps://liberty.io/average-smallest-prime-factors.htmlI looked at the average smallest prime factor (ASPF) for the numbers up to N: $\text{ASPF}(N) = \frac{1}{N-1}\ \Sigma_{k=2}^N \text{SPF}(k)$ ASPF(100) = 13 ASPF(1,000) = 79 ASPR(10,000) = 578 ......Wed, 12 Aug 2026 11:53:40 -0400https://liberty.io/how-to-really-understand-the-tensor-algebra.htmlhttps://liberty.io/how-to-really-understand-the-tensor-algebra.htmlIf $V$ is a vector space over $F$, then we define $T^r_0(V)=V^{\otimes r}$, then we define the algebra of contravariant tensors to be $$T(V)=\bigoplus_{r=0}^\infty T^r_0(V)$$ together with the te......Wed, 12 Aug 2026 11:53:19 -0400https://liberty.io/derivative-of-function-raised-to-a-power-using-the-chain-rule.htmlhttps://liberty.io/derivative-of-function-raised-to-a-power-using-the-chain-rule.htmlHow do I find the derivative of the function $f(x)= (2x+1)^2$? I've tried doing this problem and am not fully sure that I am correct. I found the derivative to be $f'(x) = 8x+4$. Is that correct?...Wed, 12 Aug 2026 11:32:54 -0400https://liberty.io/lusin-spaces-issues-on-the-definition-and-on-spaces-used-in-analysis-t.htmlhttps://liberty.io/lusin-spaces-issues-on-the-definition-and-on-spaces-used-in-analysis-t.htmlI have two questions concerning Lusin spaces. Before the questions, the definition: Definition (Lusin Space): Given a Hausdorff space $(X, \tau)$, that space is said to be Lusin if there is a st......Wed, 12 Aug 2026 11:27:42 -0400https://liberty.io/how-to-find-terminal-point-coordinates-on-a-unit-circle.htmlhttps://liberty.io/how-to-find-terminal-point-coordinates-on-a-unit-circle.htmlHey everyone I am working on a homework assignment which covers unit circles. However I am really confused and having a lot of trouble locating terminal point coordinates. Everything I have read on......Wed, 12 Aug 2026 11:25:06 -0400https://liberty.io/adding-two-polar-vectors.htmlhttps://liberty.io/adding-two-polar-vectors.htmlIs there a way of adding two vectors in polar form without first having to convert them to cartesian or complex form?...Wed, 12 Aug 2026 11:24:12 -0400https://liberty.io/taylor-series-for-exponential-function.htmlhttps://liberty.io/taylor-series-for-exponential-function.htmlThe Taylor series for $e^x = \sum_{i=0}^\infty \frac{x^i}{i!}$. Then as $e^0 = 1$, if one evaluates the Taylor series at $x=0$ we find that $e^0 = \sum_{i=0}^\infty \frac{0^i}{i!} = \frac{0^0}{0!} + \...Wed, 12 Aug 2026 11:19:07 -0400https://liberty.io/how-is-boundary-of-a-set-defined-is-boundary-of-an-open-set-s-included.htmlhttps://liberty.io/how-is-boundary-of-a-set-defined-is-boundary-of-an-open-set-s-included.htmlHow is boundary of a set defined? Is boundary of an open set $S$, included in $S$?...Wed, 12 Aug 2026 11:18:34 -0400https://liberty.io/helix-around-helix-around-circle.htmlhttps://liberty.io/helix-around-helix-around-circle.htmlI'm trying to find the parametric equations for a helix around a helix around a circle (helix on helix on circle) That is: I would like to start with a circle, add a helix around it and a helix aro......Wed, 12 Aug 2026 11:14:48 -0400https://liberty.io/proof-that-every-polynomial-of-odd-degree-has-one-real-root.htmlhttps://liberty.io/proof-that-every-polynomial-of-odd-degree-has-one-real-root.htmlI want to prove that every real polynomial of odd degree has at least one real root, using the intermediate value theorem. Let $P(x) = x^{2n+1} + a_n x^{2n} + . . . + a_0$ for each $a_i \in \math......Wed, 12 Aug 2026 11:12:17 -0400https://liberty.io/liate-how-does-it-work.htmlhttps://liberty.io/liate-how-does-it-work.htmlWhen doing Integration By Parts, I know that using LIATE can be a useful guide most of the time. For those not familiar, LIATE is a guide to help you decide which term to differentiate and which ......Wed, 12 Aug 2026 11:11:33 -0400https://liberty.io/notation-what-is-the-element-wise-max-notation.htmlhttps://liberty.io/notation-what-is-the-element-wise-max-notation.htmlFor example I have 3 vectors. $A_1 = (1, 3, 2, 6)$ $A_2 = (6, 1, 9, 1)$ $A_3 = (3, 8, 4, 0)$ $max A_1 = 6$, $max A_2 = 9$, $max A_3 = 8$ What is the proper max notation for element-wise sit......Wed, 12 Aug 2026 11:00:10 -0400https://liberty.io/calculating-real-and-imaginary-part-of-a-complex-number.htmlhttps://liberty.io/calculating-real-and-imaginary-part-of-a-complex-number.htmlConsider the complex numbers $a = \frac{(1+i)^5}{(1-i)^3}$ and $b = e^{3-\pi i}$. How do I calculate the real and imaginary part of these numbers? What is the general approach to calculate these p......Wed, 12 Aug 2026 10:59:44 -0400https://liberty.io/how-to-factor-quadratic-ax-2-bx-c.htmlhttps://liberty.io/how-to-factor-quadratic-ax-2-bx-c.htmlHow do I shorten this? How do I have to think? $$ x^2 + x - 2$$ The answer is $$(x+2)(x-1)$$ I don't know how to get to the answer systematically. Could someone explain? Does anyone have a link......Wed, 12 Aug 2026 10:58:06 -0400https://liberty.io/1-to-the-power-of-infinity-why-is-it-indeterminate-duplicate.htmlhttps://liberty.io/1-to-the-power-of-infinity-why-is-it-indeterminate-duplicate.htmlI've been taught that $1^\infty$ is undetermined case. Why is it so? Isn't $1*1*1...=1$ whatever times you would multiply it? So if you take a limit, say $\lim_{n\to\infty} 1^n$, doesn't it converg......Wed, 12 Aug 2026 10:48:35 -0400https://liberty.io/triangle-inequality-theorem.htmlhttps://liberty.io/triangle-inequality-theorem.htmltriangle inequality theorem states Sum of any $2$ sides must be greater than $3$rd side. I am doing exercises in my text book. For example, I am checking if $10,3,5$ can form a triangle. $10+3 &g......Wed, 12 Aug 2026 10:46:42 -0400https://liberty.io/measure-theory-for-self-study-duplicate.htmlhttps://liberty.io/measure-theory-for-self-study-duplicate.htmlI have good knowledge of Elementary Real analysis. Now I'd like to study measure theory by myself (self-study). So please give me direction for where to start? Which book is good for starting? I have ...Wed, 12 Aug 2026 10:30:19 -0400https://liberty.io/what-does-it-mean-sequence-with-infinite-range.htmlhttps://liberty.io/what-does-it-mean-sequence-with-infinite-range.htmlI'm trying to understand this phrase Find a sequence with infinite range that converges only to $0$. What does it mean "sequence with infinite range"? Thanks...Wed, 12 Aug 2026 10:23:45 -0400https://liberty.io/what-does-this-subset-symbol-mean.htmlhttps://liberty.io/what-does-this-subset-symbol-mean.htmlI just came across this "subset" symbol in a PDF: $$\Omega \subsetneq T$$ I've never seen it before, and I tried looking for it via Detexify (to no avail). What does it mean?...Wed, 12 Aug 2026 10:21:30 -0400https://liberty.io/finding-parametric-equations-from-lines.htmlhttps://liberty.io/finding-parametric-equations-from-lines.htmlI have a question that reads: Find a parametric equation of each of the following lines: A. $3 x_1 + 4 x_2 = 6$ D. the line through $A=(-2,1)$ parallel to $x = (1,4) + t(3,5)$ E. ......Wed, 12 Aug 2026 10:21:05 -0400https://liberty.io/how-to-find-median-from-a-probability-distribution.htmlhttps://liberty.io/how-to-find-median-from-a-probability-distribution.htmlHaving trouble on something that should be really, really easy. I need to find the median of the following probability distribution...but according to the website I linked below...I'm doing it ...Wed, 12 Aug 2026 10:20:01 -0400https://liberty.io/solving-bernoulli-differential-equation.htmlhttps://liberty.io/solving-bernoulli-differential-equation.htmlHow to solve $$t \frac{dy}{dt} + y = t^4 y^3$$ First I divided by $t$ to get $$\frac{dy}{dt} + \frac{y}{t} = t^3 y^3$$ Then I multiplied through by $y^{-3}$ to get $$y^{-3} \frac{dy}{dt} + \frac1......Wed, 12 Aug 2026 10:17:38 -0400https://liberty.io/is-there-really-no-way-to-integrate-e-x-2.htmlhttps://liberty.io/is-there-really-no-way-to-integrate-e-x-2.htmlToday in my calculus class, we encountered the function $e^{-x^2}$, and I was told that it was not integrable. I was very surprised. Is there really no way to find the integral of $e^{-x^2}$? Grap......Wed, 12 Aug 2026 10:11:45 -0400https://liberty.io/how-to-find-the-volume-of-a-tetrahedron.htmlhttps://liberty.io/how-to-find-the-volume-of-a-tetrahedron.htmlFor this question, how do we find the volume of a tetrahedron in $\Bbb R^4$ if we have a 4 by 3 matrix? The cross product and determinant only work for square matrices. I'm not sure what to do afte......Wed, 12 Aug 2026 10:06:46 -0400https://liberty.io/vector-dot-product-0-for-perpendicular-vectors.htmlhttps://liberty.io/vector-dot-product-0-for-perpendicular-vectors.htmlI'm looking for the intuition behind this property, hopefully one that would be at a high-school level of reasoning. Looking through several of the answers here, I came onto this one, whose answers......Wed, 12 Aug 2026 09:59:40 -0400https://liberty.io/showing-an-isometry-preserves-angles.htmlhttps://liberty.io/showing-an-isometry-preserves-angles.htmlIf we have two surfaces in $\mathbb{R}^3 ,$ say $S_1, S_2 $. And we have an isometry $f:S_1 \rightarrow S_2$. Then we have two curves $\alpha $ and $\beta $ on $S_1$ that intersect at $t=0$. I am t......Wed, 12 Aug 2026 09:58:07 -0400https://liberty.io/when-is-it-allowed-to-take-apart-a-limit-multiplication-of-limits.htmlhttps://liberty.io/when-is-it-allowed-to-take-apart-a-limit-multiplication-of-limits.htmlWhen is it allowed to "take apart" a limit? Here's an example to show what I mean: $\displaystyle\lim_{n\to\infty}\frac{\frac 1 ne^{\frac 1n}(e-1)}{e^{\frac 1 n}-1}=e-1$ since we can "take apart"......Wed, 12 Aug 2026 09:46:09 -0400https://liberty.io/what-do-you-get-when-you-take-the-square-root-of-a-negative-imaginary-.htmlhttps://liberty.io/what-do-you-get-when-you-take-the-square-root-of-a-negative-imaginary-.htmlSimply, what do I get when I take the square root of negative imaginary number? But, it cant be an imaginary number since the answer to the $2^{\mathrm{nd}}$ power must equal the original negative ...Wed, 12 Aug 2026 09:41:40 -0400https://liberty.io/prove-that-sin-x-cos-x-sqrt2-sin-left-x-frac-pi4-right.htmlhttps://liberty.io/prove-that-sin-x-cos-x-sqrt2-sin-left-x-frac-pi4-right.html$$\sqrt2\sin\left(x+\frac\pi4\right)=\sqrt2\left(\sin(x)\cos\left(\frac\pi4\right)+\cos(x)\sin\left(\frac\pi4\right)\right)=\sin(x)+\cos(x)$$ Could you solve it from opposite? $$\sin(x)+\cos (x)=\s......Wed, 12 Aug 2026 09:37:11 -0400https://liberty.io/why-we-use-multiplication-rule-in-an-independent-event-in-probability.htmlhttps://liberty.io/why-we-use-multiplication-rule-in-an-independent-event-in-probability.htmlI know the independent concept and multiplication rules. Example Flip Coin 5 times flip, how many times success of Head? Ans : $\frac{1}{2} \cdot \frac{1}{2} \cdot \frac{1}{2} \cdot \frac......Wed, 12 Aug 2026 09:21:54 -0400https://liberty.io/limit-of-the-ratio-of-consecutive-fibonacci-numbers-duplicate.htmlhttps://liberty.io/limit-of-the-ratio-of-consecutive-fibonacci-numbers-duplicate.htmlI have read in a book that the limit of the ratio of consequent Fibonacci numbers is the golden ratio. However, it was just mentioned thus not justified. So, my question is how would you derive the ...Wed, 12 Aug 2026 09:11:01 -0400https://liberty.io/definition-of-continuity.htmlhttps://liberty.io/definition-of-continuity.htmlIt has been a year or so I took my course of real analysis, still could not understand these two definitions of continuity-(These two definitions are given as chapter 9 and 10 in the classroom reso......Wed, 12 Aug 2026 09:05:49 -0400https://liberty.io/what-is-the-meaning-of-evaluating-the-divergence-at-a-point.htmlhttps://liberty.io/what-is-the-meaning-of-evaluating-the-divergence-at-a-point.htmlReading this first, Divergence (div) is “flux density”—the amount of flux entering or leaving a point. Think of it as the rate of flux expansion (positive divergence) or flux contraction (negative ...Wed, 12 Aug 2026 09:02:27 -0400https://liberty.io/create-unique-number-from-2-numbers.htmlhttps://liberty.io/create-unique-number-from-2-numbers.htmlis there some way to create unique number from 2 positive integer numbers? Result must be unique even for these pairs: 2 and 30, 1 and 15, 4 and 60. In general, if I take 2 random numbers result mu......Wed, 12 Aug 2026 09:01:44 -0400https://liberty.io/intuitive-proof-of-multivariable-changing-of-variables-formula-jacobia.htmlhttps://liberty.io/intuitive-proof-of-multivariable-changing-of-variables-formula-jacobia.htmlWhat is an intuitive proof of the multivariable changing of variables formula (Jacobian) without using mapping and/or measure theory? I think that textbooks overcomplicate the proof. If possible, use ...Wed, 12 Aug 2026 09:00:25 -0400https://liberty.io/understanding-e-and-e-to-the-power-of-imaginary-number.htmlhttps://liberty.io/understanding-e-and-e-to-the-power-of-imaginary-number.htmlHow did the value of $e$ come from compound interest equation. What does the value of $e$ really mean... Capacitors and inductors charge and discharge exponentially, radioactive elements decay ...Wed, 12 Aug 2026 08:54:14 -0400https://liberty.io/suppose-that-a-and-b-are-positive-integers-such-that-a-2-3b-2-show-tha.htmlhttps://liberty.io/suppose-that-a-and-b-are-positive-integers-such-that-a-2-3b-2-show-tha.html...Use these and the Well-Ordering Principle to prove that no such $a$ and $b$ exist. From this it follows that $\sqrt{3} \notin \mathbb{Q}$ I have no idea where well-ordering principle comes in for ...Wed, 12 Aug 2026 08:46:55 -0400https://liberty.io/find-the-infinite-sum-of-the-series-sum-n-1-infty-frac-1-n-2-1.htmlhttps://liberty.io/find-the-infinite-sum-of-the-series-sum-n-1-infty-frac-1-n-2-1.htmlThis is a homework question whereby I am supposed to evaluate: $$\sum_{n=1}^\infty \frac{1}{n^2 +1}$$ Wolfram Alpha outputs the answer as $$\frac{1}{2}(\pi \coth(\pi) - 1)$$ But I have no idea......Wed, 12 Aug 2026 08:44:39 -0400https://liberty.io/why-the-complex-number-6-sin-240-circ-i-cos-240-circ-lies-in-first-qua.htmlhttps://liberty.io/why-the-complex-number-6-sin-240-circ-i-cos-240-circ-lies-in-first-qua.htmlproblem : find the quadrant in which $6(\sin(240^{\circ}) + i \cos(240^{\circ}))$ lies and find the principal argument then rewrite the complex number my attempt : $6(\sin(240^{\circ}) + i \co......Wed, 12 Aug 2026 08:12:35 -0400https://liberty.io/second-derivative-of-a-polar-coordinate.htmlhttps://liberty.io/second-derivative-of-a-polar-coordinate.htmlhttps://snag.gy/buiJve.jpg I'm not sure what the issue is, but I've calculated this several times and have been unable to produce the correct answer. I have followed the procedures. I went to cal......Wed, 12 Aug 2026 08:09:52 -0400https://liberty.io/prove-n-is-o-n-log-n-closed.htmlhttps://liberty.io/prove-n-is-o-n-log-n-closed.htmlIn order to prove that $n$ is $O(n\log n)$, as per my understanding if we have to say $f(n)$ is $O(g(n))$ then $\lim\limits_{n \to \infty} \frac{f(n)}{g(n)}= C$ Then in that case when I am taking......Wed, 12 Aug 2026 07:44:49 -0400https://liberty.io/how-to-intuitively-understand-eigenvalue-and-eigenvector.htmlhttps://liberty.io/how-to-intuitively-understand-eigenvalue-and-eigenvector.htmlI'm learning multivariate analysis and I have learnt linear algebra for two semester when I was a freshman. Eigenvalue and eigenvector is easy to calculate and the concept is not difficult to unde......Wed, 12 Aug 2026 07:41:51 -0400https://liberty.io/to-convert-rotation-vector-to-rotation-matrix-the-rotation-vector-must.htmlhttps://liberty.io/to-convert-rotation-vector-to-rotation-matrix-the-rotation-vector-must.htmlI have a following formula to convert rotation vector($K\in \Bbb R^3$) to rotation matrix ($R \in SO(3)$) where $I$ is an identity matrix and K could be uniquely acquired from the conversion of an ......Wed, 12 Aug 2026 07:41:25 -0400https://liberty.io/what-does-it-mean-to-be-affinely-independent-and-why-is-it-important-t.htmlhttps://liberty.io/what-does-it-mean-to-be-affinely-independent-and-why-is-it-important-t.htmlI was studying linear optimization and i saw the term Affine independence. I came across this http://www.cis.upenn.edu/~cis610/geombchap2.pdf while trying to get a better understanding of the topic. ...Wed, 12 Aug 2026 07:38:35 -0400https://liberty.io/2d-rubik-039-s-cube.htmlhttps://liberty.io/2d-rubik-039-s-cube.htmlThere is a $3\times3$ matrix filled by numbers 1~9 that might look like this $$\begin{bmatrix}3 & 8 & 2 \\ 4 & 1 & 6 \\ 7 & 5 & 9\end{bmatrix}$$ All its rows and columns c......Wed, 12 Aug 2026 07:37:58 -0400https://liberty.io/solve-x-2-5x-3-x-2-x-in-mathbb-r.htmlhttps://liberty.io/solve-x-2-5x-3-x-2-x-in-mathbb-r.htmlSolve $x^2-|5x-3|-x<2,\ \ x\in \mathbb{R} $ I tried $x^2-|5x-3|-x<2$ , case $1$ , $x^2-(5x-3)-x<2,\ x\geq 0 \\ x^2-6x+1<0 \\ 3-2\sqrt2 < 3+2\sqrt2 \\ 0.17<x<5.8\\ $ $x^2......Wed, 12 Aug 2026 07:31:33 -0400https://liberty.io/rate-of-change-of-cylindrical-roll-039-s-volume-as-it-unrolls.htmlhttps://liberty.io/rate-of-change-of-cylindrical-roll-039-s-volume-as-it-unrolls.htmlThis is purely a "for-fun" question. I was minding my own business in the washroom this morning when I began to unroll some toilet paper from the roll, and in typical Breaking Bad fashion (sorry if......Wed, 12 Aug 2026 07:22:31 -0400https://liberty.io/what-did-tao-prove-in-the-paper-almost-all-orbits-of-the-collatz-map-a.htmlhttps://liberty.io/what-did-tao-prove-in-the-paper-almost-all-orbits-of-the-collatz-map-a.htmlLast year, Terence Tao published a paper entitled "Almost All Orbits of the Collatz Map Attain Almost Bounded Values" (via arXiv). In layman's terms, could somebody please explain what this ...Wed, 12 Aug 2026 07:17:53 -0400https://liberty.io/equation-for-line-parallel-to-z-axis-and-intersects-x-axis-at-x-k-y-0-.htmlhttps://liberty.io/equation-for-line-parallel-to-z-axis-and-intersects-x-axis-at-x-k-y-0-.htmlHow would I pick $a,b,c$ to create a line that is parallel to $z$-axis and intersects $x$-axis at point $x=k, y=0, z=0$? $$ ax+by+cz=d $$...Wed, 12 Aug 2026 07:12:32 -0400https://liberty.io/derivative-method-of-proving-of-sqrt-x-ln-x.htmlhttps://liberty.io/derivative-method-of-proving-of-sqrt-x-ln-x.htmlI have a question about a proof I read, linked here, a proof that $\sqrt{x}>\ln x$. The second answerer considers the derivative of the function,$f\left(x\right)=\sqrt{x}-\ln x$. Now the deriv......Wed, 12 Aug 2026 07:09:47 -0400https://liberty.io/why-is-the-maximum-rayleigh-quotient-equal-to-the-maximum-eigenvalue.htmlhttps://liberty.io/why-is-the-maximum-rayleigh-quotient-equal-to-the-maximum-eigenvalue.html(Note: I'm only interested in real-valued matrices here, so I'm using "transpose" and "symmetric" instead of the more general "transjugate" and "Hermitian" in the hope that it will simplify the pro......Wed, 12 Aug 2026 06:57:15 -0400https://liberty.io/roll-1-die-and-square-or-roll-2-dice-and-multiply-which-has-higher-mea.htmlhttps://liberty.io/roll-1-die-and-square-or-roll-2-dice-and-multiply-which-has-higher-mea.htmlI am wondering if there is a fast way to tell which expected value is higher: "Roll 1 die and take the square of the number that comes up or roll 2 dice and multiply them". Thank you!...Wed, 12 Aug 2026 06:46:51 -0400https://liberty.io/circular-arrangements-at-a-round-table.htmlhttps://liberty.io/circular-arrangements-at-a-round-table.htmlQ: If six people, designated as $A,B,...,F,$ are seated about a round table how many different circular arrangements are possible, if arrangements are considered the same when one can be obtained f......Wed, 12 Aug 2026 06:46:31 -0400https://liberty.io/find-the-value-of-k-if-the-roots-of-an-equation-differ-by-2.htmlhttps://liberty.io/find-the-value-of-k-if-the-roots-of-an-equation-differ-by-2.htmlI need some help, Im trying to solve the below but to no avail, would appreciate your guidance :) Find the value of $k$ if the roots of $$3x^2+5x-k=0,$$ differ by two....Wed, 12 Aug 2026 06:43:57 -0400https://liberty.io/sum-of-2-consecutive-integers-is-x-then-their-product-will-be-closed.htmlhttps://liberty.io/sum-of-2-consecutive-integers-is-x-then-their-product-will-be-closed.htmlAccording to a source, the answer is $x^2-\frac{1}{4}$ Please explain...Wed, 12 Aug 2026 06:43:26 -0400https://liberty.io/integral-from-zero-to-zero.htmlhttps://liberty.io/integral-from-zero-to-zero.htmlit seems obvious that this integral is zero and so is the limit but what theorem we are using here? I see it's connected to Riemann sums with an interval=zero Right ? The function $\mathrm{f}$ is ...Wed, 12 Aug 2026 06:42:15 -0400https://liberty.io/is-there-a-way-to-solve-for-an-unknown-in-a-factorial.htmlhttps://liberty.io/is-there-a-way-to-solve-for-an-unknown-in-a-factorial.htmlI don't want to do this through trial and error, and the best way I have found so far was to start dividing from 1. $n! = \text {a really big number}$ Ex. $n! = 9999999$ Is there a way to ...Wed, 12 Aug 2026 06:41:52 -0400https://liberty.io/at-what-rate-is-the-angle-theta-changing-when-10-ft-of-rope-is-out.htmlhttps://liberty.io/at-what-rate-is-the-angle-theta-changing-when-10-ft-of-rope-is-out.htmlA dinghy is pulled toward a dock by a rope from the bow through a ring on the dock 6 ft. above the bow. The rope is hauled in a rate of 2 ft/s. At what rate is the angle $\theta$ changing when 10 ......Wed, 12 Aug 2026 06:31:28 -0400https://liberty.io/how-to-demonstrate-derivability.htmlhttps://liberty.io/how-to-demonstrate-derivability.htmlIf given a function $f$ that I am supposed to demonstrate is derivable, is it enough for me to try to compute its derivative $f'$ and if the result is continuous, can I then state that the orig......Wed, 12 Aug 2026 06:23:22 -0400https://liberty.io/creating-a-bijection-to-check-if-graphs-are-isomorphic.htmlhttps://liberty.io/creating-a-bijection-to-check-if-graphs-are-isomorphic.htmlTo prove that two graphs are isomorphic I was taught to first consider the bijection between the two graphs. I was never taught however the rules when coming up with the bijection. Is my only rule......Wed, 12 Aug 2026 06:20:45 -0400https://liberty.io/how-to-find-critical-points-of-multidimensional-function.htmlhttps://liberty.io/how-to-find-critical-points-of-multidimensional-function.htmlI have this function: $$ f(x,y) = x^2 - 2xy+ 4y^3$$ I calculated the gradient without problems: $$\nabla f(x,y) = \left(2x-2y , -2x + 12y^3\right)^T$$ This is where kind of got stuck, I know f......Wed, 12 Aug 2026 06:16:48 -0400https://liberty.io/find-the-number-of-ordered-pairs-of-positive-integers-x-y-that-satisfy.htmlhttps://liberty.io/find-the-number-of-ordered-pairs-of-positive-integers-x-y-that-satisfy.htmlafter some simplification, we get (2004y)/(y-2004)=x and by looking at the prime factorization of 2004 we observe there are only 12 ways in which 2004 can be expressed as the product of two intege......Wed, 12 Aug 2026 06:07:12 -0400https://liberty.io/prove-that-a-set-is-countable.htmlhttps://liberty.io/prove-that-a-set-is-countable.htmlShow using a proper theorem that the set {2, 3, 4, 8, 9, 16, 27, 32, 64, 81, … } is a countable set. Im lost, this is for school, but there is a huge language barrier between students and professo......Wed, 12 Aug 2026 06:07:03 -0400https://liberty.io/finding-the-points-of-intersection-in-polar-equations.htmlhttps://liberty.io/finding-the-points-of-intersection-in-polar-equations.htmlI must find the area of the region that lies inside the first curve and outside the second curve for: $ r^2 = 8cos2\theta, r = 2$ How do I find $a$ and $b$? My solution manual tells me that the c......Wed, 12 Aug 2026 06:05:45 -0400https://liberty.io/calculating-flux-across-s-a-graph-of-function-g-x-y-over-unit-disc-in-.htmlhttps://liberty.io/calculating-flux-across-s-a-graph-of-function-g-x-y-over-unit-disc-in-.htmlLet $~F = \langle x,y,z \rangle~$ be a vector field on $ R^3$ . Let $S$ be the graph of $ g(x,y)=1- y^2 + x^2 $ over the unit disc $D$ in $ R^2$ and let $S$ be equipped with the upward orientation. ...Wed, 12 Aug 2026 05:54:52 -0400https://liberty.io/proving-absolute-value-inequalities.htmlhttps://liberty.io/proving-absolute-value-inequalities.htmlI need help proving the last two cases for the following inequality: $\bigl|\lvert x\rvert-\lvert y\rvert\bigr| \le \lvert x-y\rvert$. Case 1: $x > 0$ and $y > 0$: the inequality simplif......Wed, 12 Aug 2026 05:49:36 -0400https://liberty.io/why-combining-two-quadratic-equations-of-a-circle-and-a-parabola-creat.htmlhttps://liberty.io/why-combining-two-quadratic-equations-of-a-circle-and-a-parabola-creat.htmlWe have a parabola and a circle with the following equations and their graph placed at the end of my question. Parabola: $y^2 = 4x -4$ Circle: $(x-2)^2 + y^2 = 9$ My goal was to calculate their ...Wed, 12 Aug 2026 05:39:50 -0400