https://liberty.ioCurated star highlights with fresh energy.en-usSun, 06 Sep 2026 00:35:45 -0400https://liberty.io/what-is-the-geometric-meaning-of-singular-matrix.htmlhttps://liberty.io/what-is-the-geometric-meaning-of-singular-matrix.htmlCould anyone help explain what is the geometric meaning of singular matrix ? What's the difference between singular and non-singular matrix ? I know the definition, but couldn't understand it very ......Sun, 06 Sep 2026 04:33:11 -0400https://liberty.io/what-is-the-correct-way-to-read-f-circ-g.htmlhttps://liberty.io/what-is-the-correct-way-to-read-f-circ-g.htmlLet $f : X \to Y$ and $g : Y \to Z$ be functions. We define the composition $g \circ f : X \to Z$ by $g \circ f(x) = g(f(x))$ for each $x \in X$. I have also heard the composition read out like th......Sun, 06 Sep 2026 04:32:23 -0400https://liberty.io/what-is-the-definition-of-a-positive-integer.htmlhttps://liberty.io/what-is-the-definition-of-a-positive-integer.htmlI am reading the book "The Number-System of Algebra (2nd edition)". At the starting of page-4 the author writes: A positive integer is a symbol for the number of things in a group of distinct ...Sun, 06 Sep 2026 04:27:08 -0400https://liberty.io/should-i-use-ds-or-dr.htmlhttps://liberty.io/should-i-use-ds-or-dr.htmlThe question states "Integrate $f(x, y, z) = x + \sqrt y -z^2$ over the patch $C = C_1 + C_2$ and they proceed to give $C_1$ and $C_2$ as $r_1(t)$ and $r_1(t)$. I've been dealing with 2 forms of line ...Sun, 06 Sep 2026 04:25:50 -0400https://liberty.io/gre-math-subject-test-closed.htmlhttps://liberty.io/gre-math-subject-test-closed.htmlI am studying for GRE Math. I am looking for specific tips. What types of questions usually come up? Does anyone know any tricks (e.g. integration tricks) that might be helpful? Which theorems are ...Sun, 06 Sep 2026 04:17:09 -0400https://liberty.io/why-is-x-5-6x-3-irreducible.htmlhttps://liberty.io/why-is-x-5-6x-3-irreducible.htmlIn my math course, it is written that since over $\mathbb{F}_5[X]$, $P$ has no root in $\mathbb{F}_{25}$, it is irreducible over $\mathbb{Q}[X]$ by Gauss Lemma. I don't really understand… If we p......Sun, 06 Sep 2026 04:14:58 -0400https://liberty.io/the-introduction-rule-for-implication.htmlhttps://liberty.io/the-introduction-rule-for-implication.htmlThe introduction rule for implication says that you can prove a statement of the form $P \implies Q$ by assuming $P$ and deducing $Q$. (Btw. I don't know why it's called an introduction rule - you ......Sun, 06 Sep 2026 04:14:02 -0400https://liberty.io/indefinite-integral-of-absolute-value-of-x-is-there-a-closed-form-solu.htmlhttps://liberty.io/indefinite-integral-of-absolute-value-of-x-is-there-a-closed-form-solu.htmlBeen searching the net for awhile and everything just comes back about doing the definite integral. So just thought to ask here. Title says it all. Is there a closed form solution for the indefin......Sun, 06 Sep 2026 04:11:30 -0400https://liberty.io/what-is-null-quantification.htmlhttps://liberty.io/what-is-null-quantification.htmlI want to know the definition of null quantification and when and why we need to use it. In a book I was learning, it says: "establish rules for null quantification that we can use when a ...Sun, 06 Sep 2026 03:58:59 -0400https://liberty.io/what-exactly-happens-when-i-do-a-bit-rotation.htmlhttps://liberty.io/what-exactly-happens-when-i-do-a-bit-rotation.htmlI'm sitting in my office and having difficulties to get to know that exactly happens, when I do a bit rotation of a binary number. An example: I have the binary number 1110. By doing bit rotatio......Sun, 06 Sep 2026 03:56:15 -0400https://liberty.io/odds-of-drawing-an-ace-on-the-first-draw-or-a-two-on-the-second.htmlhttps://liberty.io/odds-of-drawing-an-ace-on-the-first-draw-or-a-two-on-the-second.htmlI'm working on a puzzle which describes a casino game where you draw 13 cards. You win if your first card is an ace, or your second is a two, or... all the way up to the 13th draw being a king. In ...Sun, 06 Sep 2026 03:51:52 -0400https://liberty.io/applications-of-the-mean-value-theorem.htmlhttps://liberty.io/applications-of-the-mean-value-theorem.htmlWhat are some interesting applications of the Mean Value Theorem for derivatives? Both the 'extended' or 'non-extended' versions as seen here are of interest. So far I've seen some trivial applica......Sun, 06 Sep 2026 03:24:47 -0400https://liberty.io/interview-question-rolling-dice-of-posterior-probability.htmlhttps://liberty.io/interview-question-rolling-dice-of-posterior-probability.htmlYou have two 6-sided dice. You roll them and get a total of 10. You roll again and get a total of 9. If you get paid 1$ for each 5 that is rolled what is your expected winnings？ I guess it is the ...Sun, 06 Sep 2026 03:05:23 -0400https://liberty.io/sketch-the-graph-of-x-3-27.htmlhttps://liberty.io/sketch-the-graph-of-x-3-27.htmlNot sure if this is the right way to solve. The question ask to sketch the graph of each function by transforming the graph of an appropriate function of the form $y=x^n$. Indicate all $x$- and $......Sun, 06 Sep 2026 03:01:25 -0400https://liberty.io/peterson-graph-non-hamiltonian-proof-explanation.htmlhttps://liberty.io/peterson-graph-non-hamiltonian-proof-explanation.htmlI'm working on graph theory and I'm trying to find a generalised elegant proof to non-Hamiltonian graphs. I stumbled onto this proof from D. West, which is simple, but I'm having trouble understand......Sun, 06 Sep 2026 02:55:11 -0400https://liberty.io/how-can-i-find-the-gradient-vector-for-this-function.htmlhttps://liberty.io/how-can-i-find-the-gradient-vector-for-this-function.html$ x^4 + y^4 + z^4 = x^2 + y^2 +z^2$ How can I find the gradient vector of $f(x,y,z)$ at $(1,1,1)$? Can I rewrite the equation by just moving everything to one side? So: $x^4 - x^2 + y^4 -y^2 +z^......Sun, 06 Sep 2026 02:54:40 -0400https://liberty.io/differentiability-implies-continuity-a-question-about-the-proof.htmlhttps://liberty.io/differentiability-implies-continuity-a-question-about-the-proof.htmlI have a question, to aid my understanding, about the proof that differentiability implies continutity.$\mathstrut$ Differentiability Definition When we say a function is differentiable at $x_0......Sun, 06 Sep 2026 02:48:18 -0400https://liberty.io/the-jacobian-of-x-y-mapsto-x-y-2-y-x-2-under-the-substitution-u-x-y-2-.htmlhttps://liberty.io/the-jacobian-of-x-y-mapsto-x-y-2-y-x-2-under-the-substitution-u-x-y-2-.htmlI am given the map $(x,y)\mapsto (x+y^2,y+x^2)$. I am unable to find the Jacobian by making the substitution $u=x+y^2$ and $v=y+x^2$. Any hints would be appreciated. (I am trying to find whether t......Sun, 06 Sep 2026 02:47:00 -0400https://liberty.io/how-many-elements-of-a-3x3-rotation-matrix-are-redundant.htmlhttps://liberty.io/how-many-elements-of-a-3x3-rotation-matrix-are-redundant.htmlI read the following and got curious: A rotation matrix is an array of nine numbers. These are subject to the six norm and orthogonality constraints, so only three degrees of freedom are lef......Sun, 06 Sep 2026 02:42:02 -0400https://liberty.io/how-to-change-the-sign-of-the-dirac-delta-function-argument.htmlhttps://liberty.io/how-to-change-the-sign-of-the-dirac-delta-function-argument.htmlHow to proof the latter equality of, $$ f(a) = \int f(x)\,\delta(x-a)\,dx =\int f(x)\,\delta(a-x)\,dx. $$...Sun, 06 Sep 2026 02:30:05 -0400https://liberty.io/moment-generating-function-of-sample-mean-of-bernoulli-random-variable.htmlhttps://liberty.io/moment-generating-function-of-sample-mean-of-bernoulli-random-variable.htmlLet $p \in (0,1)$ and $n \in \mathbb{N}$. We consider a sample of $n$ i.i.d. Bernoulli variables $X_1,\dots,X_n$ with parameter p. Computer $E[e^{\lambda\bar{X_n}}]$ such that $\bar{X_n}= \frac{1}......Sun, 06 Sep 2026 02:30:01 -0400https://liberty.io/proper-normal-subgroup.htmlhttps://liberty.io/proper-normal-subgroup.htmlDefinition: A subgroup $N$ is a proper normal subgroup of a group $G$ IFF $Ng=gN \; \; \forall g \in G$ A proper normal subgroup $N$ of $G$ is a normal subgroup $N$ of $G$ that is not the whole of......Sun, 06 Sep 2026 02:29:04 -0400https://liberty.io/calculus-percentage-of-an-interval-for-which-a-function-is-increasing.htmlhttps://liberty.io/calculus-percentage-of-an-interval-for-which-a-function-is-increasing.htmlSo I have a question that asks, Given $f(x) = 7 + 10x ^ 3 - 5x ^ 4 - 2x ^ 5$ calculate the percentage of the interval $[-6, 4]$ on which $f(x)$ is increasing. So from this I determined that I m......Sun, 06 Sep 2026 02:28:40 -0400https://liberty.io/what-are-all-the-elements-of-a-5.htmlhttps://liberty.io/what-are-all-the-elements-of-a-5.htmlWhat are all the elements of $A_5?$ $$\{1,(1,2,3),(1,3,2),(1,2,4),(1,4,2),(1,2,5),(1,5,2),(1,3,4),(1,4,3),(1,3,5),(1,5,3),(1,4,5),(1,5,4),(2,3,4),(2,4,3),(2,3,5),(2,5,3),(2,4,5),(2,5,4)(3,4,5),(3,......Sun, 06 Sep 2026 02:08:57 -0400https://liberty.io/quad-to-quad-transformation.htmlhttps://liberty.io/quad-to-quad-transformation.htmlI am trying to find a transformation that maps each internal point of my first quadrilateral domain into each point of my second quadrilateral domain. Please see the following image: Each domain is ...Sun, 06 Sep 2026 02:04:27 -0400https://liberty.io/inversely-proportional-and-inversely-related.htmlhttps://liberty.io/inversely-proportional-and-inversely-related.htmlSuppose that we have a formula like $y=\frac{1}{\sqrt{x}}$. Is it correct to say that y is inversely proportional to x? what about y is inversely related to x? If not, what other phrase should we use?...Sun, 06 Sep 2026 02:04:01 -0400https://liberty.io/double-integral-over-area-bounded-by-ellipse-closed.htmlhttps://liberty.io/double-integral-over-area-bounded-by-ellipse-closed.htmlEvaluate the double integral of $\int \int(x+y)^{2} dxdy$ over the area bounded by the ellipse $(\frac{x}{a})^2+(\frac{y}{b})^2=1$ Please check answer = $\frac{ab\pi}{4}(a^{2}+b^{2})$...Sun, 06 Sep 2026 02:03:46 -0400https://liberty.io/quadratic-equation-has-integral-roots.htmlhttps://liberty.io/quadratic-equation-has-integral-roots.htmlFind all positive numbers $p$ for which the equation $x^2+px+3p = 0$ has integral roots. We have by the quadratic formula $$x = \dfrac{-p \pm \sqrt{p^2-12p}}{2}.$$ Thus, $p^2-12p = p(p-12)$ must b......Sun, 06 Sep 2026 01:59:57 -0400https://liberty.io/vectors-representing-the-principal-axes-of-an-ellipse.htmlhttps://liberty.io/vectors-representing-the-principal-axes-of-an-ellipse.htmlThe principal axes of the ellipse given by the equation $ 9x^2+6y^2-4xy=4$ are along the directions of the vectors a) $2i+3j$ and $3i-2j$ b) $2i+j$ and $i-2j$ c) $3i+2j$ and $2i-3j$ d) $i+2j$ and ......Sun, 06 Sep 2026 01:57:41 -0400https://liberty.io/how-to-make-a-given-matrix-symmetric.htmlhttps://liberty.io/how-to-make-a-given-matrix-symmetric.htmlGiven the real square block matrix $ M = \begin{pmatrix} A & D^t & 0\\ D & c E & c F\\ 0 & H & J \end{pmatrix} $ how can I make it symmetric? Note that the dimension of $......Sun, 06 Sep 2026 01:46:28 -0400https://liberty.io/experimental-analysis-of-an-algorithm-how-to-prove-that-the-graph-is-o.htmlhttps://liberty.io/experimental-analysis-of-an-algorithm-how-to-prove-that-the-graph-is-o.htmlThis question is probably stupid, but I've been trying to figure this out for hours and I still couldn't find anything about it. Probably I'm just too lost. So basically, I'm analysing an algorithm......Sun, 06 Sep 2026 01:36:05 -0400https://liberty.io/probability-of-not-occurring-either-a-or-b.htmlhttps://liberty.io/probability-of-not-occurring-either-a-or-b.htmlIn an event, $P(A)=0.6$, $P(B)=0.2$, $P(A ∩ B)=0.1$. what is the probability of not occurring either A or B. I am facing problem with the language. I don't understand what "not occurring either ......Sun, 06 Sep 2026 01:24:53 -0400https://liberty.io/find-the-ratio-of-area-to-volume-of-a-partial-cylinder.htmlhttps://liberty.io/find-the-ratio-of-area-to-volume-of-a-partial-cylinder.htmlConsider a cylindrical container of negligible thickness with radius $r$ and height $h$, placed sideways containing some liquid (with volume $V$) inside as shown below. Define a function $f(r,h,V)......Sun, 06 Sep 2026 01:21:58 -0400https://liberty.io/how-to-find-the-determinant-of-a-5x5-matrix.htmlhttps://liberty.io/how-to-find-the-determinant-of-a-5x5-matrix.htmlHow do I find the determinant of this? $$\begin{bmatrix} 0& 6& −2& −1& 5\\ 0& 0& 0& −9& −7\\ 0& 15& 35& 0& 0\\ 0 &−1 &−11& −2& 1\\ −2......Sun, 06 Sep 2026 01:21:12 -0400https://liberty.io/proving-that-the-cone-is-not-a-smooth-manifold.htmlhttps://liberty.io/proving-that-the-cone-is-not-a-smooth-manifold.htmlShow that the cone given by $C = \{(x, y, z) \in \mathbb{R}^3 \mid z = \sqrt{x^2 + y^2}\}$ is not a smooth manifold In the definition I'm using of a smooth manifold, each point $x \in M \subseteq \...Sun, 06 Sep 2026 01:19:11 -0400https://liberty.io/show-that-the-graph-of-y-x-3-sin-pi-x-extends-to-a-smooth-arc.htmlhttps://liberty.io/show-that-the-graph-of-y-x-3-sin-pi-x-extends-to-a-smooth-arc.htmlHere's the problem: Let $y(x)$ be a real-valued function defined on the interval $x\in [0,1]$ by means of the equation $$y(x)= \left\{ \begin{array}{lr} x^3\sin(\frac{\pi}{x}) \;\;\;\ma......Sun, 06 Sep 2026 01:17:25 -0400https://liberty.io/pythagorean-triples-and-perfect-squares.htmlhttps://liberty.io/pythagorean-triples-and-perfect-squares.htmlThis problem is giving me difficulty: Show that in any Pythagorean triple there exist at most a single perfect square So far I've been working with the equations for primitive Pythagorean Tripl......Sun, 06 Sep 2026 01:16:15 -0400https://liberty.io/trigonometric-functions-sin-cos-tan-differences-actual-applications.htmlhttps://liberty.io/trigonometric-functions-sin-cos-tan-differences-actual-applications.htmlBeforehand, I want to point out that I'm studying some basic geometric function and I'm pretty behind with this topic, hence may write something not too clever or my question might be silly; I ask ......Sun, 06 Sep 2026 01:10:03 -0400https://liberty.io/derivative-of-softmax-loss-function.htmlhttps://liberty.io/derivative-of-softmax-loss-function.htmlI am trying to wrap my head around back-propagation in a neural network with a Softmax classifier, which uses the Softmax function: \begin{equation} p_j = \frac{e^{o_j}}{\sum_k e^{o_k}} \end{equat......Sun, 06 Sep 2026 01:02:36 -0400https://liberty.io/solve-f-x-f-2x-2-f-2x-3-x.htmlhttps://liberty.io/solve-f-x-f-2x-2-f-2x-3-x.htmlThe full question is to solve $f(x)f(2x^2) = f(2x^3+x)$, prove there are at most one solution per degree of $f(x)$ and then find all the solutions. (Here $f(x)$ is assumed to be polynomial.) So fa......Sun, 06 Sep 2026 01:00:16 -0400https://liberty.io/connecting-midpoints-of-sides-of-a-triangle.htmlhttps://liberty.io/connecting-midpoints-of-sides-of-a-triangle.htmlIn triangle $\triangle ABC$, $AB=8$, $BC=14$ and $CA=10$. Points $M$, $N$, and $P $ are the midpoints of sides $AB$, $BC$, and $CA$, respectively. If $M$, $N$, and $P$ are connected to form a trian......Sun, 06 Sep 2026 00:58:34 -0400https://liberty.io/brownian-motion-characteristic-function.htmlhttps://liberty.io/brownian-motion-characteristic-function.htmlLet me remind first the construction of Brownian motion. Fix a vector $x \in \mathbb{R}^n$ and define $p(t,x,y) := (2\pi t )^{-\frac{n}{2}} \cdot \exp{\left( - \frac{|x-y|^2}{2t} \right)},$ for $y......Sun, 06 Sep 2026 00:36:49 -0400https://liberty.io/determine-all-functions-f-satisfying-the-functional-relation-f-x-f-lef.htmlhttps://liberty.io/determine-all-functions-f-satisfying-the-functional-relation-f-x-f-lef.htmlThe question is : Determine all functions f satisfying the functional relation $f(x)+f\left(\frac{1}{1-x}\right)=\frac{2(1-2x)}{x(1-x)}$ where $x$ is a real number . Also $x$ is not equal to $0$ or......Sun, 06 Sep 2026 00:35:02 -0400https://liberty.io/evaluating-int-0-2-tan-1-pi-x-tan-1-x-mathrm-d-x.htmlhttps://liberty.io/evaluating-int-0-2-tan-1-pi-x-tan-1-x-mathrm-d-x.htmlHint given: Write the integrand as an integral. I'm supposed to do this as double integration. My attempt: $$\int_0^2 [\tan^{-1}y]^{\pi x}_{x}$$ $$= \int_0^2 \int_x^{\pi x} \frac { \mathrm{d}y \...Sun, 06 Sep 2026 00:35:00 -0400https://liberty.io/what-does-the-long-039-039-mean-duplicate.htmlhttps://liberty.io/what-does-the-long-039-039-mean-duplicate.htmlI have often seen the long '|' symbol with a subscript expression afterward. What does this mean in mathematics? Here is an example I found from Wikipedia: $$\large\left.\frac{dy}{dx}\right|_{x=......Sun, 06 Sep 2026 00:28:54 -0400https://liberty.io/what-does-mathrm-re-x-mean.htmlhttps://liberty.io/what-does-mathrm-re-x-mean.htmlI see this all the time in Mathematica output as well as in text, such as near the top of the Wikipedia Beta function page....Sun, 06 Sep 2026 00:27:28 -0400https://liberty.io/how-to-find-a-data-value-using-only-standard-deviation.htmlhttps://liberty.io/how-to-find-a-data-value-using-only-standard-deviation.htmlI have a homework problem with the following data values: x, 17, 24, 13, 21. I need to find x, but I do not have the mean. I only have the standard deviation, which is square root of 14, or 3.745. ......Sun, 06 Sep 2026 00:20:07 -0400https://liberty.io/calculus-of-angular-and-linear-velocity.htmlhttps://liberty.io/calculus-of-angular-and-linear-velocity.htmlI have a question regarding the following forms in the case that both angular velocity and radius change per $dt$. Wiki defines angular velocity in two dimensions as: $$ \omega = \frac {d \phi} {d......Sun, 06 Sep 2026 00:11:32 -0400https://liberty.io/does-set-difference-distribute-over-set-intersection.htmlhttps://liberty.io/does-set-difference-distribute-over-set-intersection.htmlI am asked to prove that, if $A, B$ and $C$ are sets, then $$A-(B\cap C)=(A-B)\cap(A-C).$$ However, I think that either I have made an error in my working, or the wording of the problem contains a ...Sun, 06 Sep 2026 00:07:39 -0400https://liberty.io/if-two-one-sided-limits-exist-the-two-sided-limit-exists-duplicate.htmlhttps://liberty.io/if-two-one-sided-limits-exist-the-two-sided-limit-exists-duplicate.htmlprove that if the limit as x approaches a from the positive side of the x axis, f(x) = L, and if the limit as x approaches a from the negative side of the x axis, f(x) = L, then the limit as x appr......Sun, 06 Sep 2026 00:05:50 -0400https://liberty.io/gossip-problem-proof-by-induction.htmlhttps://liberty.io/gossip-problem-proof-by-induction.htmlQuestion Suppose there are $n$ people in a group, each aware of a scandal no one else in the group knows about. These people communicate by telephone; when two people in the group talk, they ...Sun, 06 Sep 2026 00:03:54 -0400https://liberty.io/how-to-find-the-arc-measure-of-the-arc-cut-by-one-side-the-of-a-circum.htmlhttps://liberty.io/how-to-find-the-arc-measure-of-the-arc-cut-by-one-side-the-of-a-circum.htmlFirst, circumscribed means inside a circle, right? What does it exactly mean by cut by one side? A regular hexagon has side angles each $120 ^o$ so it has $240^o$ arc measure. But why is the answer......Sat, 05 Sep 2026 23:54:44 -0400https://liberty.io/a-confusion-about-the-second-connection-axiom-of-euclidean-geometry.htmlhttps://liberty.io/a-confusion-about-the-second-connection-axiom-of-euclidean-geometry.htmlIn the book of Foundations of Geometry by Hilbert, at page 2, it is stated that I, 1. Two distinct points A and B always completely determine a straight line a. We write AB = a or BA = a. ......Sat, 05 Sep 2026 23:42:45 -0400https://liberty.io/asymptotic-expansion-of-tanh-at-infinity.htmlhttps://liberty.io/asymptotic-expansion-of-tanh-at-infinity.htmlDoes $\tanh(x)$ have an asymptotic expansion for $x \rightarrow \infty$?...Sat, 05 Sep 2026 23:40:20 -0400https://liberty.io/the-foci-of-an-ellipse.htmlhttps://liberty.io/the-foci-of-an-ellipse.htmlIn Wikipedia the definition of and ellipse is the following. An ellipse is a set of points, such that for any point $P$ of the set, the sum of the distances $|PF_{1}|, |PF_{2}|$ to two fixed ...Sat, 05 Sep 2026 23:39:32 -0400https://liberty.io/the-high-power-integral.htmlhttps://liberty.io/the-high-power-integral.htmlIm trying to solve the indefinite integral $$\int\frac{x}{(x^2+4)^3} \, \mathrm{d}x $$ I tried applying polynimial division and breaking to partial fractions but it didnt help...are there any ot......Sat, 05 Sep 2026 23:33:15 -0400https://liberty.io/solving-a-fractional-quadratic-equation-problem-by-completing-the-squa.htmlhttps://liberty.io/solving-a-fractional-quadratic-equation-problem-by-completing-the-squa.htmlI have the following problem to solve using the method of completing the square. $$2x^2-3x-1 = 0$$ Here is where I've gotten to so far on this problem. $$2x^2-3x = 1$$ $$x^2-\frac{3}{2}x = \frac{1......Sat, 05 Sep 2026 23:23:22 -0400https://liberty.io/if-you-know-all-angles-and-the-area-of-the-triangle-how-do-you-find-th.htmlhttps://liberty.io/if-you-know-all-angles-and-the-area-of-the-triangle-how-do-you-find-th.htmlIf we know the angles and area of the triangle, what would be the formula to find the side lengths? For example, we know the area is 400 ft and the angles are 40, 30, and 110 degrees, how could we ......Sat, 05 Sep 2026 23:18:15 -0400https://liberty.io/why-the-inverse-of-a-matrix-involves-division-by-the-determinant.htmlhttps://liberty.io/why-the-inverse-of-a-matrix-involves-division-by-the-determinant.htmlI was studying inverse matrix. Suddenly I stumbled on the inverse of 3×3 K. And it involved a division by the determinant (Well, only with numbers it was). And it was also said, about involving a ...Sat, 05 Sep 2026 22:57:22 -0400https://liberty.io/integral-center-of-mass-of-a-right-angled-triangle.htmlhttps://liberty.io/integral-center-of-mass-of-a-right-angled-triangle.htmlHow to find the center of mass in the $x$ direction? (answer $h/3$) $$dm=\sigma (d-y)\,\mathrm dx$$ $$\frac{hx}{d}=y$$ $$\begin{eqnarray} x_{cm}&=&\frac{2}{dh\sigma} \int x \sigma\left(d-......Sat, 05 Sep 2026 22:44:30 -0400https://liberty.io/what-does-the-associative-property-actually-mean.htmlhttps://liberty.io/what-does-the-associative-property-actually-mean.htmlI look at $(a+b)+c = a+(b+c)$ for $a,b,c \in \mathbb{R}$ and think this tells me if I see three addends and two of them are in parentheses I can shift them without changing the sum. It obvious, at ......Sat, 05 Sep 2026 22:43:31 -0400https://liberty.io/if-the-determinant-of-a-coefficient-matrix-is-0-can-you-use-inverse-ma.htmlhttps://liberty.io/if-the-determinant-of-a-coefficient-matrix-is-0-can-you-use-inverse-ma.htmlIf the determinant of a coefficient matrix is 0, can you use inverse matrices to solve the system of equations? Explain your answer. I believe the answer is no, because if our determinant is equal......Sat, 05 Sep 2026 22:38:11 -0400https://liberty.io/where-does-the-symbol-for-a-partial-deriviate-come-from.htmlhttps://liberty.io/where-does-the-symbol-for-a-partial-deriviate-come-from.htmlDoes anybody know where the symbol $\partial$ comes from? (preferably with sources or with a document where it was used first) Symbol in context: $$f\colon \mathbb{R}^2\rightarrow \mathbb{R}$$ ......Sat, 05 Sep 2026 22:37:02 -0400https://liberty.io/why-is-arctan-x-x-x-3-3-x-5-5-x-7-7-dots.htmlhttps://liberty.io/why-is-arctan-x-x-x-3-3-x-5-5-x-7-7-dots.htmlWhy is $\arctan(x)=x-x^3/3+x^5/5-x^7/7+\dots$? Can someone point me to a proof, or explain if it's a simple answer? What I'm looking for is the point where it becomes understood that trigonometric ...Sat, 05 Sep 2026 22:21:22 -0400https://liberty.io/find-the-length-of-the-parametric-curve.htmlhttps://liberty.io/find-the-length-of-the-parametric-curve.htmlI am asked to find the length of the parametric curve and given: $x = 5t,\ y=4t^2,\ z = 3t^2, 0 < t < 2 $ Following the formula for length of a parametric curve I got : $L = \int_0^2 \sqrt......Sat, 05 Sep 2026 22:17:37 -0400https://liberty.io/common-chord-of-two-circles.htmlhttps://liberty.io/common-chord-of-two-circles.htmlFind length of the common chord of two circles of radii 15 cm and 20 cm, distance between the centers being 25cm. I applied the formula $\frac{2r_1r_2}{d}$, d being distane between the centers. An......Sat, 05 Sep 2026 22:16:55 -0400https://liberty.io/applied-math-foundational-books.htmlhttps://liberty.io/applied-math-foundational-books.htmlI have a BA in mathematics from a pretty good school, where I effectively exhausted the mathematics sequence. The sequence mostly focused on pure math (including measure theoretic real analysis an......Sat, 05 Sep 2026 22:15:34 -0400https://liberty.io/find-6-s-complement-of-n-268-10.htmlhttps://liberty.io/find-6-s-complement-of-n-268-10.htmlQuestion Find $6^{s}$ complement of $N=(268)_{10}$ My approach First i converted $N=(268)_{10}$ to $(1124)_{6}$ then , i found the $6^{s}$ complement by subtrating $N$ from $r^{n}$ where $r$......Sat, 05 Sep 2026 22:09:55 -0400https://liberty.io/on-the-meaning-of-a-linear-combination-of-simplexes.htmlhttps://liberty.io/on-the-meaning-of-a-linear-combination-of-simplexes.htmlI'm struggling a bit to understand the concept of a chain in Geometry/Topology, as a linear combination of simplexes, and even more to understand it geometrically (if it possible). So, let's start ......Sat, 05 Sep 2026 22:06:38 -0400https://liberty.io/how-to-find-the-euclidean-norm-of-a-complex-number.htmlhttps://liberty.io/how-to-find-the-euclidean-norm-of-a-complex-number.htmlI would like to know how to find the Euclidean norm of a complex number, like $10+i$ or $2-i$? I would appreciate a clear and easy explanation with the necessary details....Sat, 05 Sep 2026 22:01:48 -0400https://liberty.io/amount-of-work-required-for-pulling-rope-problem.htmlhttps://liberty.io/amount-of-work-required-for-pulling-rope-problem.html50 m rope with 8 millimeters in diameter is dangling from an edge. density of rope =40 g/m. how much work to pull it up to edge? // I've seen different variations of this problem, but I am unsure ......Sat, 05 Sep 2026 21:52:05 -0400https://liberty.io/removing-a-fraction-in-the-denominator.htmlhttps://liberty.io/removing-a-fraction-in-the-denominator.htmlWhat steps do I need to take to simplify $\frac{1}{\frac{1}{\sqrt{2}}}$ to $\frac{\sqrt{2}}{1}$? Can you please explain the steps i need to take in this problem and explain generally how to remove ...Sat, 05 Sep 2026 21:48:33 -0400https://liberty.io/help-understanding-the-span-of-a-set-in-r-2.htmlhttps://liberty.io/help-understanding-the-span-of-a-set-in-r-2.htmlI feel so lost in this section regarding vector spaces, sub spaces, spanning sets and basis. I understand the basic concepts regarding vectors where basically a vector gives magnitude and directio......Sat, 05 Sep 2026 21:47:14 -0400https://liberty.io/multiplying-three-binomials.htmlhttps://liberty.io/multiplying-three-binomials.htmlIs there any known way to multiply three binomials using a method similar to the FOIL method? I have searched the internet and have not found any such method....Sat, 05 Sep 2026 21:45:06 -0400https://liberty.io/difference-between-equivalent-matrices-and-similar-matrices.htmlhttps://liberty.io/difference-between-equivalent-matrices-and-similar-matrices.htmlWe say that $A$ and $B$ are equivalent if there are $P$ and $Q$ invertible s.t. $$A=PFQ^{-1}.$$ We say that they are similar if there is $P$ invertible s.t. $$A=PBP^{-1}.$$ In wikipedia, they say......Sat, 05 Sep 2026 21:38:57 -0400https://liberty.io/definition-of-pointwise-convergence.htmlhttps://liberty.io/definition-of-pointwise-convergence.htmlI'm reading about the definition of exponential function on $\Bbb R$ and I came across a definition of pointwise convergence which I don't understand: We say that a sequence of functions $(f_n)_n$ ...Sat, 05 Sep 2026 21:32:30 -0400https://liberty.io/what-does-s-t-mean.htmlhttps://liberty.io/what-does-s-t-mean.htmlEnglish is my second language and I have a question. What does "s.t." mean? $ \text{min} \quad f(x) = (x_1−2)^2+(x_2−1)^2 $ $ \text{s.t.}\qquad g_{1}(x) = x_{1} - 2x_{2} + 1 = 0 $ $ \qquad\qquad......Sat, 05 Sep 2026 21:28:22 -0400https://liberty.io/equation-for-a-smooth-step-function.htmlhttps://liberty.io/equation-for-a-smooth-step-function.htmlhttp://i.stack.imgur.com/t1PlV.jpg The equation must be smooth and satisfy the following two conditions: When $x \leq -10, y=-50$ When $x \geq 10, y=50$ Is there an equation (or multiple equatio......Sat, 05 Sep 2026 21:22:10 -0400https://liberty.io/how-are-these-triangles-similar.htmlhttps://liberty.io/how-are-these-triangles-similar.htmlI am trying to understand a proof of Pythagorean's Theorem. I am trying to see how all three triangles are similar. So far I have: (1) $\triangle ABC$ and $\triangle DBA$ have side $AB$ in commo......Sat, 05 Sep 2026 21:22:06 -0400https://liberty.io/why-dividing-by-zero-still-works.htmlhttps://liberty.io/why-dividing-by-zero-still-works.htmlToday, I was at a class. There was a question: If $x = 2 +i$, find the value of $x^3 - 3x^2 + 2x - 1$. What my teacher did was this: $x = 2 + i \;\Rightarrow \; x - 2 = i \; \Rightarrow \; (x ......Sat, 05 Sep 2026 21:19:21 -0400https://liberty.io/could-someone-please-explain-double-angle-identities.htmlhttps://liberty.io/could-someone-please-explain-double-angle-identities.htmlI don't understand how to do maths, mostly because I don't understand why formulae work they way they do, or the reasoning behind equations, etc. I tried to explain the $\sin(2\theta)$ double-angle ...Sat, 05 Sep 2026 21:12:23 -0400https://liberty.io/how-many-ways-can-the-word-arranger-be-arranged.htmlhttps://liberty.io/how-many-ways-can-the-word-arranger-be-arranged.htmlI have tried this problem by doing 8 factorial over two times two factorial, which I got 10080, which doesn't seem to be right since the answer on the answer key is pretty far from my answer, can s......Sat, 05 Sep 2026 21:11:58 -0400https://liberty.io/stokes-039-s-first-problem-rayleigh-plate-u-0-t-t-n.htmlhttps://liberty.io/stokes-039-s-first-problem-rayleigh-plate-u-0-t-t-n.htmlSo basically we have Stoke's first Problem/Rayleigh plate, meaning an impulsively starting Plate with the boundary conditions $u(0,t)=t^n$ and $u(\infty,t)=0$. In theory it is easy to solve the PDE......Sat, 05 Sep 2026 21:08:40 -0400https://liberty.io/why-is-the-number-of-ways-of-choosing-0-items-from-n-items-1.htmlhttps://liberty.io/why-is-the-number-of-ways-of-choosing-0-items-from-n-items-1.htmlIt seems easy to grasp that number of ways of choosing $n$ items from $n$ items is 1. But I am unable to understand why is it 1 for choosing 0 items....Sat, 05 Sep 2026 21:08:37 -0400https://liberty.io/express-sin-4-theta-by-formulae-involving-sin-and-cos-and-its-powers.htmlhttps://liberty.io/express-sin-4-theta-by-formulae-involving-sin-and-cos-and-its-powers.htmlI have an assignment question that says "Express $\sin 4\theta$ by formulae involving $\sin$ and $\cos$ and its powers." I'm told that $\sin 2\theta = 2 \sin\theta \cos\theta$ but I don't know how......Sat, 05 Sep 2026 21:06:05 -0400https://liberty.io/number-of-possible-combinations-of-5-from-the-set-of-numbers-1-10.htmlhttps://liberty.io/number-of-possible-combinations-of-5-from-the-set-of-numbers-1-10.htmlI am trying to figure out how to set up this problem or any similar one. If you have 10 balls numbered from 1 to 10, and you pick 5 balls, what is the probability that you will have picked ball#1? In ...Sat, 05 Sep 2026 21:01:30 -0400https://liberty.io/is-1-divided-by-3-equal-to-0-333-duplicate.htmlhttps://liberty.io/is-1-divided-by-3-equal-to-0-333-duplicate.htmlI have been taught that $\frac{1}{3}$ is 0.333.... However, I believe that this is not true, as 1/3 cannot actually be represented in base ten; even if you had infinite threes, as 0.333... is suppo......Sat, 05 Sep 2026 20:57:33 -0400https://liberty.io/vectors-vs-cartesian-coordinates.htmlhttps://liberty.io/vectors-vs-cartesian-coordinates.htmlWhat is the difference, if any, between a Cartesian coordinate and a vector? Is it that a vector always has direction and magnitude, whilst Cartesian coordinates do not?...Sat, 05 Sep 2026 20:55:45 -0400https://liberty.io/evaluating-the-given-integral-by-changing-to-polar-coordinates-why-is-.htmlhttps://liberty.io/evaluating-the-given-integral-by-changing-to-polar-coordinates-why-is-.htmlEvaluate the given integral by changing to polar coordinates. $\int\int_D x^2y \, dA$, where $D$ is the top half of the disk with center the origin and radius $5$. when i plugged everything in, I ......Sat, 05 Sep 2026 20:52:32 -0400https://liberty.io/basic-examples-of-monoids.htmlhttps://liberty.io/basic-examples-of-monoids.htmlWhat are some (simple/elementary) examples of noncommutative monoids with no additional structure? I'm having a hard time thinking of examples of "pure" monoids that aren't monoids simply ...Sat, 05 Sep 2026 20:46:56 -0400https://liberty.io/show-that-blaschke-factors-satisfy-inequality-duplicate.htmlhttps://liberty.io/show-that-blaschke-factors-satisfy-inequality-duplicate.htmlTheorem: Let $w, z \in \mathbb{C}$ such that $\bar{z}w \neq 1$. Then $$ |z| < 1 \land |w| < 1 \Rightarrow \left| \frac{w-z}{1-\bar{w}z}\right| < 1$$ I tried to make $w = u +iv, z = x + iy......Sat, 05 Sep 2026 20:43:00 -0400https://liberty.io/trigonometric-proof-involving-sec-and-cosec.htmlhttps://liberty.io/trigonometric-proof-involving-sec-and-cosec.htmlHow should I proceed to prove the following: $$\sec^2 A +\csc^2 A = \sec^2 A \cdot \csc^2 A$$ This is what I've attempted: $$LHS = 1 + \tan^2 A + 1 + \cot^2= 2+ 1\cot^2 + \cot^2$$ However, this ......Sat, 05 Sep 2026 20:38:05 -0400https://liberty.io/why-are-equations-written-by-equating-something-to-zero.htmlhttps://liberty.io/why-are-equations-written-by-equating-something-to-zero.htmlA linear equation is $$ ax + b = 0 ; \,\, \,\, a\neq 0 $$ A quadratic equation is $$ax^2 + bx + c = 0 ; \,\, a\neq 0 $$ And so on... Why are all these equations written as $\dots = 0 $? Why do ...Sat, 05 Sep 2026 20:37:38 -0400https://liberty.io/prove-set-inclusivity.htmlhttps://liberty.io/prove-set-inclusivity.htmlI need help proving the following Suppose $A,B \subseteq X$. Prove that if $X \setminus B \subseteq X \setminus A$, then $A \subseteq B$. I really need to see an example of how to do these types of ...Sat, 05 Sep 2026 20:33:38 -0400https://liberty.io/an-example-of-kirchhoff-equation.htmlhttps://liberty.io/an-example-of-kirchhoff-equation.htmlI'm studying about a simply kind of Kirchhoff equation in one-dimensional that means $$\begin{cases} u_{tt}(x,t)-\left(1+\int_{0}^{1}{u^2_x(x,t)dx}\right)u_{xx}(x,t)=|u(x,t)|^{p-2}u(x,t),0<x<......Sat, 05 Sep 2026 20:31:46 -0400https://liberty.io/finding-the-associated-unit-eigenvector.htmlhttps://liberty.io/finding-the-associated-unit-eigenvector.htmlBackground Find the eigenvalues $λ_1<λ_2$ and two associated unit eigenvectors of the symmetric matrix $$A = \begin{bmatrix}-7&12\\12&11\end{bmatrix}$$ My work so far $$A = \begin{bmatri......Sat, 05 Sep 2026 20:29:32 -0400https://liberty.io/user-carsten-s-mathematics-stack-exchange.htmlhttps://liberty.io/user-carsten-s-mathematics-stack-exchange.htmlQ&A for people studying math at any level and professionals in related fields...Sat, 05 Sep 2026 20:18:57 -0400https://liberty.io/drawing-an-ellipse.htmlhttps://liberty.io/drawing-an-ellipse.htmlYou are given a pen, pencil, pair of compasses, scissors, a ruler and a piece of paper. I require a natural way to draw an ellipse. This can be simple, from using a circle piece of paper, to a co......Sat, 05 Sep 2026 20:13:29 -0400https://liberty.io/calculating-inverse-of-trig-functions-without-calculator-closed.htmlhttps://liberty.io/calculating-inverse-of-trig-functions-without-calculator-closed.htmlWe have $\sin 25°\approx 0.42.$ How to calculate $\arcsin0. 42\approx 25°$ without the help of calculator? And how does calculator evaluate the inverse functions value....Sat, 05 Sep 2026 20:12:15 -0400https://liberty.io/qr-factorization-of-complex-matrix.htmlhttps://liberty.io/qr-factorization-of-complex-matrix.htmlIf you have two complex numbers $a,b$ how can you find the QR factorization of $ M = \begin{bmatrix} aI_n\\ bI_n \end{bmatrix} $, I can't seem to be able to do it. I tried an ear......Sat, 05 Sep 2026 20:10:22 -0400