https://liberty.ioCurated star highlights with fresh energy.en-usWed, 26 Aug 2026 12:57:34 -0400https://liberty.io/how-to-find-functional-square-root-of-sin-x.htmlhttps://liberty.io/how-to-find-functional-square-root-of-sin-x.htmlMaybe I am overlooking something, but is there some easy way to find a function $x\to f(x)$ so that $$(f\circ f)(x) = f(f(x)) = \sin(x)$$ on some interval, say $x\in [-\pi,\pi]\subset \mathbb R$ of ...Wed, 26 Aug 2026 16:51:30 -0400https://liberty.io/prove-that-the-product-of-a-non-zero-rational-and-irrational-number-is.htmlhttps://liberty.io/prove-that-the-product-of-a-non-zero-rational-and-irrational-number-is.htmlCould you please confirm if this proof is correct? Theorem: If $q \neq 0$ is rational and $y$ is irrational, then $qy$ is irrational. Proof: Proof by contradiction, we assume that $qy$ is rational. ...Wed, 26 Aug 2026 16:44:55 -0400https://liberty.io/quickest-way-to-find-characteristic-polynomial-from-a-given-matrix.htmlhttps://liberty.io/quickest-way-to-find-characteristic-polynomial-from-a-given-matrix.htmlFind the Rational form of $$ A=\begin{pmatrix} 1&2&0&4\\ \:\:\:4&1&2&0\\ \:\:\:0&4&1&2\\ \:\:\:2&0&4&1\end{pmatrix} $$ I don't wanna the solution, ...Wed, 26 Aug 2026 16:44:26 -0400https://liberty.io/given-a-byte-in-2-s-complement-what-is-0x56-in-decimal-closed.htmlhttps://liberty.io/given-a-byte-in-2-s-complement-what-is-0x56-in-decimal-closed.htmla) -13 b) -110 c) 38 d) none of the above Im not sure on how to do this question. What I have done so far is convert the hexadecimal number into a decimal, but since it's already in 2's complement......Wed, 26 Aug 2026 16:40:46 -0400https://liberty.io/area-of-trapezoids-vs-area-of-parallelograms.htmlhttps://liberty.io/area-of-trapezoids-vs-area-of-parallelograms.htmlBy definition parallelograms are special type of trapezoids. Given a trapezoid with known sides one can calculate its area according to wikipedia. But in order to calculate the area of a parallelog......Wed, 26 Aug 2026 16:30:11 -0400https://liberty.io/statistics-calculating-mean-given-standard-deviation-and-percentage.htmlhttps://liberty.io/statistics-calculating-mean-given-standard-deviation-and-percentage.htmlThis is one of our homework problems for Statistics. I'm really stuck on this one, and it would be great if someone could show me how to solve this, or at least set up the equation. The weights of ...Wed, 26 Aug 2026 16:27:12 -0400https://liberty.io/how-can-i-construct-the-centroid-of-a-quadrilateral.htmlhttps://liberty.io/how-can-i-construct-the-centroid-of-a-quadrilateral.htmlHow can I construct the centroid of a quadrilateral? I suppose that it is the intersection between the lines that join the middles of opposite sides....Wed, 26 Aug 2026 16:25:59 -0400https://liberty.io/calculate-inverse-discrete-time-fourier-transform.htmlhttps://liberty.io/calculate-inverse-discrete-time-fourier-transform.htmlCalculate Inverse Discrete Time Fourier Transform of the following where $|a| < 1$: $$ X(e^{j\omega}) = \frac{1-a^2}{(1-ae^{-j\omega})(1-ae^{j\omega})} $$ Plugging this directly into the IDTFT ...Wed, 26 Aug 2026 16:09:18 -0400https://liberty.io/what-comes-before-precalculus-closed.htmlhttps://liberty.io/what-comes-before-precalculus-closed.htmlI studied in Europe and followed mostly European academic ways. I'm tutoring a young person , and I will to know what math class comes before pre calculus. Because soon we will be starting some ...Wed, 26 Aug 2026 16:05:56 -0400https://liberty.io/baby-papa-mama-big-rudin.htmlhttps://liberty.io/baby-papa-mama-big-rudin.htmlRecently, I was looking for the reviews of some Analysis books while encountered terms such as Baby/Papa/Mama/Big Rudin. Firstly, I thought that these are the names of a book! But it turned out that ...Wed, 26 Aug 2026 16:01:51 -0400https://liberty.io/relationships-between-det-a-b-and-a-b.htmlhttps://liberty.io/relationships-between-det-a-b-and-a-b.htmlWhen computing $\det(A+B)$ we notice that there is no relation between $\det A + \det B$. However does the $\det(A+B)$ have any relation to the matrices $A+B$ as they stand?...Wed, 26 Aug 2026 16:01:05 -0400https://liberty.io/what-is-the-largest-positive-n-for-which-n-3-100-is-divisible-by-n-10.htmlhttps://liberty.io/what-is-the-largest-positive-n-for-which-n-3-100-is-divisible-by-n-10.htmlWhat is the largest positive $n$ for which $n^3+100$ is divisible by $n+10$ I tried to factorize $n^3+100$, but $100$ is not a perfect cube. I wish it were $1000$....Wed, 26 Aug 2026 15:55:43 -0400https://liberty.io/prove-the-principle-of-strong-induction.htmlhttps://liberty.io/prove-the-principle-of-strong-induction.htmlThe following is from Analysis with an Introduction to Proof by Steven Lay Prove the principle of strong induction: Let $P(n)$ be a statement that is either true or false for each $n \in \mathbb{N}$ ...Wed, 26 Aug 2026 15:48:46 -0400https://liberty.io/what-is-meant-by-the-double-vertical-line-notation-here.htmlhttps://liberty.io/what-is-meant-by-the-double-vertical-line-notation-here.htmlWhat do the double vertical lines around $\vec i$ and $\vec j$ in this equation actually mean? $$ sim(i,j) = cos(\vec i, \vec j) = \frac{\vec i \cdot \vec j}{\lVert \vec i\rVert^2 * \lVert\vec j\r......Wed, 26 Aug 2026 15:46:52 -0400https://liberty.io/conditions-for-a-unique-root-of-a-fifth-degree-polynomial.htmlhttps://liberty.io/conditions-for-a-unique-root-of-a-fifth-degree-polynomial.htmlFifth degree polynomials cannot generally be solved analytically, but at least one solution always exists. Given the normal form $$ax^5+bx^4+cx^3+dx^2+ex+f=0,$$ is it possible to find sufficient ...Wed, 26 Aug 2026 15:43:00 -0400https://liberty.io/symbolic-operation-on-ti-84-calculator-closed.htmlhttps://liberty.io/symbolic-operation-on-ti-84-calculator-closed.htmlI want the calculator to answer with the parameter answer, not with a numeric value, I have tried to clear "M" variable with no success. Any idea? http://imgur.com/cHjfLln ( sorry I don't have ......Wed, 26 Aug 2026 15:35:36 -0400https://liberty.io/x-x-y-how-to-solve-for-x-duplicate.htmlhttps://liberty.io/x-x-y-how-to-solve-for-x-duplicate.htmlI tried looking for ways to solve this equation and came across something like Lambert's W function, which, by the way, I did not understand a bit, because I've never learned it nor do I have a dec......Wed, 26 Aug 2026 15:17:02 -0400https://liberty.io/possible-polyhedra-from-euler-039-s-formula.htmlhttps://liberty.io/possible-polyhedra-from-euler-039-s-formula.htmlI'm not very clear with the euler's formula, and I couldn't find it anywhere. I'm sorry if it is a double post. F + V - E = 2 Is the euler's formula. If the equation balances, is it polyhedra al......Wed, 26 Aug 2026 15:03:32 -0400https://liberty.io/the-chain-rule-and-squared-trig-functions.htmlhttps://liberty.io/the-chain-rule-and-squared-trig-functions.htmlI am trying to do this homework problem and I have no idea how to approach it. I have tried many methods, all resulting in failure. I went to the books website and it offers no help. I am trying to......Wed, 26 Aug 2026 14:59:04 -0400https://liberty.io/moment-of-inertia-around-z-axis.htmlhttps://liberty.io/moment-of-inertia-around-z-axis.htmlHello I am having difficulty with the following; I am wanting to find I, the moment of inertia about the z axis of the region that is bounded by the paraboloid $z=x^{2}+y^{2}$ and the $z=1$ plane,......Wed, 26 Aug 2026 14:55:40 -0400https://liberty.io/use-the-given-graph-f-over-the-interval-0-7-to-find-the-following.htmlhttps://liberty.io/use-the-given-graph-f-over-the-interval-0-7-to-find-the-following.html(a) The open intervals on which $f$ is increasing. I answered $(0, 1), (3, 5), (5,7)$ (b) The open intervals on which $f$ is concave upward. I answered $(1, 4)$ (c) The open intervals on which ......Wed, 26 Aug 2026 14:54:39 -0400https://liberty.io/using-l-039-hopital-039-s-rule-with-the-indeterminate-form-of-infinity.htmlhttps://liberty.io/using-l-039-hopital-039-s-rule-with-the-indeterminate-form-of-infinity.htmlI'm aware that with L'Hopital's rule, we're dealing with indeterminate forms of $ \lim_{x\to a} \frac{f(x)}{g(x)} $, and that includes re-writing $\lim_{x\to a} \infty - \infty$ into that format. ...Wed, 26 Aug 2026 14:51:16 -0400https://liberty.io/how-to-graph-the-parametric-without-calculator-x-sin-t-y-t-2.htmlhttps://liberty.io/how-to-graph-the-parametric-without-calculator-x-sin-t-y-t-2.htmlQuestion How to graph the parametric without calculator: $x=\sin t$ , $y=t^2$ I first make t by itself by doing the following: $\sqrt{y}=t$ and plugging into the other function to get $x=\sin(\sq......Wed, 26 Aug 2026 14:43:45 -0400https://liberty.io/prove-that-there-exists-a-shortest-distance-between-two-skew-lines.htmlhttps://liberty.io/prove-that-there-exists-a-shortest-distance-between-two-skew-lines.htmlIn every geometry textbook it states that the shortest distance between two skew lines (lines that are not co-planar) is given by the unique line that runs perpendicular to both skew lines. This is ...Wed, 26 Aug 2026 14:41:41 -0400https://liberty.io/finding-the-angles-of-a-triangle-using-algebraic-expressions.htmlhttps://liberty.io/finding-the-angles-of-a-triangle-using-algebraic-expressions.htmlI think I have solved this question but want to check if I am correct with my formula I used the theorem (rule) : the measure of the exterior angle of a triangle is equal to the sum of the remote (...Wed, 26 Aug 2026 14:40:57 -0400https://liberty.io/positive-definite-and-block-matrix.htmlhttps://liberty.io/positive-definite-and-block-matrix.htmlSuppose a $n \times n$ symmetric matrix $\mathbf{M}$ is positive definite. Its block matrix form is written as follow. \begin{align} \mathbf{M} \; = \; \begin{pmatrix} \mathbf{A} &am......Wed, 26 Aug 2026 14:40:17 -0400https://liberty.io/difference-between-spe-and-spne-in-game-theory.htmlhttps://liberty.io/difference-between-spe-and-spne-in-game-theory.htmlI am slightly confused as to what the difference between Subgame Perfect Equilibrium and Subgame Perfect Nash Equilibrium is. I was told there is a difference, but now I found many lecture notes t......Wed, 26 Aug 2026 14:34:22 -0400https://liberty.io/definition-singular-manifold.htmlhttps://liberty.io/definition-singular-manifold.htmlI'm looking for the definition of a singular manifold. I haven't found it yet. For instance, in $\mathbb{R}^4$, with $f(x,y,z,t)=xy-zt$, $f^{-1}(0)$ is a singular submanifold. I only found a few ex......Wed, 26 Aug 2026 14:20:58 -0400https://liberty.io/why-do-we-draw-the-xyz-coordinate-system-like-this.htmlhttps://liberty.io/why-do-we-draw-the-xyz-coordinate-system-like-this.htmlUsually people (including, for instance, Calculus teachers) draw the $xyz$ coordinate system in such a way that the $y$ and $z$ axes are perpendicular to each other: xyz http://pad1.whstatic.com/im......Wed, 26 Aug 2026 14:17:54 -0400https://liberty.io/derivation-of-normal-equation-for-linear-least-squares-in-matrix-form.htmlhttps://liberty.io/derivation-of-normal-equation-for-linear-least-squares-in-matrix-form.htmlThe derivation can be found on wikipedia but it's not clear how each step follows. We have $y=X\beta+\epsilon$, and want to minimize $\epsilon^2$. We write objective function as $S(\beta)=||y-X\b......Wed, 26 Aug 2026 14:15:43 -0400https://liberty.io/simplifying-division-of-integrals.htmlhttps://liberty.io/simplifying-division-of-integrals.htmlThe $\overline{x}$ coordinate of the center of mass of a plane region is calculated as $$ \overline{x} = \frac{M_y}{M} = \frac{\int_a^b xf(x) \, \textrm{d}x}{\int_a^b f(x) \, \textrm{d}x} $$ And ......Wed, 26 Aug 2026 14:11:43 -0400https://liberty.io/what-is-a-number.htmlhttps://liberty.io/what-is-a-number.htmlA dictionary I consulted said a 'number' is a 'quantity', so I looked up what quantity means and the same dictionary said it is an amount or number of some material or thing. Since quantity and nu......Wed, 26 Aug 2026 13:43:03 -0400https://liberty.io/how-to-find-the-number-of-x-per-second-given-the-time-elapsed.htmlhttps://liberty.io/how-to-find-the-number-of-x-per-second-given-the-time-elapsed.htmlI'm benchmarking a websocket server and I am very poor at math so please forgive me. I am recording the amount of messages sent, and the elapsed time: Players Connected: 2. Messages Sent: 3528, E......Wed, 26 Aug 2026 13:39:42 -0400https://liberty.io/justifying-why-0-0-is-indeterminate-and-1-0-is-undefined.htmlhttps://liberty.io/justifying-why-0-0-is-indeterminate-and-1-0-is-undefined.html$\dfrac 00=x$ $0x=0$ $x$ can be any value, therefore $\dfrac 00$ can be any value, and is indeterminate. $\dfrac 10=x$ $0x=1$ There is no such $x$ that satisfies the above, therefore $\dfrac 10$......Wed, 26 Aug 2026 13:30:14 -0400https://liberty.io/find-the-absolute-maximum-and-minimum-of-f-x-frac-1-x-frac-2-x-2-on-2-.htmlhttps://liberty.io/find-the-absolute-maximum-and-minimum-of-f-x-frac-1-x-frac-2-x-2-on-2-.htmlNote: Turns out this is a pretty tricky problem where we need to use techniques not developed in the chapter. We will go over this problem in class on Tuesday. Find the absolute maximum and minimu......Wed, 26 Aug 2026 13:30:11 -0400https://liberty.io/methods-of-solving-a-problem-regarding-withdrawing-and-depositing-from.htmlhttps://liberty.io/methods-of-solving-a-problem-regarding-withdrawing-and-depositing-from.htmlI have an imaginary bank account with an initial balance $x$. I can only withdraw $0\% < w\% < 100\%$ (of the current balance) or deposit $0\% < d\% < 100\%$ (of the current balance) at a ...Wed, 26 Aug 2026 13:30:02 -0400https://liberty.io/double-integral-of-graph.htmlhttps://liberty.io/double-integral-of-graph.htmlI am currently trying to do this question but I am having trouble getting the correct answer of 25. I am also having trouble understanding whether my limits are right and whether my method of doin......Wed, 26 Aug 2026 13:15:30 -0400https://liberty.io/explicitly-calculate-shape-operator-for-graph-of-f-x-y-xy.htmlhttps://liberty.io/explicitly-calculate-shape-operator-for-graph-of-f-x-y-xy.htmlThis seems trivial but I am stuck. To gain intuition for a bigger problem, I am trying to compute the shape operator of the graph of $f(x,y)=xy$ at the point $p=(0,0,0)$, call this surface $\Sigma$......Wed, 26 Aug 2026 12:51:37 -0400https://liberty.io/strong-induction-example-using-all-of-p-1-and-and-p-k-1-and-p-k-to-pro.htmlhttps://liberty.io/strong-induction-example-using-all-of-p-1-and-and-p-k-1-and-p-k-to-pro.htmlI'm trying to understand how to do "real" strong induction, but my textbook seems to be of no help. It defines strong induction as follows: Let $P(n)$ be a property that is defined for integers $n......Wed, 26 Aug 2026 12:45:13 -0400https://liberty.io/nonsingular-z-matrix-nonsingular-m-matrix.htmlhttps://liberty.io/nonsingular-z-matrix-nonsingular-m-matrix.htmlHere I'm considering only M-matrices that are also Z-matrices, so all the off diagonal elements are negative in all matrices I consider. If I have a Z-Matrix with real, positive eigenvalues, is it......Wed, 26 Aug 2026 12:45:11 -0400https://liberty.io/how-can-i-know-that-the-graph-is-symmetric-with-respect-to-the-x-axis-.htmlhttps://liberty.io/how-can-i-know-that-the-graph-is-symmetric-with-respect-to-the-x-axis-.htmlHow can I know that the graph is symmetric with respect to the x axis or y axis? $y=x^4-2x^2-21$ $9x^2+4y^2=36$ $y^2=x-4$...Wed, 26 Aug 2026 12:38:59 -0400https://liberty.io/a-4-regular-graph-with-7-vertices-is-non-planar.htmlhttps://liberty.io/a-4-regular-graph-with-7-vertices-is-non-planar.htmlA Graph with 7 vertices each having degree 4 cannot be planar. Any hints on the proof?...Wed, 26 Aug 2026 12:38:24 -0400https://liberty.io/proof-of-derivative-of-an-exponential-function.htmlhttps://liberty.io/proof-of-derivative-of-an-exponential-function.htmlI was told to assume that $$\ln b=\lim_{h\to 0} \frac{\left(b^h-1\right)}{h}$$ where b is a positive, real, base. Unfortunately, being told to assume something isn't good enough. When using L'...Wed, 26 Aug 2026 12:38:03 -0400https://liberty.io/equation-of-chord-of-a-parabola-whose-midpoint-is-given.htmlhttps://liberty.io/equation-of-chord-of-a-parabola-whose-midpoint-is-given.htmlHow to prove $$ T = S1 $$ $$ i.e \qquad yy_1 - 2a(x+x_1) = y_1^2 - 4ax_1=0$$ as the equation of chord for a parabola y$^2$ = 4ax whose midpoint (x$_1,y_1$) is given. $$$$ I couldn't understand ......Wed, 26 Aug 2026 12:36:49 -0400https://liberty.io/how-do-you-show-that-the-three-incidence-axioms-are-independent-of-eac.htmlhttps://liberty.io/how-do-you-show-that-the-three-incidence-axioms-are-independent-of-eac.htmlHow do you show that the three incidence axioms: Incidence Axiom 1. For every pair of distinct points P and Q there is exactly one line l such that P and Q lie on l. Incidence Axiom 2. For every ......Wed, 26 Aug 2026 12:35:05 -0400https://liberty.io/what-is-multiplication-by-juxtaposition.htmlhttps://liberty.io/what-is-multiplication-by-juxtaposition.htmlI was reading http://www.purplemath.com/modules/orderops2.htm it shows = 16 ÷ 2[2] + 1 (**) ... = 5 The general consensus among math people is that "multiplication by juxtaposition" (......Wed, 26 Aug 2026 12:23:53 -0400https://liberty.io/generalizing-sherman-morrison-woodbury-formula-to-more-than-2-matrices.htmlhttps://liberty.io/generalizing-sherman-morrison-woodbury-formula-to-more-than-2-matrices.htmlIs there a generalization of Sherman–Morrison–Woodbury formula to find inverses of sums of more than 2 matrices? https://en.wikipedia.org/wiki/Sherman%E2%80%93Morrison_formula seems to be for 2 mat......Wed, 26 Aug 2026 12:19:54 -0400https://liberty.io/does-fermat-039-s-last-theorem-imply-the-modularity-theorem.htmlhttps://liberty.io/does-fermat-039-s-last-theorem-imply-the-modularity-theorem.htmlThe Wikipedia article on the proof of Fermat's Last Theorem has this sentence If the link identified by Frey could be proven, then in turn, it would mean that a proof or disproof of either of Fe......Wed, 26 Aug 2026 12:16:25 -0400https://liberty.io/what-does-it-mean-to-be-functorial-in-something.htmlhttps://liberty.io/what-does-it-mean-to-be-functorial-in-something.htmlIs there a definition of the word functorial? Often I read that certain things are functorial and often I have an intuition of what is meant. But is there a precise definition of what is meant by t......Wed, 26 Aug 2026 11:57:56 -0400https://liberty.io/is-it-acceptable-to-cite-commonly-used-definitions-from-mathworld-wolf.htmlhttps://liberty.io/is-it-acceptable-to-cite-commonly-used-definitions-from-mathworld-wolf.htmlI am writing a document/article for the average layman, with the aim of explaining certain well known results in mathematics while glossing over the more gritty details. I'm going to include several ...Wed, 26 Aug 2026 11:53:28 -0400https://liberty.io/if-a-is-an-element-of-b-does-that-imply-that-a-is-a-subset-of-b.htmlhttps://liberty.io/if-a-is-an-element-of-b-does-that-imply-that-a-is-a-subset-of-b.htmlQuestion: Suppose that $A\in B$, does this imply that $A\subset B$? If so, why? I have tried to look at using the following example: Suppose that $A = 4$ and $B = \{4\}$. We definitely have $A \i......Wed, 26 Aug 2026 11:52:11 -0400https://liberty.io/find-the-least-integer-n-such-that-f-x-is-o-x-n-for-given-functions.htmlhttps://liberty.io/find-the-least-integer-n-such-that-f-x-is-o-x-n-for-given-functions.htmlI am to find the least integer $n$ such that $f(x)$ is $O(x^n)$ for the following functions: $\begin{equation} (a) f(x) = 2x^2 + x^7 log(x) \end{equation}$ $\begin{equation} (b) f(x) = 3x^9 + (lo......Wed, 26 Aug 2026 11:51:23 -0400https://liberty.io/linear-algebra-polynomial-subspace.htmlhttps://liberty.io/linear-algebra-polynomial-subspace.htmlLet $P_n$ be the set of polynomials of degree at most $n$ with real coefficients. Is the set of polynomials of the form $p(t) = a + t^2,$ such that '$a$' is an element of $\mathbb{R},$ a subspace o......Wed, 26 Aug 2026 11:50:14 -0400https://liberty.io/find-the-unique-x-in-the-interval-0-pi-with.htmlhttps://liberty.io/find-the-unique-x-in-the-interval-0-pi-with.htmlI don't understand what I should calculate if is said: "find the unique x", e.g. in the following exercise: "Find the unique x in the interval [0,pi] with cos(x) = 0,5 * sqrt(3)"...Wed, 26 Aug 2026 11:37:15 -0400https://liberty.io/dividing-a-rectangle-into-4-parts-in-the-ratio-1-2-3-4-with-only-2-lin.htmlhttps://liberty.io/dividing-a-rectangle-into-4-parts-in-the-ratio-1-2-3-4-with-only-2-lin.htmlI have a rectangle made up of 30 identical squares (5 tall and 6 wide). By only drawing two lines on the rectangle, split the rectangle into 4 parts where the areas are in the ratio 1:2:3:4. How ......Wed, 26 Aug 2026 11:33:48 -0400https://liberty.io/dihedral-angles-between-tetrahedron-faces-from-triangles-039-angles-at.htmlhttps://liberty.io/dihedral-angles-between-tetrahedron-faces-from-triangles-039-angles-at.htmlSay I have a general tetrahedron, or in fact only the tip of a tetrahedron, being three triangles in 3-space sharing a single vertex and sharing edges pairwise. Given that I only know the three an......Wed, 26 Aug 2026 11:31:22 -0400https://liberty.io/what-is-model-theory.htmlhttps://liberty.io/what-is-model-theory.htmlI've been reading up on some Model Theory lately. My question is simple. What is the purpose of the theory to begin with? How does it enrich mathematics? A-priori, it doesn't seem like we're doing ......Wed, 26 Aug 2026 11:21:52 -0400https://liberty.io/calculate-the-degree-of-convergence.htmlhttps://liberty.io/calculate-the-degree-of-convergence.htmlI was studying about numerical methods to find root of functions. In the False Position Method, I came to the following claim: $$\text{Degree of convergence } = P = \frac{1+\sqrt5}{2}$$ I cannot fi......Wed, 26 Aug 2026 11:20:47 -0400https://liberty.io/first-de-rham-cohomology-group-of-the-2-torus.htmlhttps://liberty.io/first-de-rham-cohomology-group-of-the-2-torus.htmlShow that for the de Rham cohomology, $H^{1}_{dR}({T^{2}})$ is isomorphic to $\mathbb{R}^{2}$ by showing that the following map: $[\alpha] \to (\int_{S^{1}}f^{*}_{1}\alpha,\int_{S^{1}}f^{*}_{1}\alp......Wed, 26 Aug 2026 11:09:29 -0400https://liberty.io/find-all-x-in-mathbb-z-with-x-equiv-2-mod-4-x-equiv-8-mod9-and-x-equiv.htmlhttps://liberty.io/find-all-x-in-mathbb-z-with-x-equiv-2-mod-4-x-equiv-8-mod9-and-x-equiv.htmlFind all $x\in \mathbb{Z}$ with $x \equiv 2 \mod 4$ , $x \equiv 8\mod9$ and $x \equiv 1\mod5$ My attempt: $\gcd\left(9,5,4\right) = 1 = 9r+5s+4\times0$ $9 = 1\times5+4$ $5 = 1\times4+1$ so $1 ......Wed, 26 Aug 2026 11:07:01 -0400https://liberty.io/what-is-the-difference-between-an-unbounded-solution-and-infinite-opti.htmlhttps://liberty.io/what-is-the-difference-between-an-unbounded-solution-and-infinite-opti.htmlI am currently studying Linear Programming, and one of the few things that I haven't found a solution for (pun obviously intended) is the difference between a solution being either unbounded or hav......Wed, 26 Aug 2026 11:02:28 -0400https://liberty.io/simplify-the-expression-by-using-a-double-angle-formula-or-a-half-angl.htmlhttps://liberty.io/simplify-the-expression-by-using-a-double-angle-formula-or-a-half-angl.htmlSimplify the expression by using a Double-Angle Formula or a Half-Angle Formula. (a) $\cos^2 \left( \cfrac{θ}{2} \right)− \sin^2 \left( \cfrac{θ}{2} \right)$ (b) $2 \sin \left( \cfrac{θ}{2} \right)......Wed, 26 Aug 2026 11:02:15 -0400https://liberty.io/why-are-the-coefficients-of-the-equation-of-a-plane-the-normal-vector-.htmlhttps://liberty.io/why-are-the-coefficients-of-the-equation-of-a-plane-the-normal-vector-.htmlWhy are the coefficients of the equation of a plane the normal vector of a plane? I borrowed the below picture from Pauls Online Calculus 3 notes: http://tutorial.math.lamar.edu/Classes/CalcIII/...Wed, 26 Aug 2026 11:01:13 -0400https://liberty.io/what-is-the-difference-between-zero-vector-and-null-vector-closed.htmlhttps://liberty.io/what-is-the-difference-between-zero-vector-and-null-vector-closed.htmlWhat is the difference between zero vector and null vector? I think it is not a duplicate....Wed, 26 Aug 2026 10:57:38 -0400https://liberty.io/the-difference-between-inverse-function-and-a-function-that-is-inverti.htmlhttps://liberty.io/the-difference-between-inverse-function-and-a-function-that-is-inverti.htmlMy discrete mathematics book says: But I read an answer, https://math.stackexchange.com/a/2415543/390226, said: ["...]There are invertible functions which are not bijective,[..."] And to the s......Wed, 26 Aug 2026 10:54:31 -0400https://liberty.io/is-9-1-an-improper-fraction.htmlhttps://liberty.io/is-9-1-an-improper-fraction.htmlMy son took a test in school. The teacher told them that they did not need to simplify improper fractions in their answers. On one question, for example, the answer of 28/3 was marked as correct.......Wed, 26 Aug 2026 10:48:28 -0400https://liberty.io/cauchy-integral-formula-when-f-z-is-not-analytic-everywhere-inside-c.htmlhttps://liberty.io/cauchy-integral-formula-when-f-z-is-not-analytic-everywhere-inside-c.htmlLet $\alpha$ be a given complex number satisfying $0<|\alpha|<1$, and let $\gamma$ be the circle $|z|=2$ oriented in the counterclockwise direction. Determine the value of $$ \int_{\gamma} \f......Wed, 26 Aug 2026 10:44:14 -0400https://liberty.io/determine-an-interval-in-which-the-solution-to-the-given-ivp-exists-di.htmlhttps://liberty.io/determine-an-interval-in-which-the-solution-to-the-given-ivp-exists-di.htmlDetermine an interval in which the solution to the given IVP exists. $$(x-3){dy\over dx}+(\ln x)y=2x; (1,2)$$ I was wondering if I needed to to separation of variables or the thing where I multipl......Wed, 26 Aug 2026 10:42:47 -0400https://liberty.io/what-does-arc-in-arcsin-arccos-arctan-stands-for.htmlhttps://liberty.io/what-does-arc-in-arcsin-arccos-arctan-stands-for.htmlI was just wondering, what does the "arc" in arcsin, arccos, arctan stands for? Is there any particular reason why it is named the way it is?...Wed, 26 Aug 2026 10:41:34 -0400https://liberty.io/show-that-m-le-2e-v-le-m.htmlhttps://liberty.io/show-that-m-le-2e-v-le-m.html$G$ be a graph with $v$ vertices and $e$ edges.Let $M$ be maximum degree of the vertices of $G$, and let $m$ be the minimum degree of the vertices $G$.Show that $$m \le 2e/v \le M$$ I am completely ...Wed, 26 Aug 2026 10:33:07 -0400https://liberty.io/express-solution-in-simplest-radical-form-4x-1-3x-7-5x-1-2x-3.htmlhttps://liberty.io/express-solution-in-simplest-radical-form-4x-1-3x-7-5x-1-2x-3.htmlI need help with the problem. I'm in grade 11 math. $(4x-1)(3x+7)=(5x-1)(2x+3)$ So far I think I have to factor it and I got $(12x^2+25x-7)=(10x^2-13x-3)$ Thanks!...Wed, 26 Aug 2026 10:29:13 -0400https://liberty.io/determine-the-expected-value-and-variance-of-bar-x.htmlhttps://liberty.io/determine-the-expected-value-and-variance-of-bar-x.htmlLet $X_1,..,X_n$ be independent, identically distributed (continuous) random variables. Let $$\bar{X} = \frac{1}{n} \cdot \left(X_1+...+X_n\right)$$ Determine the expected value and variance......Wed, 26 Aug 2026 10:14:15 -0400https://liberty.io/nullity-of-t-vs-nullity-of-t-2.htmlhttps://liberty.io/nullity-of-t-vs-nullity-of-t-2.htmlFor a finite dimensional vector space, $V$, and linear transformation, $T$, what is the relationship between the nullity of $T$ and the nullity of $T^2$? I read somwhere that if $nullity(T)=k$, th......Wed, 26 Aug 2026 10:07:28 -0400https://liberty.io/good-video-lectures-for-studying-calculus.htmlhttps://liberty.io/good-video-lectures-for-studying-calculus.htmlI am looking for a good online video resource to start studying Calculus. I am studying it alone, not part of any school or university. Trying to learn and enhance my mathematical skills. Thanks!...Wed, 26 Aug 2026 10:02:12 -0400https://liberty.io/how-to-find-what-a-given-series-sequence-converges-to.htmlhttps://liberty.io/how-to-find-what-a-given-series-sequence-converges-to.htmlGiven a function series, what are the ways often used to find the function that the series converges to? For example, how do you construct the function on the right hand side from the series on th......Wed, 26 Aug 2026 09:43:39 -0400https://liberty.io/basic-tensor-derivation.htmlhttps://liberty.io/basic-tensor-derivation.htmlLet $D$ be a tensor derivation on a mnaifold $M$. I have to show that if $D(\partial_i)=\sum F_i^j \partial_j$, then $D(dx^j)=-\sum F_i^j dx^i$. Any help on how to do this? Thanks in advance....Wed, 26 Aug 2026 09:43:09 -0400https://liberty.io/proof-of-vector-calculus-identities.htmlhttps://liberty.io/proof-of-vector-calculus-identities.htmlHere is the all identities : http://en.wikipedia.org/wiki/Vector_calculus_identities I need help concerning vector functions and indexing notations. Let $\overrightarrow{a}$ be a (smooth) vector ...Wed, 26 Aug 2026 09:35:32 -0400https://liberty.io/what-is-convex-combination-of-two-points.htmlhttps://liberty.io/what-is-convex-combination-of-two-points.htmlI am studying algorithms and i saw a definition like the following: Given $3$ points $p_1 = (x_1, y_1)$, $p_2 = (x_2, y_2)$ and $p_3 = (x_3, y_3)$, $p_3$ is a convex combination of $p_1$ and $p_2$......Wed, 26 Aug 2026 09:33:26 -0400https://liberty.io/eigenvectors-of-similar-matrices.htmlhttps://liberty.io/eigenvectors-of-similar-matrices.htmlIf $A$ and $B$ are similar matrices then every eigenvector of $A$ is an eigenvector of $B$. Is the above statement is true? I know that similar matrices have same eigenvalue, but I'm not sure about......Wed, 26 Aug 2026 09:32:14 -0400https://liberty.io/is-a-positively-homogeneous-convex-function-lower-semi-continuous.htmlhttps://liberty.io/is-a-positively-homogeneous-convex-function-lower-semi-continuous.htmlHere the definition of positively homogeneous function $f$ is \begin{equation} f(\lambda x) = \lambda f(x), \quad \lambda \ge 0, \end{equation} and $f$ is defined on linear space $\mathbb{R}^n$. I ...Wed, 26 Aug 2026 09:24:42 -0400https://liberty.io/square-roots-of-complex-number-3-4i.htmlhttps://liberty.io/square-roots-of-complex-number-3-4i.htmlWhat I did $z^2=3-4i$ $(a+bi)^2 = 3-4i$ $a^2-b^2+2abi = 3-4i$ Then got 2 simultaneous equations $a^2-b^2=3$ and $2ab=-4$ Solve for $a^2$ in 1st equation: $a^2=3+b^2$ Subbed into 2nd equation......Wed, 26 Aug 2026 09:19:32 -0400https://liberty.io/equivalence-relation-and-sets.htmlhttps://liberty.io/equivalence-relation-and-sets.htmlThe question: Let $X$ and $Y$ be two sets, and let $S$ be an equivalence relation on set $X$ and $T$ be an equivalence relation on set $Y$. Define a relation $R$ on $X ×Y$ by $(a,b)R(c,d)$ if and ......Wed, 26 Aug 2026 09:18:27 -0400https://liberty.io/what-comes-after-seconds.htmlhttps://liberty.io/what-comes-after-seconds.htmlAngles can be measured in different ways. For example, one can measure angles in degrees/minutes/seconds. So $1^\circ$ is divded into $60$ min. and $1$ min is divided into $60$ sec. That way a tent......Wed, 26 Aug 2026 09:17:31 -0400https://liberty.io/prove-that-the-square-root-of-any-irrational-number-is-irrational.htmlhttps://liberty.io/prove-that-the-square-root-of-any-irrational-number-is-irrational.htmlThe problem I'm having with this proof is that I'm not sure if my proof actually proves the theorem correct or if I'm using circular reasoning. Theorem: Prove that the square root of any irratio......Wed, 26 Aug 2026 09:00:17 -0400https://liberty.io/why-does-an-integral-change-signs-when-flipping-the-boundaries.htmlhttps://liberty.io/why-does-an-integral-change-signs-when-flipping-the-boundaries.htmlLet us define a very simple integral: $f(x) = \int_{a}^{b}{x}$ for $a,b\ge 0$. Why do we have the identity $\int_{a}^{b}{x} = -\int_{b}^{a}{x}$? I drew the graphs and thought about it but to me ...Wed, 26 Aug 2026 08:51:07 -0400https://liberty.io/intuitive-and-convincing-argument-that-functions-are-vectors.htmlhttps://liberty.io/intuitive-and-convincing-argument-that-functions-are-vectors.htmlBack to school time again. As I'm discussing all the mathy stuff and insights gained over the summer, I cannot help but notice that many of my peers in second or third year undergrad cannot bridge ......Wed, 26 Aug 2026 08:48:24 -0400https://liberty.io/define-a-binary-composition-operation-star-such-that-langle-g-star-ran.htmlhttps://liberty.io/define-a-binary-composition-operation-star-such-that-langle-g-star-ran.htmlLet $G$ be a group and $a\in G $ be any non identity element i.e. $a\neq e$. Define a binary composition ( operation ) $\star$ such that $\langle G,\star \rangle$ is a group with $a$ as its identit......Wed, 26 Aug 2026 08:32:26 -0400https://liberty.io/how-to-prove-that-the-matrix-is-not-invertible.htmlhttps://liberty.io/how-to-prove-that-the-matrix-is-not-invertible.html$A$ and $B$ are $3\times 3$ matrices ($A,B \in \mathcal{M}_{3 \times 3}(\mathbb{R})$). There are two equations: $$A^2+3BA=I$$ $$A^2=AB$$ I want to prove that $A$ does not have an inverse. I tri......Wed, 26 Aug 2026 08:24:24 -0400https://liberty.io/lower-limit-topology-properties.htmlhttps://liberty.io/lower-limit-topology-properties.htmlI am reading topology from Munkres book. While reading the countability and Separation axioms, I came across several references to Lower limit topology ($\mathbb{R}_l$) which essentially comprises of ...Wed, 26 Aug 2026 08:15:07 -0400https://liberty.io/how-many-solutions-exists-for-this-equation-duplicate.htmlhttps://liberty.io/how-many-solutions-exists-for-this-equation-duplicate.html$$x_1 + x_2 + x_3 + x_4 = 28$$ I tried to solve it with generating functions. Is it correct to get to the form of $${(1 + x + {x^2} + {x^3} + ....)^4}$$ and this equals to: $${(1 - x)^{ - 4}}$$......Wed, 26 Aug 2026 08:08:12 -0400https://liberty.io/find-center-of-rotation-after-object-rotated-by-known-angle-2d.htmlhttps://liberty.io/find-center-of-rotation-after-object-rotated-by-known-angle-2d.htmlI need to be able to calculate and find the true center of rotation (x,y) of an object after it has been rotated by a known angle. Previously I would simply find the center of the object, rotate 180 ...Wed, 26 Aug 2026 08:07:07 -0400https://liberty.io/negation-of-biconditional-statements.htmlhttps://liberty.io/negation-of-biconditional-statements.htmlLet $p$ and $q$ be two sub statements of the compound biconditional statement given as $p$⇔$q$. The negation of this biconditional statement is given as ($p$^~$q$)∨($q$^~$p$) In the above statem......Wed, 26 Aug 2026 07:58:31 -0400https://liberty.io/camera-rotation-draws-twoards-z-animation-script-built-from-mathformul.htmlhttps://liberty.io/camera-rotation-draws-twoards-z-animation-script-built-from-mathformul.htmlI have constructed a script in FME which calculates the position and rotation of the camera along a track "railroad track". I use math when I write my script so no standard functions. The software ......Wed, 26 Aug 2026 07:56:41 -0400https://liberty.io/proving-x-n-y-n-x-y-x-n-1-x-n-2-y-x-y-n-2-y-n-1.htmlhttps://liberty.io/proving-x-n-y-n-x-y-x-n-1-x-n-2-y-x-y-n-2-y-n-1.htmlIn Spivak's Calculus 3rd Edition, there is an exercise to prove the following: $$x^n - y^n = (x-y)(x^{n-1} + x^{n-2} y + ... + x y^{n-2} + y^{n-1})$$ I can't seem to get the answer. Either I've g......Wed, 26 Aug 2026 07:48:00 -0400https://liberty.io/how-to-perform-wedge-product.htmlhttps://liberty.io/how-to-perform-wedge-product.htmlI have heard all kinds of great things about Clifford/Geometric algebra, but I can't find any good resources. I have been looking EVERYWHERE for just one actual example of a wedge product being ...Wed, 26 Aug 2026 07:46:31 -0400https://liberty.io/how-is-the-subgradient-method-different-from-gradient-descent-in-pract.htmlhttps://liberty.io/how-is-the-subgradient-method-different-from-gradient-descent-in-pract.htmlI'm trying to learn about the subgradient descent optimization method. I'm having trouble understanding how it differs from basic gradient descent in a practical sense. According to this lecture,......Wed, 26 Aug 2026 07:42:50 -0400https://liberty.io/solving-inequalities-with-variables-in-the-denominator.htmlhttps://liberty.io/solving-inequalities-with-variables-in-the-denominator.htmlOk so I've got an exam in 4 hours and I can't ever figure out these problems. Here's the problem I'm struggling with now: $$\frac{x}{3x - 5} \leq \frac{2}{x - 1}.$$ I've learned this so many tim......Wed, 26 Aug 2026 07:37:34 -0400https://liberty.io/why-are-natural-numbers-called-natural-numbers.htmlhttps://liberty.io/why-are-natural-numbers-called-natural-numbers.htmlWhen we say $1,2,3...$ are natural numbers, why don't we include rational and irrational numbers? Isn't $\pi$ something natural? Shouldn't we say all real numbers the Natural numbers? Shouldn't......Wed, 26 Aug 2026 07:18:53 -0400https://liberty.io/determine-whether-the-series-frac-e-frac-1-n-n-2-converges-or-diverges.htmlhttps://liberty.io/determine-whether-the-series-frac-e-frac-1-n-n-2-converges-or-diverges.htmlThe problem states Determine whether the series converges or diverges: $$\sum_{n=1}^\infty\frac{e^{1/n}}{n^2}$$ Immediately I always like to try the test for divergence: $$\lim_{n\to\infty} \s......Wed, 26 Aug 2026 07:17:55 -0400https://liberty.io/find-the-six-trigonometric-functions-given-a-point-with-y-2x.htmlhttps://liberty.io/find-the-six-trigonometric-functions-given-a-point-with-y-2x.htmlThe terminal side of the angle θ is standard position lies on the given line and satisfies the given condition. Determine the 6 trigonometric functions of said angle. $$y = 2x, \quad \sec θ &gt......Wed, 26 Aug 2026 07:08:44 -0400