https://liberty.ioCurated star highlights with fresh energy.en-usThu, 20 Aug 2026 01:26:05 -0400https://liberty.io/what-039-s-the-formula-to-work-out-number-of-fixtures-for-any-number-o.htmlhttps://liberty.io/what-039-s-the-formula-to-work-out-number-of-fixtures-for-any-number-o.htmlIf there are $20$ teams in a league, the total number of fixtures are $380$ if every team plays each other twice. How would you come to that number where there are any number of teams?...Thu, 20 Aug 2026 05:24:18 -0400https://liberty.io/number-of-line-segments-to-approximate-a-circle.htmlhttps://liberty.io/number-of-line-segments-to-approximate-a-circle.htmlThis is part of a homework question, so naturally, not looking for a solution. But I have no idea how to approach the problem. The question is: how many line segments are necessary to approximate a ...Thu, 20 Aug 2026 05:12:42 -0400https://liberty.io/how-do-i-show-that-the-set-of-natural-number-n-has-the-same-size-as-nx.htmlhttps://liberty.io/how-do-i-show-that-the-set-of-natural-number-n-has-the-same-size-as-nx.htmlHow do I show that the set of natural numbers $\mathbb{N}$ has the same size as $\mathbb{N}\times\mathbb{N}$? I know that for two sets to have the same size, there must be an injection from the set $\...Thu, 20 Aug 2026 05:02:03 -0400https://liberty.io/how-to-prove-that-the-midpoint-of-the-given-line-segment-lies-on-anoth.htmlhttps://liberty.io/how-to-prove-that-the-midpoint-of-the-given-line-segment-lies-on-anoth.htmlI have the following question with me: In a triangle ABC D and E lie on sides BC and AB respectively. F lies on AC such that EF is parallel to BC, G lies on side BC such that EG lies parallel to A......Thu, 20 Aug 2026 05:01:27 -0400https://liberty.io/how-to-solve-equations-like-8-2n-1-32-n-1.htmlhttps://liberty.io/how-to-solve-equations-like-8-2n-1-32-n-1.htmlI have stumbled on this question, and there are a few questions after it of the same type. How do I solve it and what is the right approach for this kind of question? $$8^{2n+1} = 32^{n+1}$$ I ne......Thu, 20 Aug 2026 04:55:32 -0400https://liberty.io/sum-of-the-alternating-harmonic-series-sum-k-1-infty-frac-1-k-1-k-frac.htmlhttps://liberty.io/sum-of-the-alternating-harmonic-series-sum-k-1-infty-frac-1-k-1-k-frac.htmlI know that the harmonic series $$\sum_{k=1}^{\infty}\frac{1}{k} = \frac{1}{1} + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \frac{1}{5} + \frac{1}{6} + \cdots + \frac{1}{n} + \cdots \tag{I}$$ diver......Thu, 20 Aug 2026 04:55:01 -0400https://liberty.io/how-to-prove-whether-a-polynomial-function-is-even-or-odd.htmlhttps://liberty.io/how-to-prove-whether-a-polynomial-function-is-even-or-odd.htmlWe know that a function is even if $f(-x) = f(x)$ and odd if $f(-x) = -f(x)$. With this reasoning is it possible to prove that a polynomial function such as $f(x) = a_{2n}x^{2n} + a_{2n-2}x^{2n-2}......Thu, 20 Aug 2026 04:36:17 -0400https://liberty.io/i-think-i-can-complete-the-square-of-any-quadratic-is-it-true-any-reas.htmlhttps://liberty.io/i-think-i-can-complete-the-square-of-any-quadratic-is-it-true-any-reas.htmlI was taught that you could only complete the square of a quadratic if the coefficient on the $x^2$ term is 1. However, playing a little bit with other quadratics, I've found that it's just not t......Thu, 20 Aug 2026 04:34:52 -0400https://liberty.io/failed-calc-2-it-039-s-the-algebra-stupid-closed.htmlhttps://liberty.io/failed-calc-2-it-039-s-the-algebra-stupid-closed.htmlWell, I took "Multidimensional Math" w/ Linear Algebra and Calculus 2 at the same time over a 6 week period for the Summer session. It was a disaster. I don't think my Algebra and Trig are good en......Thu, 20 Aug 2026 04:31:01 -0400https://liberty.io/calculate-the-derivative-of-sin5x-using-limits.htmlhttps://liberty.io/calculate-the-derivative-of-sin5x-using-limits.htmlWell, that's it. How do you calculate $\frac{d}{dx} {\sin5x}$ using the limit formula for derivatives? $$\lim_{h \to 0} \frac {f(x+h)-f(x)}h$$ I managed to get a lot of sines and cosines using ...Thu, 20 Aug 2026 04:19:34 -0400https://liberty.io/finding-the-sum-of-the-infinite-series.htmlhttps://liberty.io/finding-the-sum-of-the-infinite-series.htmlHi I am trying to solve the sum of the series of this problem: $$ 11 + 2 + \frac 4 {11} + \frac 8 {121} + \cdots $$ I know its a geometric series, but I cannot find the pattern around this....Thu, 20 Aug 2026 04:13:39 -0400https://liberty.io/what-does-conserved-mean-in-mathematics.htmlhttps://liberty.io/what-does-conserved-mean-in-mathematics.htmlHope this is not a stupid "what is X" question. I read a book from applied mathematics and I failed to find any reference for the concept "conserved". I am not sure if this is a mathematical jarg......Thu, 20 Aug 2026 04:03:47 -0400https://liberty.io/finding-vector-equation-of-a-plane-from-its-cartesian-equation.htmlhttps://liberty.io/finding-vector-equation-of-a-plane-from-its-cartesian-equation.htmlThe Cartesian equation is $x-3y-4z=1$. Here is what I have tried: Finding three points on the plane by setting two variables equal to 0: $x=0$, $y=0$; $z=\frac{-1}{4}$ $y=0$, $z=0$; $x=1$ $x=0$, $......Thu, 20 Aug 2026 04:01:40 -0400https://liberty.io/how-to-calculate-the-volume-of-tetrahedron-given-by-4-points.htmlhttps://liberty.io/how-to-calculate-the-volume-of-tetrahedron-given-by-4-points.htmlI want to calculate the volume of the tetrahedron defined by those 4 points: $$ P_1 = (-0.0865403, -0.122347, 0.898904)\\ P_2 = (-0.436523, -0.30131, 1.92251)\\ P_3 = (-0.459102, -0.06......Thu, 20 Aug 2026 03:59:20 -0400https://liberty.io/can-an-indefinite-integral-be-expressed-as-a-definite-integral-with-va.htmlhttps://liberty.io/can-an-indefinite-integral-be-expressed-as-a-definite-integral-with-va.htmlIf I have a function $f(t)$, and an indefinite integral of this function, $g(x) = \int f(t)\, dt$, is there any way I can express $g(x)$ as a definite integral whose bounds depend on $x$? I though......Thu, 20 Aug 2026 03:43:53 -0400https://liberty.io/is-the-sum-of-two-exponential-function-can-be-equivalent-to-a-third-ex.htmlhttps://liberty.io/is-the-sum-of-two-exponential-function-can-be-equivalent-to-a-third-ex.htmlWhat will be the sum of two exponential functions $2\exp(4 x) + 3 \exp(5 x)$ equivalent to a third exponential function? Is it possible?...Thu, 20 Aug 2026 03:42:13 -0400https://liberty.io/why-can-039-t-the-polynomial-ring-be-a-field.htmlhttps://liberty.io/why-can-039-t-the-polynomial-ring-be-a-field.htmlI'm currently studying Polynomial Rings, but I can't figure out why they are Rings, not Fields. In the definition of a Field, a Set builds a Commutative Group with Addition and Multiplication. This ...Thu, 20 Aug 2026 03:39:49 -0400https://liberty.io/strongly-connected-graph-equivalent-formulation.htmlhttps://liberty.io/strongly-connected-graph-equivalent-formulation.htmlA digraph(=directed graph = graph with directed/oriented edges) $X$ is said to be strongly connected if for any distinct vertices $v,w$, there is a directed path from $v$ to $w$. In particular in......Thu, 20 Aug 2026 03:34:41 -0400https://liberty.io/equation-for-a-plane-perpendicular-to-a-line-through-two-given-points-.htmlhttps://liberty.io/equation-for-a-plane-perpendicular-to-a-line-through-two-given-points-.htmlThe following type of question is quite popular with examiners at the institution where I study. Find an equation of the plane containing the point $(0, 1, 1)$ and perpendicular to the line passing ...Thu, 20 Aug 2026 03:30:30 -0400https://liberty.io/how-can-i-find-which-function-corresponds-to-a-set-of-data-points.htmlhttps://liberty.io/how-can-i-find-which-function-corresponds-to-a-set-of-data-points.htmlSuppose I have a set of data points like this: 1;1 2;4 3;9 4;16 5;25 6;36 ... The first column is the input of the function and the second one is the result. I can tell if I look at it that the fu......Thu, 20 Aug 2026 03:28:12 -0400https://liberty.io/eigenvalues-of-second-order-ordinary-differential-equation.htmlhttps://liberty.io/eigenvalues-of-second-order-ordinary-differential-equation.htmlThe question is find all the eigenvalue of the following equation $-\frac{d^2y}{dx^2}+x^2y=\lambda y$ I have found the first function which is $y=e^{\frac{-x^2}{2}}$, however I have no clue on ho......Thu, 20 Aug 2026 03:19:57 -0400https://liberty.io/what-is-the-difference-between-correlated-equilibrium-and-mixed-equili.htmlhttps://liberty.io/what-is-the-difference-between-correlated-equilibrium-and-mixed-equili.htmlWhat is the difference between correlated equilibrium and mixed equilibrium? Here's what I understand : Unlike a pure Nash equilibrium, a mixed equilibrium corresponds to when each player has a ...Thu, 20 Aug 2026 03:11:53 -0400https://liberty.io/how-to-find-the-objective-function-of-the-dual-problem.htmlhttps://liberty.io/how-to-find-the-objective-function-of-the-dual-problem.htmlConsider the linear programming Problem: Maximize $z= 2x_1 +3x_2 +x_3$ such that $4x_1+3x_2 + x_3=6$ $x_1 + 2x_2 + 5x_3 \ge 4$ $x_1, x_2 , x_3 \ge 0$ Write down the objective function of the ......Thu, 20 Aug 2026 03:06:05 -0400https://liberty.io/notation-for-combinations-and-permutations.htmlhttps://liberty.io/notation-for-combinations-and-permutations.htmlThe number of ways of obtaining an ordered subset of $k$ elements from a set of $n$ elements is given by: $$_nP_k = \frac{n!}{(n - k)!}$$ In other words, to get the permutations we form the fract......Thu, 20 Aug 2026 02:40:45 -0400https://liberty.io/inverse-of-sets.htmlhttps://liberty.io/inverse-of-sets.htmlLet $A = \mathcal{P}(\{1, 2, 3, 4\})$. Let $f$ be the following function. $f : A \rightarrow A$ defined by $f(X) = \{1, 2, 3, 4\} \setminus X$. Does $f^{−1}$ exist? What is $f^{−1}$? I'm quite p......Thu, 20 Aug 2026 02:33:54 -0400https://liberty.io/implicit-solution-to-a-differential-equation.htmlhttps://liberty.io/implicit-solution-to-a-differential-equation.htmlI'm looking at the ODE: $\frac{dY}{dX} - \frac{ X^2 + 2 Y^2 - 1 }{ ( Y - 2 X )X } = 0$ I'm looking for an $implicit$ solution to the above. Meaning, I want to find a relation $F(X,Y)=0$, where $\......Thu, 20 Aug 2026 02:31:53 -0400https://liberty.io/volume-of-parabolic-cylinder-bound-by-plane.htmlhttps://liberty.io/volume-of-parabolic-cylinder-bound-by-plane.htmlMake a sketch of the solid in the first octant bounded by the plane $x + y = 1$ and the parabolic cylinder $x^{2} + z = 1$ Calculate the volume of the solid.. I have no idea where to even star......Thu, 20 Aug 2026 02:28:32 -0400https://liberty.io/unit-impulse-step-response-of-a-1st-order-differential-equation.htmlhttps://liberty.io/unit-impulse-step-response-of-a-1st-order-differential-equation.htmlYou are given the equation $10v'(t) + 0.6 v(t) = f(t)$ $v(t)$ is the velocity of the object Determine the unit impulse response AND the unit step response. How would i approach this question? do......Thu, 20 Aug 2026 02:24:51 -0400https://liberty.io/creating-a-rotating-partner-round-robin-schedule.htmlhttps://liberty.io/creating-a-rotating-partner-round-robin-schedule.htmlFirst, I'm going to apologize for a lack of proper terminology on this post. It's been well over a decade since my days in the classroom, so I've regressed back to a simple math layman here. (As a ...Thu, 20 Aug 2026 02:22:44 -0400https://liberty.io/automorphism-and-inner-automorphism-of-d-4.htmlhttps://liberty.io/automorphism-and-inner-automorphism-of-d-4.htmlI want to find $Aut(D_4)$ and $Inn(D_4)$. The group $D_4 = \{1,r,r^2,r^3, s,rs,r^2s,r^3s\}$. An automorphism of a group $G$ is an isomorphism from the group to itself. An inner automorphism of a g......Thu, 20 Aug 2026 01:54:55 -0400https://liberty.io/notation-what-does-this-summation-mean.htmlhttps://liberty.io/notation-what-does-this-summation-mean.htmlI need to understand what the summation below mean. I specifically do not understand the $i<j$ below the summation. $$\sum_{\substack{ i<j}}^{N} d_{ij}$$ Insight about the equation There......Thu, 20 Aug 2026 01:53:28 -0400https://liberty.io/what-is-a-level-set.htmlhttps://liberty.io/what-is-a-level-set.htmlI have the ellipsoid $x^2 + 2y^2 + z^2 = 4$. It is mentioned that I can rewrite this equation as a level set $f(x,y) = x^2 + 2y^2 + z^2$. Of course I can look at Wikipedia where a level set is ...Thu, 20 Aug 2026 01:32:43 -0400https://liberty.io/online-video-course-on-topology.htmlhttps://liberty.io/online-video-course-on-topology.htmlI am at the beginning of learning topology, I found PDFs like topology without tears, but I wonder if there is a full university-like course online, on MIT open courseware I did not found anything ......Thu, 20 Aug 2026 01:19:52 -0400https://liberty.io/d-dx-notation-explanation-please.htmlhttps://liberty.io/d-dx-notation-explanation-please.htmlI know how to derive, I know how to integrate. I know what to do when I see $\frac{d}{dx}$ and such but what does it really mean? I know it means something like derive in terms of $x$, but whats the ...Thu, 20 Aug 2026 01:18:29 -0400https://liberty.io/a-slightly-problematic-integral-int-1-x-4-1-1-4-mathrm-d-x.htmlhttps://liberty.io/a-slightly-problematic-integral-int-1-x-4-1-1-4-mathrm-d-x.htmlQuestion. To find the integral of- $$\int {\frac{1}{(x^4+1)^\frac{1}{4}} \, \mathrm{d}x}$$ I have tried substituting $x^4+1$ as $t$, and as $t^4$, but it gives me an even more complex integral. An......Thu, 20 Aug 2026 01:14:11 -0400https://liberty.io/heron-039-s-formula.htmlhttps://liberty.io/heron-039-s-formula.htmlHeron's Formula gives the area of a triangle when the length of all three sides are known. There is no need to calculate angles or other distances in the triangle first to determine its area. The f......Thu, 20 Aug 2026 01:14:00 -0400https://liberty.io/how-do-i-solve-y-039-039-4y-0.htmlhttps://liberty.io/how-do-i-solve-y-039-039-4y-0.htmlThis problem is in Penney's Elementary Differential Equations, listed as a reducible, 2nd-order DE. The chapter has taught two techniques to be used, which are for when either $x$ or $y(x)$ is mis......Thu, 20 Aug 2026 01:11:51 -0400https://liberty.io/probability-of-a-given-neg-b.htmlhttps://liberty.io/probability-of-a-given-neg-b.htmlIf we have the probabilities of $P(A)$, $P(B)$ and $P(A\mid B)$, how can we calculate the probability of $P(A\mid\neg B)$ ? Does $A$ depends on $\neg B$ if it Actually depends on $B$ ?...Thu, 20 Aug 2026 01:06:29 -0400https://liberty.io/n-th-derivative-of-e-1-x.htmlhttps://liberty.io/n-th-derivative-of-e-1-x.htmlI am trying to find the $n$'th derivative of $f(x)=e^{1/x}$. When looking at the first few derivatives I noticed a pattern and eventually found the following formula $$\frac{\mathrm d^n}{\mathrm d......Thu, 20 Aug 2026 01:05:25 -0400https://liberty.io/determinant-of-matrix-exponential.htmlhttps://liberty.io/determinant-of-matrix-exponential.htmlSuppose $A$ is a $n \times n$ constant matrix. How can I prove $\det(e^A) = e^{\displaystyle \sum_{\lambda_i\in\sigma(A)} \lambda_i}$, where $\sigma(A)$ is the multiset of eigenvalues of $A$? The ...Thu, 20 Aug 2026 01:02:53 -0400https://liberty.io/is-littlewood-039-s-law-true.htmlhttps://liberty.io/is-littlewood-039-s-law-true.htmlLittlewood's law states that a person can expect to experience an event with odds of one in a million (defined by the law as a "miracle") at the rate of about one per month. This is attributed to the ...Thu, 20 Aug 2026 01:00:29 -0400https://liberty.io/finding-the-area-of-a-15-75-90-triangle-with-the-length-of-the-hypoten.htmlhttps://liberty.io/finding-the-area-of-a-15-75-90-triangle-with-the-length-of-the-hypoten.htmlSo there is a right triangle $ABC$ with $m∠C=90°$, $m∠B=75°$, and $BC\ (the \ hypotenuse)=12 cm$. I want to find the area of this triangle It would look something like this: Note: I have already ...Thu, 20 Aug 2026 00:58:39 -0400https://liberty.io/how-do-i-calculate-the-indefinite-integral-with-a-square-root.htmlhttps://liberty.io/how-do-i-calculate-the-indefinite-integral-with-a-square-root.htmlFor example, how do I calculate the indefinite integral of the square root of x? I understand how the power rule works, but im confused by the square root. I would appreciate any help given!...Thu, 20 Aug 2026 00:53:09 -0400https://liberty.io/calculating-inverse-tangent.htmlhttps://liberty.io/calculating-inverse-tangent.htmlWhen calculating the inverse tangent of a degree, the calculator will always give an angle between 90 degrees and –90 degrees. But I want to find the positive value of the negative angle. Do I add ......Thu, 20 Aug 2026 00:40:57 -0400https://liberty.io/what-is-the-significance-of-convergence-divergence-of-series.htmlhttps://liberty.io/what-is-the-significance-of-convergence-divergence-of-series.htmlThere are many ways and mathematical methods to test the convergence or Divergence of a series. What does Convergence or divergence indicate practically in the real world? Why is the convergence of ...Thu, 20 Aug 2026 00:37:40 -0400https://liberty.io/choosing-4-men-from-6-and-4-women-from-7-with-a-restriction.htmlhttps://liberty.io/choosing-4-men-from-6-and-4-women-from-7-with-a-restriction.htmlA team of 4 men and 4 women has to be chosen from 6 men and 7 women, so total combinations: $^6C_{4} \times ^7C_{4} = 525$ If a restriction is placed that some two women will always be chosen tog......Thu, 20 Aug 2026 00:37:25 -0400https://liberty.io/logic-symbol-forall-before-or-after-statement.htmlhttps://liberty.io/logic-symbol-forall-before-or-after-statement.htmlI have question about what is difference about two statements below. $$\forall a,b,c, \ \ a+(b+c)=(a+b)+c $$ or $$a+(b+c)=(a+b)+c , \ \forall a,b,c$$ I usaually see second in textbooks but my ...Thu, 20 Aug 2026 00:16:52 -0400https://liberty.io/epsilon-delta-proof-of-a-quadratic-limit.htmlhttps://liberty.io/epsilon-delta-proof-of-a-quadratic-limit.htmlI just started learning about limits and their definition. I'm practicing with some proofs that involve the epsilon-delta definition of a limit.While I absolutely have no problem with linear functi......Wed, 19 Aug 2026 23:58:27 -0400https://liberty.io/what-is-the-actual-curvature-of-earth-either-per-kilometer-km-of-sea-l.htmlhttps://liberty.io/what-is-the-actual-curvature-of-earth-either-per-kilometer-km-of-sea-l.htmlFirst of all, the following question with it's answers are the main reason I put this question, because as I will show the question wording is bad. How to calculate curvature of Earth per surface ...Wed, 19 Aug 2026 23:57:26 -0400https://liberty.io/theorem-of-existence-and-uniqueness-for-first-order-linear-differentia.htmlhttps://liberty.io/theorem-of-existence-and-uniqueness-for-first-order-linear-differentia.htmlThe theorem of existence and uniqueness is: Let $ y'+p(x)y=g(x) $, $ y(x_{0})=y_{0} $ be a first order linear differential equation such that $ p(x) $ and $ g(x) $ are both continuous for $ a<x......Wed, 19 Aug 2026 23:45:46 -0400https://liberty.io/conditions-for-taking-a-limit-into-an-infinite-sum.htmlhttps://liberty.io/conditions-for-taking-a-limit-into-an-infinite-sum.htmlSuppose $f\left(x\right)={\displaystyle \sum_{n=0}^{\infty}g_{n}\left(x\right)}$ under what conditions is it true that: $$\lim_{x\to c}f\left(x\right)={\displaystyle \sum_{n=0}^{\infty}\lim_{x\to c......Wed, 19 Aug 2026 23:30:51 -0400https://liberty.io/full-and-reduced-svd-of-a-3x3-matrix.htmlhttps://liberty.io/full-and-reduced-svd-of-a-3x3-matrix.htmlI currently studying for an exam, and I'm currently working my way through some old exam problems and I'm currently at the following. First, we have a matrix $A= \begin{bmatrix} 2&0&0\\2&......Wed, 19 Aug 2026 23:29:52 -0400https://liberty.io/what-would-base-1-be.htmlhttps://liberty.io/what-would-base-1-be.htmlBase $10$ uses these digits: $\{0,1,2,3,4,5,6,7,8,9\};\;$ base $2$ uses: $\{0,1\};\;$ but what would base $1$ be? Let's say we define Base $1$ to use: $\{0\}$. Because $10_2$ is equal to $010_2$, ......Wed, 19 Aug 2026 23:29:18 -0400https://liberty.io/dot-product-of-a-vector-and-del-operator.htmlhttps://liberty.io/dot-product-of-a-vector-and-del-operator.htmlHow did this expansion come about. What is the physical significance of this expansion. And what is the significance of $$\vec{A} . \nabla {\vec{A}}$$ And what does this mathematically represent....Wed, 19 Aug 2026 23:25:29 -0400https://liberty.io/find-the-interpolating-polynomial-of-degree-3-that-interpolates-f-x-x-.htmlhttps://liberty.io/find-the-interpolating-polynomial-of-degree-3-that-interpolates-f-x-x-.htmlFind the interpolating polynomial of degree $3$ that interpolates $f(x) = x^3$ at the nodes $x_0=0, x_1 = 1, x_2=2, x_3 = 3$. Here are my workings below The basic Lagrange polynomials are: $$L_......Wed, 19 Aug 2026 23:12:55 -0400https://liberty.io/intuition-of-euclidean-norm.htmlhttps://liberty.io/intuition-of-euclidean-norm.htmlI know the following : A norm is Euclidean iff it respects the parallelogram law. The problem is that the parallelogram law is not very intuitive geometrically. So I am wondering if there is a ...Wed, 19 Aug 2026 23:03:23 -0400https://liberty.io/can-you-give-me-a-specific-representation-of-q-weyl-algebra.htmlhttps://liberty.io/can-you-give-me-a-specific-representation-of-q-weyl-algebra.htmlCan you give me a specific representation of q-Weyl algebra over field k,i.e $K<x,y>/<yx=qxy>$ for proving that {$x^{i}y^{j}$, i,j$\in$$Z$} is indeed a basis in $k<x,y>/<yx=qxy......Wed, 19 Aug 2026 22:56:48 -0400https://liberty.io/setting-up-an-english-pool-table.htmlhttps://liberty.io/setting-up-an-english-pool-table.htmlA friend and I were playing some English pool yesterday. When you rack the balls it has the specific set up in the picture where the 8 ball must be in that central position and the balls are laid out ...Wed, 19 Aug 2026 22:48:44 -0400https://liberty.io/what-is-the-inverse-function-of-cot-x.htmlhttps://liberty.io/what-is-the-inverse-function-of-cot-x.htmlMy Math skills got real rustic over time so bear with a (hopefully) simple question: I'm looking for a way to compute $n$ from this formula, where $\cot(x)$ is the cotangent. $$\cot(n) = \frac{\pi}......Wed, 19 Aug 2026 22:39:02 -0400https://liberty.io/why-does-a-times-its-inverse-equal-to-the-identity-matrix-duplicate.htmlhttps://liberty.io/why-does-a-times-its-inverse-equal-to-the-identity-matrix-duplicate.htmlI was trying to come up with a proof of why: $AA^{-1} = I$. If we know that: $A^{-1}A = I$, then $A(A^{-1}A) = A \implies (AA^{-1})A = A$. However I don't like setting $AA^{-1} = I$ for fear that......Wed, 19 Aug 2026 22:36:28 -0400https://liberty.io/even-numbers-have-more-factors-than-odd-numbers.htmlhttps://liberty.io/even-numbers-have-more-factors-than-odd-numbers.htmlThis was an exercise to show that, in a sense, the even numbers have more prime factors than the odds, but--if it's right-- I still have a question. As an heuristic calculation, we could take a la......Wed, 19 Aug 2026 22:32:12 -0400https://liberty.io/finding-the-real-potential-and-streamfunction-from-a-complex-potential.htmlhttps://liberty.io/finding-the-real-potential-and-streamfunction-from-a-complex-potential.htmlI'm trying to solve the following problem given on a previous exam in my fluid mechanics course: A stationary flow is given as the following complex velocity potential: $$ \beta(z) = \ln(z - a) + \......Wed, 19 Aug 2026 22:31:07 -0400https://liberty.io/what-are-the-classic-or-great-books-for-algebra-geometry-and-trigonome.htmlhttps://liberty.io/what-are-the-classic-or-great-books-for-algebra-geometry-and-trigonome.htmlThere are classic textbooks for Calculus like Spivak, Courant, Apostol, etc that do a fantastic job at explaining the fundamental concepts and theory along with great problem sets. My question is the ...Wed, 19 Aug 2026 22:29:44 -0400https://liberty.io/why-the-derivative-is-the-rate-of-change-of-the-function.htmlhttps://liberty.io/why-the-derivative-is-the-rate-of-change-of-the-function.htmlThe price of an action follow the function $f(t)=e^{-t}$. The question is, what is the rate of change of the price ? For me, the rate of change is the rate of change by unit of time $h$, i.e. $$\fr......Wed, 19 Aug 2026 22:24:56 -0400https://liberty.io/about-the-definition-of-local-fields.htmlhttps://liberty.io/about-the-definition-of-local-fields.htmlI read in a book that the definition of local fields is: A field $K$ for which with respect to a discrete valuation $v$, the residue field is finite and $K$ is complete with respect to $v$. Howeve......Wed, 19 Aug 2026 22:23:24 -0400https://liberty.io/trying-to-understand-discontinuities-and-domains.htmlhttps://liberty.io/trying-to-understand-discontinuities-and-domains.htmlI am really struggling with this concept and I feel like no math book clarifies it sufficiently. First off: I assume that if a one-sided limit exists then it is a real value and not $\pm \infty$.......Wed, 19 Aug 2026 22:22:15 -0400https://liberty.io/stars-and-bars-everyone-gets-at-least-one.htmlhttps://liberty.io/stars-and-bars-everyone-gets-at-least-one.htmlIn how many ways can we distribute $11$ pieces of candy among $4$ kids, provided that every kid gets at least $1$ piece of candy? I know how to solve this problem one way, but why does this way no......Wed, 19 Aug 2026 22:10:38 -0400https://liberty.io/what-symbol-expresses-less-than-approximately.htmlhttps://liberty.io/what-symbol-expresses-less-than-approximately.htmlSuppose, I want to state that $a$ is less than $b$. However, I do not know $b$ exactly, but only that it is approximately $c$. With other words I want to state that $a$ is less than some value that......Wed, 19 Aug 2026 21:58:16 -0400https://liberty.io/rounding-up-to-two-decimal-points.htmlhttps://liberty.io/rounding-up-to-two-decimal-points.htmlI just have a question regarding my answers in my exam paper, my teacher said to give the answer to two decimal points, I got the correct answer (2.89) but I rounded it up to 2.90. Now my teacher d......Wed, 19 Aug 2026 21:53:47 -0400https://liberty.io/solving-a-neumann-problem-in-a-disk.htmlhttps://liberty.io/solving-a-neumann-problem-in-a-disk.html$\Delta u(x,y) = x^2 $ $x^2 + y^2 <9$ On the boundary $x^2 + y^2 = 9$, $\frac {\delta u}{\delta n} = y$. I've seen similar problems solved in texts (Strauss, Bleecker, Stavroulakis), but whe......Wed, 19 Aug 2026 21:49:38 -0400https://liberty.io/let-x-n-and-y-n-be-given-sequences-and-define-z-n-to-be-the-shuffled-s.htmlhttps://liberty.io/let-x-n-and-y-n-be-given-sequences-and-define-z-n-to-be-the-shuffled-s.htmlLet $(x_n)$ and $(y_n)$ be given sequences and define $(z_n)$ to be the "shuffled" sequence $(x_1,y_1,x_2,y_2,x_3,y_3,..., x_n,y_n,...)$. Prove that $(z_n)$ is convergent if and only if $(x_n)$ and......Wed, 19 Aug 2026 21:42:53 -0400https://liberty.io/2-log-2-3-cdot-log-3-2-without-using-a-calculator.htmlhttps://liberty.io/2-log-2-3-cdot-log-3-2-without-using-a-calculator.htmlI want to express this into a single logarithm without using the calculator. $$2\log_2 3 \cdot \log_3 2$$ My calculator's log function has only log base 10. It's easy to change the base to 10 an......Wed, 19 Aug 2026 21:33:41 -0400https://liberty.io/derivation-of-formula-for-finding-median-for-grouped-data.htmlhttps://liberty.io/derivation-of-formula-for-finding-median-for-grouped-data.htmlI know the formula of formula for finding median for grouped data that is $$\mathrm{Median} = L_m + \left [ \frac { \frac{n}{2} - F_{m-1} }{f_m} \right ] \times c$$ and I know what all the letters ......Wed, 19 Aug 2026 21:31:48 -0400https://liberty.io/volumes-of-n-balls-what-is-so-special-about-n-5.htmlhttps://liberty.io/volumes-of-n-balls-what-is-so-special-about-n-5.htmlThe volume of an $n$-dimensional ball of radius $1$ is given by the classical formula $$V_n=\frac{\pi^{n/2}}{\Gamma(n/2+1)}.$$ For small values of $n$, we have $$V_1=2\qquad$$ $$V_2\approx 3.14$$ ......Wed, 19 Aug 2026 21:15:00 -0400https://liberty.io/period-of-sin-x-cos-x.htmlhttps://liberty.io/period-of-sin-x-cos-x.htmlThe period of $\sin(x)$ is $2\pi$ and $\cos(x)$ is $2\pi$. And the period of $\sin(x)+\cos(x)$ is also $2\pi$. Why it is so?...Wed, 19 Aug 2026 21:14:44 -0400https://liberty.io/difference-between-rolle-039-s-theorem-and-mean-value-theorem.htmlhttps://liberty.io/difference-between-rolle-039-s-theorem-and-mean-value-theorem.htmlThe Rolle's theorem is defined as: Suppose f is continous on $[\alpha,\beta]$ such that $f(\alpha) = f(\beta)$ $f′$ exists on $(\alpha,\beta)$ Then f has a local maximun or minimum at some $c \in (\...Wed, 19 Aug 2026 21:13:19 -0400https://liberty.io/ball-thrown-straight-up-modern-elementary-differential-equations.htmlhttps://liberty.io/ball-thrown-straight-up-modern-elementary-differential-equations.htmlI'm reading this book: Modern Elementary Differential Equations I understand that the second derivative of x (position) with respect to t is the downward effect gravity (acceleration) has on the b......Wed, 19 Aug 2026 21:10:30 -0400https://liberty.io/solve-a-pde-with-feynman-kac-formula.htmlhttps://liberty.io/solve-a-pde-with-feynman-kac-formula.htmlSo there is the following PDE given: $\frac{\partial}{\partial t}f(t,x) + rx\frac{\partial}{\partial x}f(t,x)+\frac{\sigma^2 x^2}{2}\frac{{\partial}^2}{\partial x^2}f(t,x) = rf(t,x)$ With boundary ...Wed, 19 Aug 2026 21:01:54 -0400https://liberty.io/how-many-ways-to-make-scalene-triangles-in-a-nonagon.htmlhttps://liberty.io/how-many-ways-to-make-scalene-triangles-in-a-nonagon.htmlThe title pretty much says it all, but: How many distinct scalene triangles can be formed by connecting three vertices of a regular nonagon? Any answers would be greatly appreciated, thank you!......Wed, 19 Aug 2026 20:58:31 -0400https://liberty.io/proof-question-let-v-be-a-finite-dimensional-vector-space-and-let-t-v-.htmlhttps://liberty.io/proof-question-let-v-be-a-finite-dimensional-vector-space-and-let-t-v-.htmlLet V be a finite-dimensional vector space, and let $T:V \rightarrow V$ be linear. a). if $\mathrm{rank} (T)= \mathrm{rank} (T^2)$, prove $R(T) \bigcap N(T)$={0}. $N(T)$ is defined by $T(0_V)=0_......Wed, 19 Aug 2026 20:55:57 -0400https://liberty.io/solve-z-5-32-include-complex-solutions.htmlhttps://liberty.io/solve-z-5-32-include-complex-solutions.htmlQuestion: Solve $z^5 =32$ (including complex solutions) My Working : $$ z^5=32 $$ $$ z^5=32cis(0+2k\pi) $$ $$ z=2cis(0+\frac{2k\pi}5) $$ I then went through all the solutions for $k=0\......Wed, 19 Aug 2026 20:51:53 -0400https://liberty.io/how-do-i-show-a-mapping-is-a-homomorphism.htmlhttps://liberty.io/how-do-i-show-a-mapping-is-a-homomorphism.htmlI don't want to make this question too broad, or non-specific. I'll will discuss a simple situation so we can all share a common context, but my question is less about this particular group, and m......Wed, 19 Aug 2026 20:42:38 -0400https://liberty.io/find-all-solutions-for-dy-dx-y-2.htmlhttps://liberty.io/find-all-solutions-for-dy-dx-y-2.htmlStarted to learn ODE, and the second example was of the equation $y^\prime=y^2$ We showed that the group of solutions is $\{y(x)=0 \}\cup \{ y(x)=\frac{1}{c-x} | c \in \mathbb{R} \}$ and mentioned ......Wed, 19 Aug 2026 20:35:51 -0400https://liberty.io/percentage-change-over-multiple-years.htmlhttps://liberty.io/percentage-change-over-multiple-years.htmlI have data for the number of mice in an area of 13 years. $\begin{bmatrix} \text{Year} & \text{#mice} \\ 1995 & 30 \\ 1996 & 113 \\ 1997 & 106 \\ 1998 & 50 \\ 1999 & 19 \\......Wed, 19 Aug 2026 20:30:57 -0400https://liberty.io/finding-the-mean-and-median-of-a-probability-density-function.htmlhttps://liberty.io/finding-the-mean-and-median-of-a-probability-density-function.htmlI suspect this is super-easy, but I haven't done any math in about ten years and I'm working with concepts that have been woefully explained... I need to find the mean and median of a continuous r......Wed, 19 Aug 2026 20:27:12 -0400https://liberty.io/express-if-true-then-1-else-0-in-a-formula-suitable-for-desmos-calcula.htmlhttps://liberty.io/express-if-true-then-1-else-0-in-a-formula-suitable-for-desmos-calcula.htmlIn programming, often the value of True is also 1, and False is 0. This means that: (x>5)*4 will return 4 if x is greater than 5 (because (x>5)==1), else 0. I need to accomplish a similar......Wed, 19 Aug 2026 20:26:04 -0400https://liberty.io/l-039-hospital-rule-in-multivariable-calculus.htmlhttps://liberty.io/l-039-hospital-rule-in-multivariable-calculus.htmlI was just wondering. Given functions $f(x,y), g(x,y) $, and the corresponding limit $\lim_{(x,y) \to (0,0) } \frac{f(x,y)}{g(x,y)} $ , where $\lim_{(x,y) \to (0,0) } f = \lim_{(x,y) \to (0,0) }g =......Wed, 19 Aug 2026 20:19:18 -0400https://liberty.io/what-039-s-the-intuition-behind-a-split-exact-sequence.htmlhttps://liberty.io/what-039-s-the-intuition-behind-a-split-exact-sequence.htmlI've learned the following definition for a split exact sequence: A short exact sequence of $R$-module homomorphisms $0\to A \stackrel{f}{\to} B \stackrel{g}{\to} C \to 0$ is split if there exists a ...Wed, 19 Aug 2026 20:01:58 -0400https://liberty.io/torque-and-angular-frequency-of-a-merry-go-round.htmlhttps://liberty.io/torque-and-angular-frequency-of-a-merry-go-round.htmlI have been trying to approach this by finding the gravitational torque of the person on the merry-go-round, however I can't find a way to relate that to angular velocity in the given circumstances......Wed, 19 Aug 2026 19:57:35 -0400https://liberty.io/finding-the-vertex-of-a-quadratic-function.htmlhttps://liberty.io/finding-the-vertex-of-a-quadratic-function.htmlI'm in the process of relearning math and came across a problem when calculating the vertex for $5x^2-20x+15$ function. I took the following steps to find the vertex: I divided every term by 5 to ......Wed, 19 Aug 2026 19:46:43 -0400https://liberty.io/is-there-such-a-thing-as-implicit-integration.htmlhttps://liberty.io/is-there-such-a-thing-as-implicit-integration.htmlA problem I'm working on asks me at a certain point to integrate the following: $$x=2\left(\sin\left(y\right)\right)^2.$$ I've never integrated anything in this form. I can't isolate for $y$ eith......Wed, 19 Aug 2026 19:44:56 -0400https://liberty.io/what-are-equations-with-a-degree-more-than-3-called.htmlhttps://liberty.io/what-are-equations-with-a-degree-more-than-3-called.htmlSorry if I am using the incorrect terminology. I am beginning to learn Mathematics. I had it in school and college but have forgotten most of it, so please be gentle with me if my questions are stu......Wed, 19 Aug 2026 19:43:19 -0400https://liberty.io/line-integral-around-a-triangle.htmlhttps://liberty.io/line-integral-around-a-triangle.htmlLet $R$ be the interior of the triangle with vertices $(0,0), (4,2),$ and $(0,2)$. Let $C$ be the boundary of $R$, oriented counterclockwise. Now evaluate the integral below. $$\int_C(y+e^\sqrt{x......Wed, 19 Aug 2026 19:39:05 -0400https://liberty.io/039-normally-distributed-random-numbers-039-vs-039-uniformly-distribut.htmlhttps://liberty.io/039-normally-distributed-random-numbers-039-vs-039-uniformly-distribut.htmlwhat is the difference between 'normally distributed random numbers' and 'uniformly distributed random number'? A answer in a layman language is appreciated :)...Wed, 19 Aug 2026 19:38:41 -0400https://liberty.io/biased-coin-probability-uneven-probability.htmlhttps://liberty.io/biased-coin-probability-uneven-probability.htmlGetting at least $2$ heads when flipping a coin $3$ times but the coin is biased so that heads are $3$ times more likely than tails. Can anyone explain how uneven probability works? Thank you...Wed, 19 Aug 2026 19:38:12 -0400https://liberty.io/graph-of-y-log-1-x.htmlhttps://liberty.io/graph-of-y-log-1-x.htmlTo draw the graph of $|y|=|\log(1-|x|)|$, I see that $x=0, y=0$ satisfies it. Also, the asymptotes would be $x=1, x=-1$. I have made a graph which has sort of eight curves in it but I don't know ho......Wed, 19 Aug 2026 19:31:23 -0400https://liberty.io/why-can-039-t-there-be-a-quintic-formula-duplicate.htmlhttps://liberty.io/why-can-039-t-there-be-a-quintic-formula-duplicate.htmlPossible Duplicate: Why is it so hard to find the roots of polynomial equations? For polynomials (with real coefficients), in degrees 2, 3, 4, there are the quadratic, cubic, and quartic formu......Wed, 19 Aug 2026 19:30:35 -0400https://liberty.io/solve-tan2x-1.htmlhttps://liberty.io/solve-tan2x-1.htmlAre there any errors in my work? Thanks in advance! (Sorry for the bad format. I'm still new to this) $\tan2x=1$ $\frac{2\tan x}{1-\tan^2x}=1$ $2\tan x=1-\tan^2x$ $0=1-\tan^2x-2\tan x$ $0 =-\t......Wed, 19 Aug 2026 19:17:09 -0400https://liberty.io/what-local-system-really-is.htmlhttps://liberty.io/what-local-system-really-is.htmlI know a local system is a locally constant sheaf. But why does a local system on the topological space $X$ correspond to $\tilde{X}\times_G V$, where $G$ is the fundamental group of $X$, $\tilde{X......Wed, 19 Aug 2026 19:10:34 -0400https://liberty.io/why-h-p-coprod-i-in-i-x-i-g-cong-prod-i-in-i-h-p-x-i-g.htmlhttps://liberty.io/why-h-p-coprod-i-in-i-x-i-g-cong-prod-i-in-i-h-p-x-i-g.htmlMy question is the following; why $\displaystyle H^p\big(\coprod_{i\in I}X_i;G\big)$ is isomorphic to $\displaystyle\prod_{i\in I}H^p(X_i;G)$? I have tried the following: one can prove the followin......Wed, 19 Aug 2026 19:08:01 -0400