https://liberty.ioCurated star highlights with fresh energy.en-usThu, 20 Aug 2026 05:20:54 -0400https://liberty.io/fundamental-group-of-mathbb-r-3-minus-trefoil-knot.htmlhttps://liberty.io/fundamental-group-of-mathbb-r-3-minus-trefoil-knot.htmlLet $ \ T \subset \mathbb{R}^3 \ $ be the trefoil knot. A picture is given below. I need a hint on how to calculate the fundamental group of $ \ X = \mathbb{R}^3 \setminus T \ $ using Seifert-van K......Thu, 20 Aug 2026 09:14:55 -0400https://liberty.io/proof-of-the-central-limit-theorem-using-moment-generating-functions.htmlhttps://liberty.io/proof-of-the-central-limit-theorem-using-moment-generating-functions.htmlBelow is a method of proving the Central Limit Theorem using moment generating functions. Let $$X_{1},X_{2},...,X_{n}$$ be a sequence of i.i.d. random variables with expected value and variance ......Thu, 20 Aug 2026 09:09:20 -0400https://liberty.io/matrix-determinant-lemma-derivation.htmlhttps://liberty.io/matrix-determinant-lemma-derivation.htmlWhile reading this wikipedia article on the determinant lemma, I stumbled upon this expression (in a proof section): \begin{equation} \begin{pmatrix} \mathbf{I} & 0 \\ \mathbf{v}^\mathrm{T} &am......Thu, 20 Aug 2026 08:49:58 -0400https://liberty.io/modular-arithmetic-find-the-square-root.htmlhttps://liberty.io/modular-arithmetic-find-the-square-root.htmlOkay so I am in college and in the notes it shows this example: // Example: Compute the square root of 3 modulo 143 3 modulo 143 = 3 mod (11*13) Then he jumps to this: √3 (mod 11) = ± ...Thu, 20 Aug 2026 08:20:10 -0400https://liberty.io/what-is-the-difference-between-sqrt-x-2-and-sqrt-x-2-closed.htmlhttps://liberty.io/what-is-the-difference-between-sqrt-x-2-and-sqrt-x-2-closed.html$\sqrt{x^2}\;$ and $(\sqrt{x})^2$; I mean, are they equal? because what about the following facts: $\sqrt{x^2} = (x^2)^{1/2} = x^{(2)(\frac{1}{2})} = x^{\frac{2}{2}} = x \\ (\sqrt{x})^2 = x^{\frac......Thu, 20 Aug 2026 08:08:57 -0400https://liberty.io/outer-automorphisms-of-semi-simple-lie-algebras.htmlhttps://liberty.io/outer-automorphisms-of-semi-simple-lie-algebras.htmlIt is known that outer automorphisms of semi-simple Lie algebras are automorphisms of their corresponding Dynkin diagrams. But would it be correct to say that for a semi-simple Lie algebra all outer ...Thu, 20 Aug 2026 08:01:39 -0400https://liberty.io/what-is-the-precise-definition-of-a-rigid-shape.htmlhttps://liberty.io/what-is-the-precise-definition-of-a-rigid-shape.htmlWikipedia's section on rigid shapes does not appear to actually define what a rigid shape is. Rather it defines 'same shape' and 'rigid transformations' without giving any definitions of what is ...Thu, 20 Aug 2026 08:00:26 -0400https://liberty.io/euler-039-s-identity-why-is-the-e-in-e-ix-what-if-it-were-some-other-c.htmlhttps://liberty.io/euler-039-s-identity-why-is-the-e-in-e-ix-what-if-it-were-some-other-c.html$e^{ix}$ describes a unit circle in polar coordinates on the complex plane, where x is the angle (in radians) counterclockwise of the positive real axis. My intuition behind this is that $\frac{d}......Thu, 20 Aug 2026 07:59:50 -0400https://liberty.io/finding-irrational-and-complex-roots-of-a-cubic-polynomial.htmlhttps://liberty.io/finding-irrational-and-complex-roots-of-a-cubic-polynomial.htmlI've got a question which shows short answers and no method so I'm trying to find a hand performed method of solving the cubic polynomial for the roots: $$f(x) = 2x^3 + 8x^2 + 10x - 6$$ From the ...Thu, 20 Aug 2026 07:54:33 -0400https://liberty.io/calculate-the-length-of-ac.htmlhttps://liberty.io/calculate-the-length-of-ac.htmlThe diameter $AB$ of the circle is $10\,\text{cm}$. The length of $BC$ is $6\,\text{cm}$. Calculate the length of $AC$. I'm doing a mock exam and I'm not sure how to work out the length of $AC$. Any ...Thu, 20 Aug 2026 07:53:28 -0400https://liberty.io/does-the-gamma-function-have-an-inverse.htmlhttps://liberty.io/does-the-gamma-function-have-an-inverse.htmlDoes the Gamma Function have an Inverse? (Is there an "arc-gamma" function?) Where $\Gamma(x) = y... \Gamma^{-1}(y) = x\ (arc\Gamma(y)=x)$. I've searched and found something called DiGamma Functi......Thu, 20 Aug 2026 07:50:45 -0400https://liberty.io/proof-for-if-a-is-invertible-then-ab-is-invertible.htmlhttps://liberty.io/proof-for-if-a-is-invertible-then-ab-is-invertible.htmlFirst, $A$ and $B$ are square matrices So to prove if $AB$ is invertible, then $A$ is invertible: I let $C=(AB^{-1})B$ Then $CA=(AB^{-1})AB=I$ And $C=A^{-1}$ , so A is invertible. But how do I......Thu, 20 Aug 2026 07:49:19 -0400https://liberty.io/what-can-be-a-real-world-example-of-permutation-without-repetition-clo.htmlhttps://liberty.io/what-can-be-a-real-world-example-of-permutation-without-repetition-clo.htmlI know that a Permutation lock is a concrete real-world example of "permutation with repetition". What can be a concrete real-world example of Permutation without repetition?...Thu, 20 Aug 2026 07:46:03 -0400https://liberty.io/prove-or-disprove-there-exists-a-ring-homomorphism-phi-mathbb-c-righta.htmlhttps://liberty.io/prove-or-disprove-there-exists-a-ring-homomorphism-phi-mathbb-c-righta.htmlProve or disprove: There exists a ring homomorphism $\varphi: \mathbb{C}\rightarrow \mathbb{R}\times\mathbb{R}$. I think it is intuitive to try $\varphi: \mathbb{C}\rightarrow \mathbb{R}\times\mat......Thu, 20 Aug 2026 07:42:23 -0400https://liberty.io/definition-of-definition.htmlhttps://liberty.io/definition-of-definition.htmlI was wondering if there is a good way to "define" what definition means exactly in mathematics. Since the answers may be subjective or philosophical, I want to ask only for references on this topi......Thu, 20 Aug 2026 07:38:54 -0400https://liberty.io/numbers-that-can-be-expressed-as-the-sum-of-two-cubes-in-exactly-two-d.htmlhttps://liberty.io/numbers-that-can-be-expressed-as-the-sum-of-two-cubes-in-exactly-two-d.htmlIt seems known that there are infinitely many numbers that can be expressed as a sum of two positive cubes in at least two different ways (per the answer to this post: Number Theory Taxicab Number)......Thu, 20 Aug 2026 07:38:28 -0400https://liberty.io/simultaneously-solenoidal-and-irrotational-vector.htmlhttps://liberty.io/simultaneously-solenoidal-and-irrotational-vector.htmlCan we have a vector field $\vec{V}(\vec{r})$ such that $\nabla\times\vec{V}=\vec{0}$ and $\nabla\cdot\vec{V}=0$...Thu, 20 Aug 2026 07:37:37 -0400https://liberty.io/number-of-solutions-of-the-equation-3-x-4-x-5-x-in-the-set-of-positive.htmlhttps://liberty.io/number-of-solutions-of-the-equation-3-x-4-x-5-x-in-the-set-of-positive.htmlI came across the following question: Find the number of solutions of the equation $3^x+4^x=5^x$ in the set of positive real numbers. I tried the above question by taking log on both sides and then ...Thu, 20 Aug 2026 07:34:52 -0400https://liberty.io/finding-the-moment-generating-function-of-a-transformed-random-variabl.htmlhttps://liberty.io/finding-the-moment-generating-function-of-a-transformed-random-variabl.htmlSuppose I have a random variable $X$ governed by the Gamma distribution. If I apply a transformation like $Y = \ln\bigg( \frac{X}{1+X}\bigg)$, could I use properties of moment generating functions to ...Thu, 20 Aug 2026 07:30:45 -0400https://liberty.io/height-and-velocity-of-ball-thrown-vertically.htmlhttps://liberty.io/height-and-velocity-of-ball-thrown-vertically.htmlA ball is thrown upward from roof of 32 foot building with velocity of $112$ ft/sec. The height after $t$ seconds is: $s(t)=32+112t-16t^2$. (a) Find the maximum height that the ball reaches. (ans......Thu, 20 Aug 2026 07:30:21 -0400https://liberty.io/is-there-an-existing-formula-in-finding-the-area-of-a-rhombus-wherein-.htmlhttps://liberty.io/is-there-an-existing-formula-in-finding-the-area-of-a-rhombus-wherein-.htmlis there an existing formula in finding the area of a rhombus wherein only the side is given? No measure of angles, no lengths of diagonals , height, etc. is given....Thu, 20 Aug 2026 07:23:58 -0400https://liberty.io/expected-value-of-a-uniform-distribution.htmlhttps://liberty.io/expected-value-of-a-uniform-distribution.htmlLet $f(x) = 0.025x + 0.15$ for $2 < x < 6$. The expected value formula is $1/2 \cdot (b-a)$. So is the expected value just $1/2 \cdot (6-2) = 4$ or do I have to integrate $f(x)$ first?...Thu, 20 Aug 2026 07:23:48 -0400https://liberty.io/how-to-derive-poisson-distribution-from-binomial-distribution.htmlhttps://liberty.io/how-to-derive-poisson-distribution-from-binomial-distribution.htmlI am familiar with another answer about this topic: How to prove Poisson Distribution is the approximation of Binomial Distribution? , but i don't understand one part of it. Could anyone explain t......Thu, 20 Aug 2026 07:23:04 -0400https://liberty.io/is-an-anti-symmetric-and-asymmetric-relation-the-same-are-irreflexive-.htmlhttps://liberty.io/is-an-anti-symmetric-and-asymmetric-relation-the-same-are-irreflexive-.htmlI don't understand the difference between an anti symmetric and asymmetric relation. From my understanding, it is asymmetric if there is not any element where: if (x,y) (y,x). But what if you have......Thu, 20 Aug 2026 07:18:28 -0400https://liberty.io/r-and-r-radr-have-the-same-simple-left-modules.htmlhttps://liberty.io/r-and-r-radr-have-the-same-simple-left-modules.htmlLet $R$ be a ring (not necessarily commutative). $\operatorname{rad}R:=\bigcap \operatorname{ann}M$, where $M$ ranges over all the simple left $R$-modules. My question is as follow: $R$ and $R/\...Thu, 20 Aug 2026 07:17:01 -0400https://liberty.io/what-are-some-examples-of-proof-by-contrapositive.htmlhttps://liberty.io/what-are-some-examples-of-proof-by-contrapositive.htmlApplying the Modus Tollens argument to Fermat's Little Theorem really helped me to understand logical implication. I never knew that FLT was actually a compositality test. Theorem (FLT): given int......Thu, 20 Aug 2026 07:13:50 -0400https://liberty.io/definition-of-the-j-invariant-of-an-elliptic-curve.htmlhttps://liberty.io/definition-of-the-j-invariant-of-an-elliptic-curve.htmlIt seems that most introductory books on elliptic curves simply state the definition of the j-invariant of an elliptic curve without giving any background on how that definition was conceived. Of ......Thu, 20 Aug 2026 07:11:58 -0400https://liberty.io/what-is-and-how-can-i-find-an-orthogonal-component.htmlhttps://liberty.io/what-is-and-how-can-i-find-an-orthogonal-component.htmlI have the following task : Find the orthogonal projection and orthogonal component of vector $ \overline x $ relatively to linear subspace generated by vectors $ \overline a_{1}, \overl......Thu, 20 Aug 2026 07:08:58 -0400https://liberty.io/confused-about-effective-rate-of-discount-theory-of-interest.htmlhttps://liberty.io/confused-about-effective-rate-of-discount-theory-of-interest.htmlI'm currently reading Kellison's book, The Theory of Interest. I've reached the chapter on Effective Rate of Discount and it's somewhat confusing. The book explains it as a loan where interest is p......Thu, 20 Aug 2026 07:08:24 -0400https://liberty.io/why-is-the-square-root-of-a-number-not-plus-or-minus-duplicate.htmlhttps://liberty.io/why-is-the-square-root-of-a-number-not-plus-or-minus-duplicate.htmlFor example, $\sqrt{4}$. I've asked a bunch of people and I get mixed answers all the time, as to whether it is $-2$ and $+2$ or just $+2$. How about if there's a negative in front of the square r......Thu, 20 Aug 2026 07:04:38 -0400https://liberty.io/quadrilateral-inscribed-angles-calculation-with-one-arc-angle.htmlhttps://liberty.io/quadrilateral-inscribed-angles-calculation-with-one-arc-angle.htmlI am trying desperately to solve following problem. How can I solve it, the image and question is included in image...Thu, 20 Aug 2026 07:03:13 -0400https://liberty.io/x-3-3x-3-0-prove-that-10-x-127.htmlhttps://liberty.io/x-3-3x-3-0-prove-that-10-x-127.html$x$ is the real root of the equation $$3x^3-5x+8=0,\tag 1$$ prove that $$e^x>\frac{40}{237}.$$ I find this inequality in a very accidental way,I think it's very difficult,because the actual valu......Thu, 20 Aug 2026 06:54:45 -0400https://liberty.io/find-an-exponential-equation-that-passes-through-the-points-2-2-25-and.htmlhttps://liberty.io/find-an-exponential-equation-that-passes-through-the-points-2-2-25-and.htmlI am asked to find an exponential equation that passes through $(2, 2.25)$ and $(5, 60.75)$. My textbook says the solution it's $y=0.25(3)^x$ whereas I got $y=0.028(9)^x$. Here is my working: Express ...Thu, 20 Aug 2026 06:51:18 -0400https://liberty.io/what-039-s-the-difference-between-f-cdot-g-and-f-g-x.htmlhttps://liberty.io/what-039-s-the-difference-between-f-cdot-g-and-f-g-x.htmlFor example if $f(x) = x + 2$ and $g(x) = 4x - 1$ Then what would be the difference in $f \cdot g$ and $f(g(x))$?...Thu, 20 Aug 2026 06:37:33 -0400https://liberty.io/how-to-check-if-these-vectors-are-normal-or-orthogonal.htmlhttps://liberty.io/how-to-check-if-these-vectors-are-normal-or-orthogonal.htmlIf we have number of vectors. $a = (1, 0, 2, 0)$ $b = (-2, 1, 0, -1)$ $c = (-1, 1, 2, -1)$ How to check if they are normal, or orthogonal vectors?...Thu, 20 Aug 2026 06:33:50 -0400https://liberty.io/quaternions-why-does-ijk-1-and-ij-k-and-ji-k.htmlhttps://liberty.io/quaternions-why-does-ijk-1-and-ij-k-and-ji-k.htmlCurrently i am studying quaternions. I do understand that i, j and k are imaginairy numbers. so $i^2 = j^2 =k^2 = -1$. But I could not understand this: $$\begin{matrix}ij=k,&ji=-k,\\jk=i,&amp......Thu, 20 Aug 2026 06:32:41 -0400https://liberty.io/notation-for-rounding-function.htmlhttps://liberty.io/notation-for-rounding-function.htmlThe Wikipedia article http://en.wikipedia.org/wiki/Nearest_integer_function mentions the following notation for the rounding or "nearest integer" function. (That is, the function that corresponds t......Thu, 20 Aug 2026 06:15:06 -0400https://liberty.io/what-is-the-difference-between-a-polynomial-and-a-function-or-can-they.htmlhttps://liberty.io/what-is-the-difference-between-a-polynomial-and-a-function-or-can-they.htmlI have been wondering over this basic question (seems rather trivial at first sight) for a long time: What is the difference between a polynomial and function? My confusion arises form the follow......Thu, 20 Aug 2026 06:06:52 -0400https://liberty.io/christoffel-symbols-for-spherical-polar-coordinates.htmlhttps://liberty.io/christoffel-symbols-for-spherical-polar-coordinates.htmlIf we are given a line element; $$ds^2=dr^2+r^2d\theta^2+r^2sin^2\theta d\varphi^2$$ We can easily then see that the metric and the inverse metric are; $$g=\begin{pmatrix}1&0&0\\0&r^2&a......Thu, 20 Aug 2026 05:46:54 -0400https://liberty.io/pythagoras-theorem-proof.htmlhttps://liberty.io/pythagoras-theorem-proof.htmlIs my logic correct below to prove the Pythagoras Theorem? Thanks. Area Rectangle R \begin{align*} R &= WL\\ &=(2a+b)(2b+a)\\ &=4ab+2a^2+2b^2+ab\\ &=5ab+2a^2+2b^2 \end{align*}......Thu, 20 Aug 2026 05:44:09 -0400https://liberty.io/how-to-compute-homography-matrix-h-from-corresponding-points-2d-2d-pla.htmlhttps://liberty.io/how-to-compute-homography-matrix-h-from-corresponding-points-2d-2d-pla.htmlI went through this thread Mapping Irregular Quadrilateral to a Rectangle If i know the 4 corresponding points in image say p1->p1' p2->p2' p3->p3' p4->p4' then how to compute pi(x,y) from pi'(x......Thu, 20 Aug 2026 05:41:59 -0400https://liberty.io/how-is-prisoner-039-s-dilemma-different-from-chicken.htmlhttps://liberty.io/how-is-prisoner-039-s-dilemma-different-from-chicken.htmlChicken is a famous game where two people drive on a collision course straight towards each other. Whoever swerves is considered a 'chicken' and loses, but if nobody swerves, they will both crash. ......Thu, 20 Aug 2026 05:38:03 -0400https://liberty.io/homogeneous-polynomial-in-k-x-y-z-can-factor-into-linear-polynomials.htmlhttps://liberty.io/homogeneous-polynomial-in-k-x-y-z-can-factor-into-linear-polynomials.htmlMy question is quite simple. Let $k$ be an algebraically closed field and $f\in k[X,Y]$. We know that $f$ can factor into linear polynomials. I would like to know if there is some generalization o......Thu, 20 Aug 2026 05:36:59 -0400https://liberty.io/showing-a-sequence-is-convergent-using-the-epsilon-n-definition.htmlhttps://liberty.io/showing-a-sequence-is-convergent-using-the-epsilon-n-definition.htmlI need to prove that the following sequence converges: $\lim_{n\rightarrow \infty} \frac{2n^2+3n+1}{n^2+n+1}=2$ So for the proof/solution I have the following: Let $\epsilon >0$. Then let $N=......Thu, 20 Aug 2026 05:35:15 -0400https://liberty.io/two-rectangles-the-1-st-has-twice-the-perimeter-of-the-2-nd-and-the-2-.htmlhttps://liberty.io/two-rectangles-the-1-st-has-twice-the-perimeter-of-the-2-nd-and-the-2-.htmlHow can this be solved using just algebra, where the first rectangle has sides $a$ and $b$, and the second rectangle has sides $c$ and $d$? These are the two equations that follow: $$a + b = 2(c +......Thu, 20 Aug 2026 05:33:05 -0400https://liberty.io/how-many-barcodes-are-there.htmlhttps://liberty.io/how-many-barcodes-are-there.htmlA barcode is made of white and black lines. A barcode always begins and ends width a black line. Each line has is of thickness 1 or 2, and the whole barcode is of thickness 12. How many different ...Thu, 20 Aug 2026 05:28:15 -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 -0400