https://liberty.ioCurated star highlights with fresh energy.en-usFri, 28 Aug 2026 03:20:54 -0400https://liberty.io/negation-of-the-definition-of-limit-of-a-function.htmlhttps://liberty.io/negation-of-the-definition-of-limit-of-a-function.htmlQuestion Suppose we are dealing with real valued functions $f(x)$ of one real variable $x$. We say that the limit of $f(x)$ at point $a$ exists and equals to $L$ if and only if $$\exists L:\lef......Fri, 28 Aug 2026 07:11:44 -0400https://liberty.io/analyticity-and-removable-singularity.htmlhttps://liberty.io/analyticity-and-removable-singularity.htmlIn Churchill's book of Complex Analysis there are two statements that I can't match them to be consistent: In one place it says that a function must be analytic at a removable singular point : why......Fri, 28 Aug 2026 07:09:32 -0400https://liberty.io/how-should-i-study-the-matrix-cookbook.htmlhttps://liberty.io/how-should-i-study-the-matrix-cookbook.htmlI use The Matrix Cookbook by Kaare Brandt Petersen and Michael Syskind Pedersen to solve many problems (mostly matrix derivatives). In most cases, I just map the problem to one of the formula and s......Fri, 28 Aug 2026 06:53:25 -0400https://liberty.io/area-of-a-circular-segment.htmlhttps://liberty.io/area-of-a-circular-segment.htmlSee the picture below: How can I calculate the area in black, using no handy formulas which will give me the answer if I plug in the right values? I had the idea to take $\displaystyle \int_{0.5r}......Fri, 28 Aug 2026 06:52:00 -0400https://liberty.io/proof-of-the-deduction-theorem-explanation.htmlhttps://liberty.io/proof-of-the-deduction-theorem-explanation.htmlI'm reading through this proof of the deduction theorem, and there are a few things I don't understand. The basic idea is to show that if $\Gamma\cup \{A\}\vdash B$, then we have a proof of $B$ wi......Fri, 28 Aug 2026 06:33:19 -0400https://liberty.io/how-are-c-0-c-1-norms-defined.htmlhttps://liberty.io/how-are-c-0-c-1-norms-defined.htmlHow are $C^0,C^1$ norms defined? I know $L_p,L_\infty$ norms but are the former defined....Fri, 28 Aug 2026 06:26:24 -0400https://liberty.io/can-you-disprove-this-rule-pedmsa-division-before-multiplication-subtr.htmlhttps://liberty.io/can-you-disprove-this-rule-pedmsa-division-before-multiplication-subtr.htmlI know the proper teaching of PEMDAS/PEDMAS/BODMAS/BOMDAS.. is that multiplication and division have equal priority and are evaluated going through the arithmetic expression left to right (So, doing ...Fri, 28 Aug 2026 06:25:58 -0400https://liberty.io/what-is-approximation-theory.htmlhttps://liberty.io/what-is-approximation-theory.htmlWhat exactly is "Approximation Theory"? If I read the wikipedia-article I doesn't get much clearer. Why are "pure" mathematicians interested in it? I see a lot of people that do harmonic analysis a......Fri, 28 Aug 2026 06:23:41 -0400https://liberty.io/what-is-an-imaginary-line.htmlhttps://liberty.io/what-is-an-imaginary-line.htmlAn imaginary point is defined as an ordered pair of values, at least one of which is complex. My text says that if $h^2 < ab$ in the equation : $$ ax^2 + 2hxy + by^2 $$ it represents two imagi......Fri, 28 Aug 2026 06:22:13 -0400https://liberty.io/nice-examples-of-groups-which-are-not-obviously-groups.htmlhttps://liberty.io/nice-examples-of-groups-which-are-not-obviously-groups.htmlI am searching for some groups, where it is not so obvious that they are groups. In the lecture's script there are only examples like $\mathbb{Z}$ under addition and other things like that. I don't ...Fri, 28 Aug 2026 06:11:50 -0400https://liberty.io/formula-for-continuously-dividing-by-2.htmlhttps://liberty.io/formula-for-continuously-dividing-by-2.htmlI was helping my daughter with her physics homework. The problem had a bee flying back and forth between two bikes moving toward each other. The question was to see how far the bee traveled. Turned......Fri, 28 Aug 2026 06:02:34 -0400https://liberty.io/dividing-matrices.htmlhttps://liberty.io/dividing-matrices.htmlHI Im learning about matrices in school and am curious about how to divide them so i can find out different identity matrices for refecting shapes over linear functions...Fri, 28 Aug 2026 05:55:20 -0400https://liberty.io/prove-that-a-x-b-y-0-where-x-sqrt-log-a-b-y-sqrt-log-b-a-a-0-b-0-a-b-n.htmlhttps://liberty.io/prove-that-a-x-b-y-0-where-x-sqrt-log-a-b-y-sqrt-log-b-a-a-0-b-0-a-b-n.htmlI'm solving logarithm questions. I got stuck in this question. Prove that $a^x - b^y = 0$ where $x = \sqrt{\log_a b}$ & $y = \sqrt{\log_b a}$ , $a > 0$, $b > 0$ & $a, b \ne1$ I've ......Fri, 28 Aug 2026 05:47:28 -0400https://liberty.io/eigenvalues-of-product-matrix.htmlhttps://liberty.io/eigenvalues-of-product-matrix.htmlI have two matrices, both positive definite, real symmetric and one is diagonal. What can I say about lower and upper bound of the eigenvalues of the product matrix in terms the of lower and upper ......Fri, 28 Aug 2026 05:46:21 -0400https://liberty.io/what-is-the-purpose-of-the-limit.htmlhttps://liberty.io/what-is-the-purpose-of-the-limit.htmlI haven't taken Calculas yet, but I see the use of limit (approaching zero or infinity) in other classes such as physics. I just wanted some intuitive explanation of the limit. I was thinking t......Fri, 28 Aug 2026 05:39:00 -0400https://liberty.io/prove-that-two-equations-are-equivalent.htmlhttps://liberty.io/prove-that-two-equations-are-equivalent.htmlEDIT: Missed something very important! Sorry! We have $x^4+1=2(2x-1)^{1/4}$ not $x^4+1=2\sqrt{2x-1}$. One friend of mine told me that the equation $x^4+1=2(2x-1)^{1/4}$, where $x\geq \frac{1}{2}$ is ...Fri, 28 Aug 2026 05:34:59 -0400https://liberty.io/finding-d-in-rsa.htmlhttps://liberty.io/finding-d-in-rsa.htmlSuppose your RSA modulus is $55 = 5 * 11$ and your encryption exponent is $e = 3$. Find the decryption modulus d. I know $d = 40-13 = 27$ However, I get $1$. $$40 = (P_1-1)(P_2-1)$$ extended ...Fri, 28 Aug 2026 05:33:27 -0400https://liberty.io/cartesian-product-with-three-sets.htmlhttps://liberty.io/cartesian-product-with-three-sets.htmlI was given a question that asks "Let A={1,2} ,B={x,3,y},C={2,y}. Find A x B x C and C x B x C " I have an example that I could go by but i'm not sure what they were multiplying together to get it.......Fri, 28 Aug 2026 05:28:43 -0400https://liberty.io/integration-with-respect-to-dx-dy-and-dz-more-than-one-variable.htmlhttps://liberty.io/integration-with-respect-to-dx-dy-and-dz-more-than-one-variable.htmlSorry if my title was vague but i was not entirely sure what its called. Anyways i was solving some work and energy problems and encountered this integration: $$\int_{2,1,4}^{2,-3,3} 2x\sin^2y \......Fri, 28 Aug 2026 05:22:33 -0400https://liberty.io/numbers-in-a-list-which-are-perfect-squares-and-perfect-cubes-of-numbe.htmlhttps://liberty.io/numbers-in-a-list-which-are-perfect-squares-and-perfect-cubes-of-numbe.htmlHow many numbers in the list $$1,2,3,...,2001$$ are perfect squares and perfect cubes of whole numbers? My progress: Well I do know $$1,4,9,16,25,36,...$$ are perfect squares and $$1,8,27,64,...$$......Fri, 28 Aug 2026 05:14:59 -0400https://liberty.io/how-do-i-find-the-value-of-a-partial-sum-without-calculator.htmlhttps://liberty.io/how-do-i-find-the-value-of-a-partial-sum-without-calculator.htmlWithout using a calculator, how do I find: $$\sum_{k=0}^7 \left(\tfrac{2}{3}\right)^k $$ I know about the formula for the sum of a geometric series $\frac{a\cdot(1-r^n)}{(1-r)}$ but im not sure h......Fri, 28 Aug 2026 04:53:57 -0400https://liberty.io/does-anyone-know-a-simple-proof-for-the-fundamental-theorem-of-finitel.htmlhttps://liberty.io/does-anyone-know-a-simple-proof-for-the-fundamental-theorem-of-finitel.htmlI am looking for a simple/short proof for the fundamental theorem of finitely generated abelian group. Let $(G,\ +)$ be an abelian group. Assume $G$ is finitely generated. Prove that there exist ......Fri, 28 Aug 2026 04:25:46 -0400https://liberty.io/wedge-sum-example.htmlhttps://liberty.io/wedge-sum-example.htmlI am trying to understand the concept of wedge sum in the context of Algebraic Topology. Say I am given two sets, $A=[1, 2, 3]$ and $B=[4, 5]$ Now, disjoint union $A \sqcup B$ = $[(1,0), (2,0), (......Fri, 28 Aug 2026 04:23:11 -0400https://liberty.io/using-fsolve-to-solve-non-linear-equations-in-matlab.htmlhttps://liberty.io/using-fsolve-to-solve-non-linear-equations-in-matlab.htmlI'm trying to solve this system equations in matlab syms y z i x=[29.6,29.4,34.4,35.1,34.3,36.1,31.6,32.9,31.5,28.3,37.1,28.4,27.3,33.1,31.3,33.1] n = 16 eqns = sym(zeros(1, 2 * n)); for i = 1:n......Fri, 28 Aug 2026 04:21:10 -0400https://liberty.io/is-this-the-correct-notation.htmlhttps://liberty.io/is-this-the-correct-notation.html$\left [ \left \{ 1,2,3,4 \right \}! \right]^{-1}$Is this the correct notation if one wanted to obtain the factorial for each number in a sequence and then take the sequence and inverse each number......Fri, 28 Aug 2026 04:20:49 -0400https://liberty.io/why-does-the-sum-of-inverse-squares-equal-pi-2-6-duplicate.htmlhttps://liberty.io/why-does-the-sum-of-inverse-squares-equal-pi-2-6-duplicate.htmlI've heard that $1+1/4+1/9+1/16+1/25+...$ converges to $\pi^2/6$. This was very surprising to me and I was wondering if there was a reason that it converges to this number? I am also confused why......Fri, 28 Aug 2026 04:00:35 -0400https://liberty.io/inverse-of-a-step-function.htmlhttps://liberty.io/inverse-of-a-step-function.htmlHow can I write the function below $$f(x)=\left\{\begin{array}{ll} 1, & 0\leq t\leq 1, \\ 0, & t>1 \end{array}\right.$$ using the unit step function? I mean, I don't know how could I wr......Fri, 28 Aug 2026 03:58:29 -0400https://liberty.io/how-to-calculate-recurring-probability.htmlhttps://liberty.io/how-to-calculate-recurring-probability.htmlI have a very basic understanding of statistics and probability. I just passed an intro college course on it. The problem is, there's one type of problem that I'd like to know how to do, and I don'......Fri, 28 Aug 2026 03:41:12 -0400https://liberty.io/how-do-the-sets-emptyset-times-b-a-times-emptyset-emptyset-times-empty.htmlhttps://liberty.io/how-do-the-sets-emptyset-times-b-a-times-emptyset-emptyset-times-empty.htmlIf we have a function $f:A \rightarrow B$, then one way to give meaning, I think, to this function, in terms of set theory, is to say, that $f$ is actually a binary relation $f=(A,B,G_f)$, where $G......Fri, 28 Aug 2026 03:30:25 -0400https://liberty.io/find-the-12th-term-and-the-sum-of-the-first-12-terms-of-a-geometric-se.htmlhttps://liberty.io/find-the-12th-term-and-the-sum-of-the-first-12-terms-of-a-geometric-se.htmlA geometric series has a first term $\sqrt{2}$ and a second term $\sqrt{6}$ . Find the 12th term and the sum of the first 12 terms. I can get to the answers as irrational numbers using a calculato......Fri, 28 Aug 2026 03:27:47 -0400https://liberty.io/intuition-for-dense-sets-real-analysis.htmlhttps://liberty.io/intuition-for-dense-sets-real-analysis.htmlI have been having problems with dense sets as my lecturer didn't really develop an intuition for dense sets in my class. So can any of you please help me with that? And can you please tell me (the ...Fri, 28 Aug 2026 03:25:35 -0400https://liberty.io/finite-and-infinite-speed-of-propagation-for-wave-and-heat-equation.htmlhttps://liberty.io/finite-and-infinite-speed-of-propagation-for-wave-and-heat-equation.htmlWhat is the formal definition of Finite and infinite speed of propagation? I have searched for it, is the finite one means the solution is only determined by a bounded region? Also I do not under......Fri, 28 Aug 2026 03:13:44 -0400https://liberty.io/why-are-direct-proofs-often-considered-better-than-indirect-proofs-clo.htmlhttps://liberty.io/why-are-direct-proofs-often-considered-better-than-indirect-proofs-clo.htmlAs the title indicates, I'm curious why direct proofs are often more preferable than indirect proofs. I can see the appeal of a direct proof, for it often provides more insight into why and how the ...Fri, 28 Aug 2026 03:12:37 -0400https://liberty.io/finding-all-complex-z-that-satisfy-z-4-4z-8-0.htmlhttps://liberty.io/finding-all-complex-z-that-satisfy-z-4-4z-8-0.htmlI was given this task: $$z^4+4z+8=0$$ And I am totally stuck at solving this. The university handed out some solutions that are the following:$$z=−2±2i$$ And well... plugging this into the funct......Fri, 28 Aug 2026 03:03:55 -0400https://liberty.io/what-is-the-most-unusual-proof-you-know-that-sqrt-2-is-irrational.htmlhttps://liberty.io/what-is-the-most-unusual-proof-you-know-that-sqrt-2-is-irrational.htmlWhat is the most unusual proof you know that $\sqrt{2}$ is irrational? Here is my favorite: Theorem: $\sqrt{2}$ is irrational. Proof: $3^2-2\cdot 2^2 = 1$. (That's it) That is a corol......Fri, 28 Aug 2026 03:03:45 -0400https://liberty.io/solve-the-equation-x-2-frac-9x-2-x-3-2-27.htmlhttps://liberty.io/solve-the-equation-x-2-frac-9x-2-x-3-2-27.htmlProblem Statement:- Solve the equation $$x^2+\dfrac{9x^2}{(x+3)^2}=27$$ I have tried to turn it into a quadratic equation so as to be saved from solving a quartic equation, but have not been able to ...Fri, 28 Aug 2026 02:55:04 -0400https://liberty.io/4-x-4-x-1-24-what-is-the-value-of-2x-x.htmlhttps://liberty.io/4-x-4-x-1-24-what-is-the-value-of-2x-x.htmlSo, every week our teacher gives us a very difficult question worth 1 point and I can never get them right, so I would highly appreciate the person who tells and explains, in good detail, why this ...Fri, 28 Aug 2026 02:52:27 -0400https://liberty.io/logarithmic-equation-solve-for-x.htmlhttps://liberty.io/logarithmic-equation-solve-for-x.html$$\log_{3x}81 = 2$$ How would I go about solving this? This is what I tried: $$\log_{3x}81 = 2$$ $$\frac{\log81}{\log 3 + \log x }= 2$$ Where do I go from here? If I isolate $\log x$ on one si......Fri, 28 Aug 2026 02:44:48 -0400https://liberty.io/how-can-a-solid-with-semi-circular-cross-sections-be-perpendicular-to-.htmlhttps://liberty.io/how-can-a-solid-with-semi-circular-cross-sections-be-perpendicular-to-.htmlLet S be the solid with flat base, whose base is the region in the xy plane defined by the curves $$y=e^x$$,$$y=−3$$,$$x=0$$ and $$x=1$$, and whose cross-sections perpendicular to the x axis are semi-...Fri, 28 Aug 2026 02:27:33 -0400https://liberty.io/simplest-known-solution-to-world-039-s-hardest-easy-geometry-problem.htmlhttps://liberty.io/simplest-known-solution-to-world-039-s-hardest-easy-geometry-problem.htmlI've seen multiple solutions to this well-known problem (posed, among other sites, at http://thinkzone.wlonk.com/MathFun/Triangle.htm). None of the solutions are particularly simple. I was wonderin......Fri, 28 Aug 2026 02:21:49 -0400https://liberty.io/how-to-show-that-pi-pi-is-not-a-integer.htmlhttps://liberty.io/how-to-show-that-pi-pi-is-not-a-integer.htmlThis goes without saying, but, I can't use a calculator to evaluate $\pi^\pi$. I think we need to find a integer $x$ such that $$x<\pi^\pi < x+1. \tag{1}$$ However, since I have no ideia wha......Fri, 28 Aug 2026 02:19:37 -0400https://liberty.io/proving-cos-x-2-sin-x-2-1.htmlhttps://liberty.io/proving-cos-x-2-sin-x-2-1.htmlI need to prove that $\cos(x)^2+\sin(x)^2=1$ Here's how I started (using the Cauchy product): \begin{align} \cos(x)^2+\sin(x)^2 &=\sum_{k=0}^{\infty}\sum_{l=0}^k(-1)^l\frac{x^{2l}}{(2l)!}(-1)......Fri, 28 Aug 2026 02:09:52 -0400https://liberty.io/converting-the-equation-of-the-line-to-the-vector-and-symmetric-forms-.htmlhttps://liberty.io/converting-the-equation-of-the-line-to-the-vector-and-symmetric-forms-.htmlDoes anyone know how to convert the equation of the line to the vector and symmetric forms: $x=-1-2t$ $y=1+3t$ $z=2+t$...Fri, 28 Aug 2026 02:04:10 -0400https://liberty.io/is-the-number-8-special-in-turning-a-sphere-inside-out.htmlhttps://liberty.io/is-the-number-8-special-in-turning-a-sphere-inside-out.htmlSo after watching the famous video on youtube How to turn a sphere inside out I noticed that the sphere is deformed into 8 bulges in the process. Is there something special about the number 8 here?......Fri, 28 Aug 2026 02:01:58 -0400https://liberty.io/finding-the-equation-of-the-tangent-in-slope-intercept-form-at-a-parti.htmlhttps://liberty.io/finding-the-equation-of-the-tangent-in-slope-intercept-form-at-a-parti.htmlOne of the practice problems in my Calculus book is as follows: The graph of y=$8/(x^2-4)$ is called the Witch of Agnesi. (a) Find y' d/dx (u/v) = (v du/dx - u dv/dx)/($v^2$) $=((x^2+4)(0)-(8)(......Fri, 28 Aug 2026 02:01:41 -0400https://liberty.io/confusion-about-expectation-moment-generating-function-distributation.htmlhttps://liberty.io/confusion-about-expectation-moment-generating-function-distributation.html(I am currently studying a high dimensional probability course with very little background knowledge in probability theory as a whole, so I hope it's not annoying that I seem to be oblivious about ......Fri, 28 Aug 2026 02:00:43 -0400https://liberty.io/related-rates-of-change-cylinder-question.htmlhttps://liberty.io/related-rates-of-change-cylinder-question.htmlA cylindrical tank with radius 5 cm is being filled with water at rate of 3 cm^3 per min. how fast is the height of the water increasing? I dont want this question solved, but please help me corre......Fri, 28 Aug 2026 01:56:19 -0400https://liberty.io/coordinates-rotation-by-120-degree.htmlhttps://liberty.io/coordinates-rotation-by-120-degree.htmlIf I have a point on a standard grid with coordinates say: $A_1=(1000,0)$ $A_2=(707,707)$ Is there a easy way to transfer this points to $\pm 120$ degrees from the origin $(0,0)$, and kee......Fri, 28 Aug 2026 01:53:19 -0400https://liberty.io/difference-between-simplicial-and-singular-homology.htmlhttps://liberty.io/difference-between-simplicial-and-singular-homology.htmlI am having some difficulties understanding the difference between simplicial and singular homology. I am aware of the fact that they are isomorphic, i.e. the homology groups are in fact the same (......Fri, 28 Aug 2026 01:43:28 -0400https://liberty.io/relationship-between-simple-interest-and-arithmetic-progression.htmlhttps://liberty.io/relationship-between-simple-interest-and-arithmetic-progression.htmlDescribe the relationship between the mathematics of calculating simple interest, and the mathematics behind analyzing arithmetic sequences. How might simple interest be described as a problem of a......Fri, 28 Aug 2026 01:36:16 -0400https://liberty.io/definition-of-multidimensional-wiener-process.htmlhttps://liberty.io/definition-of-multidimensional-wiener-process.htmlI've been reading through Evans' excellent Introduction to Stochastic Differential Equations (used to be lecture notes available on his website a few years back, and became this book), and got some......Fri, 28 Aug 2026 01:30:10 -0400https://liberty.io/what-is-the-latest-verified-research-on-the-3x-1-problem-closed.htmlhttps://liberty.io/what-is-the-latest-verified-research-on-the-3x-1-problem-closed.htmlWikipedia : Collatz Conjecture Take any positive integer n. If n is even, divide it by $2$ to get $n / 2$. If n is odd, multiply it by $3$ and add $1$ to obtain $3n + 1$. Repeat the process (which......Fri, 28 Aug 2026 01:25:29 -0400https://liberty.io/understanding-the-determinant-of-an-infinite-matrix.htmlhttps://liberty.io/understanding-the-determinant-of-an-infinite-matrix.htmlThe fredholm determinant of an infinite matrix $A$ is defined by $$ \det(I+A) = \exp(\text{tr}(\log(I+A))). $$ I'm trying to understand the formula and how to use it. To be concrete here are my ...Fri, 28 Aug 2026 01:24:19 -0400https://liberty.io/what-are-differences-between-affine-space-and-vector-space.htmlhttps://liberty.io/what-are-differences-between-affine-space-and-vector-space.htmlI know smilar questions have been asked and I have looked at them but none of them seems to have satisfactory answer. I am reading the book a course in mathematics for student of physics vol. 1 by ......Fri, 28 Aug 2026 01:16:49 -0400https://liberty.io/solving-the-equation-ln-x-x.htmlhttps://liberty.io/solving-the-equation-ln-x-x.htmlI tried solving this equation for a long time but did not succeed. Any help is appreciated. $$\ln x=-x$$ I am not sure the tag is correct, I am not familiar with English mathematical terms. Please ...Fri, 28 Aug 2026 01:15:39 -0400https://liberty.io/most-general-solution-for-ode.htmlhttps://liberty.io/most-general-solution-for-ode.htmlI am facing some difficulties understanding the difference between a general solution and the most general solution of a 2nd order ODE for example for homogeneous constant coefficients case with 2 ......Fri, 28 Aug 2026 01:09:54 -0400https://liberty.io/need-help-determining-the-pairs-of-quaternions-that-anticommute.htmlhttps://liberty.io/need-help-determining-the-pairs-of-quaternions-that-anticommute.htmlI tried to solve another exercise and I would be grateful if someone could tell me if my answer is right. This is the exercise: Characterize the pairs $p,q \in \mathbb H$ such that $pq = -qp$. ......Fri, 28 Aug 2026 01:09:19 -0400https://liberty.io/deriving-the-rayleigh-distribution-from-the-gaussian.htmlhttps://liberty.io/deriving-the-rayleigh-distribution-from-the-gaussian.htmlTake a complex number $z=u+iv$ and call its magnitude $x$. Consider the real and imaginary parts to be Gaussian distributed: $$ f_{U}(u)=\frac{1}{\sqrt{2\pi\sigma^{2}}}\exp\left(\frac{-u^{2}}{2\s......Fri, 28 Aug 2026 01:03:46 -0400https://liberty.io/solving-modulus-equation-systems.htmlhttps://liberty.io/solving-modulus-equation-systems.htmlI am studying for a test in discrete math and I created my own question but I cannot seem to solve it. Is it possible to solve the following equation system (without brainless testing), and if so......Fri, 28 Aug 2026 01:00:53 -0400https://liberty.io/why-does-0-75-times-1-25-equal-0-93-and-not-1.htmlhttps://liberty.io/why-does-0-75-times-1-25-equal-0-93-and-not-1.htmlIf percentage points can equal decimal points, then $1.25$ should equal $125$% when multiplying $1.25$ by any number. Therefore $125$% of $0.75$ should equal $1$ if $0.75$ is also $75$% of $1$. ......Fri, 28 Aug 2026 01:00:35 -0400https://liberty.io/find-the-point-of-intersection-of-the-straight-line-frac-x-1-4-frac-y-.htmlhttps://liberty.io/find-the-point-of-intersection-of-the-straight-line-frac-x-1-4-frac-y-.htmlFind the point of intersection of the straight line $$\frac{X+1}{4}=\frac{Y-2}{-2}=\frac{Z+6}{7}$$ and plane $3X+8Y-9Z=0$ the point of the line is $M(-1,2,-6)$ and direction vector of the line is......Fri, 28 Aug 2026 00:52:23 -0400https://liberty.io/find-the-value-of-c-such-that-the-system-has-a-solution-other-than-0-0.htmlhttps://liberty.io/find-the-value-of-c-such-that-the-system-has-a-solution-other-than-0-0.htmlI am asked to the find the value $c$ such that the system has a solution other than $(0, 0, 0)$. This is the linear system: \begin{array}{rcrcrcl} cx & & & + & 4z& = &am......Fri, 28 Aug 2026 00:46:37 -0400https://liberty.io/sard-039-s-theorem-proof-detail-question-regarding-partial-derivatives.htmlhttps://liberty.io/sard-039-s-theorem-proof-detail-question-regarding-partial-derivatives.htmlSard's Theorem: Suppose $M$ and $N$ are smooth manifolds with or without boundary and $F:M\to N$ is a smooth map. Then the set of critical values of $F$ has measure zero in $N$. In what appears to......Fri, 28 Aug 2026 00:37:41 -0400https://liberty.io/intuition-on-the-curl-formula.htmlhttps://liberty.io/intuition-on-the-curl-formula.htmlI'm trying to understand the formula for the curl: $$\operatorname{Curl}(F) = \nabla \times F $$ Why the vector product? What means the vector product of a vector and a operator? What is the mean......Fri, 28 Aug 2026 00:34:49 -0400https://liberty.io/equation-for-x-axis-3d-paraboloid.htmlhttps://liberty.io/equation-for-x-axis-3d-paraboloid.htmlI've been playing around with plenty of variants of paraboloid equations. However I couldn't come up with the equation for a x-axis parallel 3D paraboloid of revolution. For a 2D parabola the equ......Fri, 28 Aug 2026 00:34:02 -0400https://liberty.io/given-log2-a-log7-b-find-log-56.htmlhttps://liberty.io/given-log2-a-log7-b-find-log-56.htmlI don't know how to solve this. Can someone help me? How do I use the information above to help me find Log 56?...Fri, 28 Aug 2026 00:27:38 -0400https://liberty.io/are-there-infinitely-many-pythagorean-triples.htmlhttps://liberty.io/are-there-infinitely-many-pythagorean-triples.htmlI believe these questions are all asking different things, but: Are there infinitely many (integer) solutions to the pythagorean theorem? Is every positive integer part of a solution to the pytha......Fri, 28 Aug 2026 00:20:50 -0400https://liberty.io/find-the-self-adjoint-form-of-the-ode-x-2y-039-039-xy-039-mu-y-0.htmlhttps://liberty.io/find-the-self-adjoint-form-of-the-ode-x-2y-039-039-xy-039-mu-y-0.htmlFind the self-adjoint form of the ODE : $ x^2y''+xy'+\mu y=0 \ $ in the form $ \frac{d}{dx} [p_0 (x) y']+ \mu [q_0(x)]y=0 \ $ Answer: The adjoint equation is $ \frac{d^2}{dx^2}(x^2y)-\frac{......Fri, 28 Aug 2026 00:19:56 -0400https://liberty.io/matrix-norm-inequality-lvert-a-rvert-infty-leq-sqrt-n-lvert-a-rvert-2.htmlhttps://liberty.io/matrix-norm-inequality-lvert-a-rvert-infty-leq-sqrt-n-lvert-a-rvert-2.htmlSo I'm trying to prove that $\lVert A\rVert_\infty \leq \sqrt{n} \lVert A\rVert_2$. I've written the right hand side in terms of rows, but this method doesn't seem to be getting me anywhere. Wh......Fri, 28 Aug 2026 00:17:43 -0400https://liberty.io/can-every-irrational-number-be-written-in-terms-of-finitely-many-ratio.htmlhttps://liberty.io/can-every-irrational-number-be-written-in-terms-of-finitely-many-ratio.htmlConsider the irrational number $\sqrt{2}$. It can be written in terms, i.e., in a closed form expression, of two rational numbers as $2^{\frac{1}{2}}$. Does it hold in general that every irrational ...Fri, 28 Aug 2026 00:11:34 -0400https://liberty.io/sum-of-2-dice-rolls-is-even-probability.htmlhttps://liberty.io/sum-of-2-dice-rolls-is-even-probability.htmli have to calculate how probably it is, that the sum of 2 dice rolls is even. From my course i know that i have N^n Possibilities = 6^2 = 36. 6 = sides of dice and 2 = number of dice rolls Know i ...Fri, 28 Aug 2026 00:08:47 -0400https://liberty.io/how-do-i-negate-the-following-statement-closed.htmlhttps://liberty.io/how-do-i-negate-the-following-statement-closed.htmlHow do I negate the following statement? What are the steps which should be taken to negate an argument such as the one given? $$\exists y \forall x ((y > 0) \land (x < y))$$...Thu, 27 Aug 2026 23:56:35 -0400https://liberty.io/element-chasing-proof-that-a-setminus-c-setminus-b-setminus-c-a-setmin.htmlhttps://liberty.io/element-chasing-proof-that-a-setminus-c-setminus-b-setminus-c-a-setmin.htmlI would like to construct a formal proof of the following: $$(A\setminus C)\setminus(B\setminus C) = (A\setminus B)\setminus C$$ Let $a∈A$ be an arbitrary element, we will show that $a\in A \cap \...Thu, 27 Aug 2026 23:56:26 -0400https://liberty.io/how-to-calculate-the-dot-product-of-two-functions.htmlhttps://liberty.io/how-to-calculate-the-dot-product-of-two-functions.htmlHopefully someone can help me out with this. I have a question by my prof that goes like: Prove that the following function is orthogonal to each other (i.e., they form a basis in the space of square ...Thu, 27 Aug 2026 23:54:23 -0400https://liberty.io/why-does-i-i-equal-more-than-one-number-duplicate.htmlhttps://liberty.io/why-does-i-i-equal-more-than-one-number-duplicate.htmlAs I understand it, $i^i$ returns more than one real number. Why does this happen? When I raise $i$ to $i$, I'm raising a specific number to itself. So why do I get multiple values of this, as if i......Thu, 27 Aug 2026 23:53:32 -0400https://liberty.io/riemann-hypothesis-and-robin-039-s-inequality-implications.htmlhttps://liberty.io/riemann-hypothesis-and-robin-039-s-inequality-implications.htmlIs the following statement true: Let $\Bbb{A}$ be the set of all Natural numbers n, greater than or equal to 5041, for which the inequality $\displaystyle \sigma(n)<e^{\gamma}n\log\log n$ is not ...Thu, 27 Aug 2026 23:49:43 -0400https://liberty.io/is-integration-from-a-to-b-same-or-b-to-a-or-is-negative.htmlhttps://liberty.io/is-integration-from-a-to-b-same-or-b-to-a-or-is-negative.htmlThe integration is generally area under the curve in $\Bbb R^2 \to \Bbb R$ Is integration in the range from $a$ to $b$ the same as $b$ to $a$ or is it negative? If it is negative, Is it merely no......Thu, 27 Aug 2026 23:41:01 -0400https://liberty.io/construct-the-cross-section-of-a-cube-by-a-plane-passing-through-three.htmlhttps://liberty.io/construct-the-cross-section-of-a-cube-by-a-plane-passing-through-three.htmlI want to find/construct the cross section of a cube that includes the three points shown below. In other words, if a plane went through the cube such that it would slice where the points are, what......Thu, 27 Aug 2026 23:39:17 -0400https://liberty.io/finding-the-length-of-a-complex-vector-duplicate.htmlhttps://liberty.io/finding-the-length-of-a-complex-vector-duplicate.htmlLet us say the vector $u = (1 - i, 2 + 3i, 5)$. How would I find $\|u\|$? I have tried solving this through: $\sqrt{(1-i)^2 + (2+3i)^2 + 5^2}$ But I ended up needing to find the square root of a ...Thu, 27 Aug 2026 23:23:41 -0400https://liberty.io/calculating-probabilities-over-longer-period-of-time.htmlhttps://liberty.io/calculating-probabilities-over-longer-period-of-time.htmlThere's a great question/answer at: Calculating probabilities over different time intervals This is an awesome answer, but I'd like to ask a related question: What if the period goes the other dire......Thu, 27 Aug 2026 23:16:34 -0400https://liberty.io/proving-int-csc-2-x-dx.htmlhttps://liberty.io/proving-int-csc-2-x-dx.htmlAccording to the standard integral results, it is known that $\int \csc^2 x \, dx = -\cot x$ since $\frac{d}{dx} \cot x = - \csc^2 x$. However, supposing that this is not known, is there a better m......Thu, 27 Aug 2026 23:16:30 -0400https://liberty.io/how-to-excel-at-math.htmlhttps://liberty.io/how-to-excel-at-math.htmlI know this question has been asked at least a hundred times earlier but before you mark this as duplicate please read this cause I seriously need some professional help here. I'm a 15 year old ki......Thu, 27 Aug 2026 23:15:57 -0400https://liberty.io/find-x-as-function-of-y.htmlhttps://liberty.io/find-x-as-function-of-y.htmlI am trying to find $y$ as function of $x$ of: $$y=\frac{x-1}{x+1}$$ I have tried $$y=\frac{x-1}{x+1}\iff (x+1)y=x-1\iff(x+1)y+1=x$$ Which is of course wrong...Thu, 27 Aug 2026 23:15:55 -0400https://liberty.io/what-is-an-acyclic-connected-graph-in-graph-theory.htmlhttps://liberty.io/what-is-an-acyclic-connected-graph-in-graph-theory.htmlI want to know What it is and whether there is a difference in the definition when looking at undirected and directed graphs?...Thu, 27 Aug 2026 23:15:13 -0400https://liberty.io/about-the-definition-of-axiomatizable-theory-and-consistency.htmlhttps://liberty.io/about-the-definition-of-axiomatizable-theory-and-consistency.htmlDefinition: If $A$ is a theory and $B \subseteq A$ then $B$ is a set of axioms for $A$ iff 1) B is recursive and 2) $B \models C$ for all $C \in A$. We say $A$ is axiomatizable iff $A$ has a se......Thu, 27 Aug 2026 23:02:53 -0400https://liberty.io/trapezoidal-pyramid.htmlhttps://liberty.io/trapezoidal-pyramid.htmlIs the following problem correct? If it is correct, please present a solution. Given a trapezoidal pyramid ABCDM. The base $ABCD$ is trapezoid (where $AB\; ||\; CD$). The planes $ADM$ and $BCM$ ......Thu, 27 Aug 2026 22:52:04 -0400https://liberty.io/why-is-the-eccentricity-of-a-parabola-1.htmlhttps://liberty.io/why-is-the-eccentricity-of-a-parabola-1.htmlWhat I know: The standard form for a parabola is $(x-h)^2=4a(y-k)$, if the axis of symmetry of the parabola is vertical, or $(y-k)^2=4a(x-h)$, if the axis of symmetry of the parabola is horizontal......Thu, 27 Aug 2026 22:46:08 -0400https://liberty.io/what-is-division-sum-closed.htmlhttps://liberty.io/what-is-division-sum-closed.htmlWhat is the meaning of "division sum". I have seen the being used a lot but haven't found an explanation defining the term. Understanding from its usage, it seems to be another term for '...Thu, 27 Aug 2026 22:34:59 -0400https://liberty.io/suppose-that-x-and-y-are-irrational-but-x-y-is-rational-prove-that-x-y.htmlhttps://liberty.io/suppose-that-x-and-y-are-irrational-but-x-y-is-rational-prove-that-x-y.htmli was wondering if someone could check my proof $Q= \{a/b , c,d : a,c ∈ \mathbb Z , b,d ∈ N>0\}$ $a/b =x+y$ $a/b -y=x$ proof by contradiction. Let $x-y$ is rational $c/d = x-y$ sub $a/b -y......Thu, 27 Aug 2026 22:32:42 -0400https://liberty.io/find-the-constant-term-of-polynomial.htmlhttps://liberty.io/find-the-constant-term-of-polynomial.htmlThere's fifth degree polynomial, it's first coefficient equals $-7$. $$-7x^5+a_4x^4+a_3x^3+a_2x^2+a_1x+a_0$$ Also: $$W(1)=-2$$ $$W(2)=-4$$ $$W(3)=-6$$ $$W(4)=-8$$ $$W(5)=-10$$ Find the value of ...Thu, 27 Aug 2026 22:30:03 -0400https://liberty.io/power-method-matlab.htmlhttps://liberty.io/power-method-matlab.htmlI am trying to find the largest eigenvalue and its corresponding eigenvector by the Power method .. I did this code and I just want to display the largest eigenvalue and its eigenvector after ......Thu, 27 Aug 2026 22:22:50 -0400https://liberty.io/what-is-the-difference-between-a-linear-n-k-code-and-a-linear-n-k-d-co.htmlhttps://liberty.io/what-is-the-difference-between-a-linear-n-k-code-and-a-linear-n-k-d-co.htmlIn my lecture notes on coding theory, I have the following 2 definitions: Definition 1: Definition 2: Where: $n$ is the number of symbols in each codeword $k$ is the dimension of the vector space ...Thu, 27 Aug 2026 22:18:50 -0400https://liberty.io/for-any-real-function-if-f-0-0-then-f-039-0-0-closed.htmlhttps://liberty.io/for-any-real-function-if-f-0-0-then-f-039-0-0-closed.htmlThis seems a little too trivial but I have been facing many problems in my intro to real analysis books about functions who at first are discontinuous at zero but then "the hole is plugged" How co......Thu, 27 Aug 2026 22:06:00 -0400https://liberty.io/what-is-the-opposite-of-logarithmic-scale.htmlhttps://liberty.io/what-is-the-opposite-of-logarithmic-scale.htmlIf one wants to plot a variable whose value changes drastically, it makes sense to plot its order of magnitude. That way, the smaller values in the plot aren't squished all the way at the bottom. T......Thu, 27 Aug 2026 22:03:06 -0400https://liberty.io/the-class-of-all-classes-not-containing-themselves.htmlhttps://liberty.io/the-class-of-all-classes-not-containing-themselves.htmlIn ZF classes are used informally to resolve Russells Paradox, that is the collection of all sets that do not contain themselves does not form a set but a proper class. But doesn't the same paradox ...Thu, 27 Aug 2026 21:49:46 -0400https://liberty.io/mass-of-wire-using-line-integral.htmlhttps://liberty.io/mass-of-wire-using-line-integral.htmlIf $\rho(x, y)$ is the density of a wire (mass per unit length), then $m = \int \rho(x,y)ds$ is the mass of the wire. Find the mass of a wire having the shape of a semicircle $x = 1 + \cos(t), y = ......Thu, 27 Aug 2026 21:43:27 -0400https://liberty.io/what-is-the-mathematical-symbol-representation-for-the-remainder-opera.htmlhttps://liberty.io/what-is-the-mathematical-symbol-representation-for-the-remainder-opera.htmlIs there an equivalent in mathematical language to the modulo (or mod) function in computing?...Thu, 27 Aug 2026 21:34:29 -0400https://liberty.io/prove-the-riemann-lebesgue-lemma-int-b-af-x-cos-nx-dx-rightarrow-0-as-.htmlhttps://liberty.io/prove-the-riemann-lebesgue-lemma-int-b-af-x-cos-nx-dx-rightarrow-0-as-.htmlI would like to prove the Riemann-Lebesgue lemma, namely that $\int^b_af(x)\cos(nx)dx\rightarrow 0$ as $n\rightarrow \infty$ for any regulated function $f$. The textbook which I'm working from say......Thu, 27 Aug 2026 21:33:16 -0400https://liberty.io/express-frac-ln-left-frac-1-2-right-80000000-as-a-negative-exponent.htmlhttps://liberty.io/express-frac-ln-left-frac-1-2-right-80000000-as-a-negative-exponent.htmlI would like to express $\dfrac{\ln\left(\frac{1}{2}\right)}{80000000}$ as a negative exponent. Using Desmos online calculator I tried this variant: Why are the results different? I expected/goal ......Thu, 27 Aug 2026 21:31:52 -0400https://liberty.io/given-a-simple-graph-and-its-complement-prove-that-either-of-them-is-a.htmlhttps://liberty.io/given-a-simple-graph-and-its-complement-prove-that-either-of-them-is-a.htmlI was tasked to prove that when given 2 graphs $G$ and $\bar{G}$ (complement), at least one of them is a always a connected graph. Well, I always post my attempt at solution, but here I'm totally ......Thu, 27 Aug 2026 21:30:08 -0400