https://liberty.ioCurated star highlights with fresh energy.en-usSun, 30 Aug 2026 18:28:34 -0400https://liberty.io/how-to-be-good-at-proving-duplicate.htmlhttps://liberty.io/how-to-be-good-at-proving-duplicate.htmlI'm starting my Discrete Math class, and I was taught proving techniques such as proof by contradiction, contrapositive proof, proof by construction, direct proof, equivalence proof etc. I know ho......Sun, 30 Aug 2026 22:27:22 -0400https://liberty.io/partial-derivative-with-constant.htmlhttps://liberty.io/partial-derivative-with-constant.htmlGiven $$c = 0.03 + 0.08a$$ What is $\dfrac{\partial c}{\partial a}$? I'm guessing $\dfrac{\partial c}{\partial a} = 0.03 + 0.08 = 0.11$? The constant is confusing me. Thanks....Sun, 30 Aug 2026 22:15:31 -0400https://liberty.io/why-is-the-definition-of-the-absolute-value-x-1-the-way-it-is.htmlhttps://liberty.io/why-is-the-definition-of-the-absolute-value-x-1-the-way-it-is.htmlIn my notebook it is given that for the above function, we would have: $f(x) = {-(x+1), x<-1; (x+1), x\geq-1}$ What I don't get is why did we take $-1$ instead of $0$ as is the case for the fu......Sun, 30 Aug 2026 21:42:19 -0400https://liberty.io/equivalence-relation-showing-that-an-operation-is-well-defined.htmlhttps://liberty.io/equivalence-relation-showing-that-an-operation-is-well-defined.htmlLet $r \in \mathbb{N}$. Observe the equivalence relation $\sim \subseteq \mathbb{Z} \times \mathbb{Z}$ defined as $x \sim y :\Leftrightarrow (\exists k \in \mathbb{Z})(y=x+kr)$. Show that an operat......Sun, 30 Aug 2026 21:41:59 -0400https://liberty.io/the-derivative-of-the-electric-field-for-a-uniformly-charged-rod.htmlhttps://liberty.io/the-derivative-of-the-electric-field-for-a-uniformly-charged-rod.htmlThe formula for the electric field at a point due to a charge $Q$ (just considering the magnitude) at some distance $x$ away from the point is $E=\dfrac{k_eQ}{x^2}$ where $k_e$ is a constant equal to ...Sun, 30 Aug 2026 21:28:12 -0400https://liberty.io/showing-ha-h-if-and-only-if-a-belongs-to-h.htmlhttps://liberty.io/showing-ha-h-if-and-only-if-a-belongs-to-h.htmlLet be $H$ a subgroup. Show $Ha=H$ if and only if $a$ belongs to $H$. Here what I have understand of the proof: $a$ must belong to $Ha$ because $ea=a$ where $e$ is the identity in $H$. So the onl......Sun, 30 Aug 2026 21:24:55 -0400https://liberty.io/summing-the-digits-of-pi.htmlhttps://liberty.io/summing-the-digits-of-pi.htmlI find $\pi$ and other transcendental numbers quite fascinating. Lately I've been playing around with sequences and $\pi$ and found something relatively intriguing. Consider $\pi$ written in base......Sun, 30 Aug 2026 21:17:29 -0400https://liberty.io/what-is-2-32-1-in-decimal-notation.htmlhttps://liberty.io/what-is-2-32-1-in-decimal-notation.htmlI came across decimal notation, and surprisingly I have never heard of it. I have heard of scientific notation and just thought this was decimal notation. However, when I want to find a definition ......Sun, 30 Aug 2026 21:16:00 -0400https://liberty.io/definition-of-a-sheaf-making-sense-of.htmlhttps://liberty.io/definition-of-a-sheaf-making-sense-of.htmlI have started to read an english translation of Serre's FAC. Immediately a sheaf is defined. The more categorical definition given on wikipedia actually makes more sense to me, but I would like to ...Sun, 30 Aug 2026 21:15:52 -0400https://liberty.io/fourier-transform-of-a-radial-function.htmlhttps://liberty.io/fourier-transform-of-a-radial-function.htmlConsider a function $f \in L^2(\mathbb{R}^n)$ such that $f$ is radial. My question is, is the Fourier transform $\hat{f}(\xi)$ automatically radial (I can see it is even in each variable $x_i$), or......Sun, 30 Aug 2026 21:14:49 -0400https://liberty.io/proving-that-the-dot-product-is-distributive.htmlhttps://liberty.io/proving-that-the-dot-product-is-distributive.htmlI know that one can prove that the dot product, as defined "algebraically", is distributive. However, to show the algebraic formula for the dot product, one needs to use the distributive property i......Sun, 30 Aug 2026 21:03:51 -0400https://liberty.io/relation-between-curvature-and-radius-of-curvature-duplicate.htmlhttps://liberty.io/relation-between-curvature-and-radius-of-curvature-duplicate.htmlWhy is the radius of curvature the reciprocal of the curvature? How to see this intuitively as well show it rigorously?...Sun, 30 Aug 2026 21:01:52 -0400https://liberty.io/in-a-triangle-abc-the-real-part-of-the-equation-acosb-bcosa-i-asinb-bs.htmlhttps://liberty.io/in-a-triangle-abc-the-real-part-of-the-equation-acosb-bcosa-i-asinb-bs.htmlIf A B C are the angles of a triangle, and a, b, c are the corresponding sides, then the real part of $ (acosB+bcosA + i(asinB-bsinA))^n $is? My book says that the real part of this is $(acosB+bc......Sun, 30 Aug 2026 21:00:20 -0400https://liberty.io/open-loop-transfer-function.htmlhttps://liberty.io/open-loop-transfer-function.htmla block diagram system is given as per the attached image. what is the open loop transfer function? i say it is found by opening the loop and multiplying all of the terms. So that: U = KFYX. Then ...Sun, 30 Aug 2026 20:51:53 -0400https://liberty.io/what-is-the-logic-rationale-behind-the-vector-cross-product.htmlhttps://liberty.io/what-is-the-logic-rationale-behind-the-vector-cross-product.htmlI don't think I ever understood the rationale behind this. I get that the dot product $\mathbf{a} \cdot \mathbf{b} =\lVert \mathbf{a}\rVert \cdot\lVert \mathbf{b}\rVert \cos\theta$ is derived fro......Sun, 30 Aug 2026 20:48:44 -0400https://liberty.io/does-every-matrix-have-a-ref.htmlhttps://liberty.io/does-every-matrix-have-a-ref.htmlI'm wondering if anyone here has experience using Wolframalpha? I can't seem to get a row echelon form (REF) or echelon form, it always does reduced row echelon form (R-REF). Anybody know how to ge......Sun, 30 Aug 2026 20:43:41 -0400https://liberty.io/calculate-arbitrary-points-from-a-plane-equation.htmlhttps://liberty.io/calculate-arbitrary-points-from-a-plane-equation.htmlI understand how one can calculate a plane equation (ax+by+cz=d) from three points but how can you go in reverse? How can you calculate arbitrary points from a plane equation?...Sun, 30 Aug 2026 20:42:19 -0400https://liberty.io/understanding-the-derivative-as-a-linear-transformation.htmlhttps://liberty.io/understanding-the-derivative-as-a-linear-transformation.htmlIt's been a while now I am studying multivariable calculus and the concept of differentiation in space (or higher dimension). I saw relative posts but one question remains. I can't understand the c......Sun, 30 Aug 2026 20:18:58 -0400https://liberty.io/evaluate-the-integral-of-sec-2x-1-dx.htmlhttps://liberty.io/evaluate-the-integral-of-sec-2x-1-dx.htmlI got $\ln|\sec(2x +1) + \tan(2x+1)| + \text C$ as an answer. I saw that the integral of $\sec x$ is $\ln|\sec x + \tan x| + \text C$. But I feel I may have left something out because that was too ......Sun, 30 Aug 2026 20:17:50 -0400https://liberty.io/why-is-the-integral-of-dx-equal-to-x.htmlhttps://liberty.io/why-is-the-integral-of-dx-equal-to-x.htmlPlease give a thorough explanation, not just "$dx$ is the derivative of $x$, so the antiderivative of $dx$ is $x$, duh"....Sun, 30 Aug 2026 20:05:13 -0400https://liberty.io/using-exponential-shift-to-find-general-solution-of-an-ode.htmlhttps://liberty.io/using-exponential-shift-to-find-general-solution-of-an-ode.htmlWe have these 2 theorems/definitions. *For each natural number n, $(D-m)^n e^{mx} y = e^{mx}D^ny$ *If f(D) is a polynomial in D with constant coefficients then $e^{mx}f(D)y=f(D-m)e^{mx}y$$\,\,\,\......Sun, 30 Aug 2026 20:02:32 -0400https://liberty.io/is-there-a-name-for-sec-x-039-s-relationship-with-tan-x.htmlhttps://liberty.io/is-there-a-name-for-sec-x-039-s-relationship-with-tan-x.htmlIn a couple of trig identities, esp to do with integrals and derivatives, you see a relationship between tan(x) and sec(x). Similarly between csc(x) and cot(x). $ \frac{d}{dx}\tan(x) = \sec^2(x) $......Sun, 30 Aug 2026 20:00:36 -0400https://liberty.io/on-the-notion-of-set-equality.htmlhttps://liberty.io/on-the-notion-of-set-equality.htmlI should start by providing some context. I want to understand the foundations of mathematics and in doing so have been looking at different texts involving ZF set theory and category theory. I ...Sun, 30 Aug 2026 19:57:28 -0400https://liberty.io/isomorphism-between-h-h-cap-n-and-hn-n.htmlhttps://liberty.io/isomorphism-between-h-h-cap-n-and-hn-n.htmlSuppose we have a group G under operation +, and let H be a subgroup and N a normal subgroup. I want to prove that $H/H\cap N$ is isomorphic to $HN/N$. Where, if I am not mistaken: $H/H\cap N = \......Sun, 30 Aug 2026 19:56:23 -0400https://liberty.io/if-x-is-domain-and-y-is-range-what-is-z.htmlhttps://liberty.io/if-x-is-domain-and-y-is-range-what-is-z.htmlThis is a really small stupid question, but I'm currently self-teaching myself Multivariable Calculus, and I'm just curious if there is a term for this. I did a small google search and didn't find ...Sun, 30 Aug 2026 19:49:06 -0400https://liberty.io/prove-every-odd-number-is-of-the-form-2k-5-is-every-even-number-of-the.htmlhttps://liberty.io/prove-every-odd-number-is-of-the-form-2k-5-is-every-even-number-of-the.htmlI know even numbers are of the form $n=2k, k \in \mathbb Z$, and that odd numbers are of the form $n=2k+1, k \in \mathbb Z$. I'm being asked to prove that all odd numbers are of the form $n=2k+......Sun, 30 Aug 2026 19:45:06 -0400https://liberty.io/how-to-derive-the-closed-form-solution-of-geometric-series.htmlhttps://liberty.io/how-to-derive-the-closed-form-solution-of-geometric-series.htmlI have the following equation: $$g(n) = 1 + c^2 + c^3 + ... + c^n$$ The closed form solution of this series is $$g(n) = \frac{c^{n+1} -1}{c-1}$$ However, I am having a difficult time seeing the ...Sun, 30 Aug 2026 19:38:55 -0400https://liberty.io/sum-of-series-question-s-n-1-1-4-1-9-1-16-1-25-1-n-2-2.htmlhttps://liberty.io/sum-of-series-question-s-n-1-1-4-1-9-1-16-1-25-1-n-2-2.htmlProve that for any nonzero natural $n$ it is true that $$S_n = 1 + 1/4 + 1/9 + 1/16 + 1/25 + … + 1/n^2 < 2.$$ I'm sort of at a loss here. I'm not sure if there exists some formula or method to......Sun, 30 Aug 2026 19:24:50 -0400https://liberty.io/what-does-r-0-mean.htmlhttps://liberty.io/what-does-r-0-mean.htmlI'm reading this paper: http://www.aaai.org/Papers/KDD/1996/KDD96-027.pdf, and the authors used the symbol $R_{0}^{+}$ in the definition of Exact Exception Problem, such as $D: P(I) \rightarrow R_{......Sun, 30 Aug 2026 19:12:20 -0400https://liberty.io/why-do-we-essentially-need-complete-measure-space.htmlhttps://liberty.io/why-do-we-essentially-need-complete-measure-space.htmlWhile reading the motivation of complete measure space on Wikipedia, what I concluded was, completeness is not really necessary when we define on one measure space and it is necessary when we wan......Sun, 30 Aug 2026 19:06:57 -0400https://liberty.io/geodesic-sphere-using-only-regular-pentagons-and-hexagons.htmlhttps://liberty.io/geodesic-sphere-using-only-regular-pentagons-and-hexagons.htmlI know geodesic approximation to a construct a spherical dome shape needs 12 pentagons and these pentagons are regular pentagons. However when I look closely hexagons are slightly different in th......Sun, 30 Aug 2026 18:53:35 -0400https://liberty.io/continuously-changing-dimensions-of-a-rectangle.htmlhttps://liberty.io/continuously-changing-dimensions-of-a-rectangle.htmlThe dimension of a rectangle are continousl changing. The width increases at a rate of 3 in/s while the length decreases at the rate of 2 in/s. At one instant the rectangle is a 20-in square. How ......Sun, 30 Aug 2026 18:40:12 -0400https://liberty.io/show-that-b-x-n-1-p-1-b-n-x-1-n-p.htmlhttps://liberty.io/show-that-b-x-n-1-p-1-b-n-x-1-n-p.htmlIn my statistics book $B$ is defined as: $$B(x; \ n, \ p) = \sum_{y = 0}^{x}b(y; \ n, \ p) = \sum_{y = 0}^{x}{n \choose y}p^{y}(1 - p)^{n - y}$$ I want to show that: $$B(x; \ n, \ 1 - p) = 1 - B(n......Sun, 30 Aug 2026 18:33:28 -0400https://liberty.io/how-to-find-explicit-formula-for-a-sequence-seems-to-be-geometric-but-.htmlhttps://liberty.io/how-to-find-explicit-formula-for-a-sequence-seems-to-be-geometric-but-.htmlI recently came across this interesting problem, where the progression was neither arithmetic nor geometric. The problem asks for an explicit formula for the following recursive formula: $a_{1} = ......Sun, 30 Aug 2026 18:28:23 -0400https://liberty.io/how-does-full-row-rank-imply-column-space-is-r-m-for-a-m-times-n-matri.htmlhttps://liberty.io/how-does-full-row-rank-imply-column-space-is-r-m-for-a-m-times-n-matri.htmlFrom Gilbert Strang's textbook Introduction to Linear Algebra (p.159) Every matrix with full row rank has these properties $Ax=b$ has a solution for every right side $b$. The column space is the......Sun, 30 Aug 2026 18:25:42 -0400https://liberty.io/expected-number-of-steps-between-states-in-a-markov-chain.htmlhttps://liberty.io/expected-number-of-steps-between-states-in-a-markov-chain.htmlSuppose I am given a state space $S=\{0,1,2,3\}$ with transition probability matrix $\mathbf{P}= \begin{bmatrix} \frac{2}{3} & \frac{1}{3} & 0 & 0 \\[0.3em] \frac{2}{......Sun, 30 Aug 2026 18:22:19 -0400https://liberty.io/confusion-regarding-fermat-039-s-theorem-and-closed-interval-method.htmlhttps://liberty.io/confusion-regarding-fermat-039-s-theorem-and-closed-interval-method.htmlIn James Stewart's Calculus (8th edition) Fermat's Theorem is stated as If $f$ has a local maximum or minimum at $c$, and if $f'(c)$ exists, then $f'(c) = 0$. The author cautions that the convers......Sun, 30 Aug 2026 18:10:59 -0400https://liberty.io/why-divide-diameter-by-square-root-of-2-to-get-a-diameter-of-a-circle-.htmlhttps://liberty.io/why-divide-diameter-by-square-root-of-2-to-get-a-diameter-of-a-circle-.htmlI would be very grateful if you can help me with this problem. I am trying to explain in the simplest terms possible the sequence of f-stops in photography. The common f-stop rounded sequence is: f......Sun, 30 Aug 2026 18:06:05 -0400https://liberty.io/definition-for-covariant-derivative.htmlhttps://liberty.io/definition-for-covariant-derivative.htmlWhat is simple definition of the covariant derivative that looks like the definition of the derivative of a function in calculus? definition of the derivative of a function in calculus is: $$\fra......Sun, 30 Aug 2026 18:01:24 -0400https://liberty.io/has-there-ever-been-an-application-of-dividing-by-0.htmlhttps://liberty.io/has-there-ever-been-an-application-of-dividing-by-0.htmlRegarding the expression $a/0$, according to Wikipedia: In ordinary arithmetic, the expression has no meaning, as there is no number which, multiplied by $0$, gives $a$ (assuming $a\not= 0$), a......Sun, 30 Aug 2026 18:01:13 -0400https://liberty.io/proving-this-limit-lim-limits-n-to-infty-sqrt-n-n-1.htmlhttps://liberty.io/proving-this-limit-lim-limits-n-to-infty-sqrt-n-n-1.htmlThis is supposed to be proven: $$\lim\limits_{n\to\infty}\sqrt [n]n=1$$ The sequence is monotone decreasing and has a lower bound of $1$. So $\epsilon=0$ With the Archimedean Property we get: $......Sun, 30 Aug 2026 17:45:50 -0400https://liberty.io/finding-maclaurin-series-of-a-function.htmlhttps://liberty.io/finding-maclaurin-series-of-a-function.htmlThe prompt is to find the Maclaurin series of the function $$f(x) = xe^{x^2-1}$$ and evaluate the 100th derivative. At what value of x does this series converge? We know that $$e^x = \sum \frac{x......Sun, 30 Aug 2026 17:28:40 -0400https://liberty.io/how-do-derivatives-work-in-the-xyz-3d-plane.htmlhttps://liberty.io/how-do-derivatives-work-in-the-xyz-3d-plane.htmlIn just the XY plane, a derivative is the slope of the tangent line. How does it work when you have a 3D shape? I recall 3D math, and the Z-axis. Was this Calc 3? Let's say you have a "mou......Sun, 30 Aug 2026 17:22:04 -0400https://liberty.io/ode-theory-are-centers-and-linear-centers-the-same-for-reversible-syst.htmlhttps://liberty.io/ode-theory-are-centers-and-linear-centers-the-same-for-reversible-syst.htmleveryone! I'm trying to prove that a linear center of a planar system IS a center when the system is reversible (invariant under the change of variables $t\mapsto -t$ and $y \mapsto -y$). I fooled ...Sun, 30 Aug 2026 17:16:12 -0400https://liberty.io/solve-the-equation-x-2-3-6x-1-3-8-0.htmlhttps://liberty.io/solve-the-equation-x-2-3-6x-1-3-8-0.htmlI'm going around in circles on this one... Solve the equation $$x^{2/3}-6x^{1/3} + 8 = 0$$ According to the book, the answer is 8, 64, but that makes no sense to me either... Thanks Gary...Sun, 30 Aug 2026 17:10:20 -0400https://liberty.io/meaning-of-commutative-diagram.htmlhttps://liberty.io/meaning-of-commutative-diagram.htmlWhat is the meaning of a commutative diagram in mathematics? For example, if a map translate an object, then rotate it around the origin and then translate it again, is this a commutative diagram?...Sun, 30 Aug 2026 17:04:02 -0400https://liberty.io/law-of-total-expectation-with-conditional-expectations.htmlhttps://liberty.io/law-of-total-expectation-with-conditional-expectations.htmlI am confused between these two equations which use the law of total expectation: $$E(X|Y)=E(X|Y,Z)P(Z)+E(X|Y,Z')P(Z')$$ $$E(X|Y)=E(X|Y,Z)P(Z|Y)+E(X|Y,Z')P(Z'|Y)$$ where Z' is the complement of Z. ......Sun, 30 Aug 2026 16:44:41 -0400https://liberty.io/subset-of-a-finite-set-is-finite.htmlhttps://liberty.io/subset-of-a-finite-set-is-finite.htmlWe define $A$ to be a finite set if there is a bijection between $A$ and a set of the form $\{0,\ldots,n-1\}$ for some $n\in\mathbb N$. How can we prove that a subset of a finite set is finite? It......Sun, 30 Aug 2026 16:43:18 -0400https://liberty.io/confused-about-the-definition-and-properties-of-a-sigma-algebra.htmlhttps://liberty.io/confused-about-the-definition-and-properties-of-a-sigma-algebra.htmlSo I am taking an intro to measure theoretic probability course and am a bit confused by how we defined sigma-algebras and measurable sets. In particular if you have a sigma-algebra is that suffic......Sun, 30 Aug 2026 16:42:53 -0400https://liberty.io/composition-of-convex-and-power-function.htmlhttps://liberty.io/composition-of-convex-and-power-function.htmlLet $g$ be a convex nonegative function, and $p\ge1$. To show: $f(x)=g(x)^p$ is convex. Let $h(x)=x^p$. Then clearly $f=h \circ g$. Denote $\tilde{h}$ as the extended-value extension of $h$, which ...Sun, 30 Aug 2026 16:37:31 -0400https://liberty.io/finding-area-of-a-triangle-with-integrals.htmlhttps://liberty.io/finding-area-of-a-triangle-with-integrals.htmlI am suppose to find the area of a triangle using integrals with vertices 0,0 1,2 and 3,1 This gives me $y= 2x$ $y=\frac{1}{3}x$ $y= \frac{-1}{2}x+\frac{5}{2}$ for my slopes I know that I can ...Sun, 30 Aug 2026 16:37:22 -0400https://liberty.io/injectivity-surjectivity-and-cardinality.htmlhttps://liberty.io/injectivity-surjectivity-and-cardinality.htmlIt is known that if $f:A\to B$ is injective, then the $\mathrm{Card}(A)\leq\mathrm{Card}(B)$. Similarly if $f$ is surjective, the opposite inequality holds. Can one please show me how to write down a ...Sun, 30 Aug 2026 16:31:22 -0400https://liberty.io/how-to-simplify-a-function-with-fractional-exponents.htmlhttps://liberty.io/how-to-simplify-a-function-with-fractional-exponents.htmlI want to understand how to simplify fractional exponents in the formula $$g(x) = (4/3)x^{-1/3}-(5/3)x^{2/3}$$ My text book says the answer is $$\frac{4 -5x}{3x^{1/3}}$$ but I can only simplify ......Sun, 30 Aug 2026 16:29:06 -0400https://liberty.io/length-of-a-spiral.htmlhttps://liberty.io/length-of-a-spiral.htmlI need to find the length of a spiral. The spiral start at a certain radius R1 (25mm) and ends at a larger radius R2(unknown). As the spiral spins outwards, the distance between each arm of the spi......Sun, 30 Aug 2026 16:19:01 -0400https://liberty.io/solving-the-trig-inequality-sin-x-cos-x-1.htmlhttps://liberty.io/solving-the-trig-inequality-sin-x-cos-x-1.html$|\sin{x} + \cos{x} |> 1$ How to solve this kind of question? Is there any websites to learn trigonometry inequalities? My teacher only taught us the simple question but not the complicated one.......Sun, 30 Aug 2026 16:15:34 -0400https://liberty.io/optimality-conditions-necessary-vs-sufficient.htmlhttps://liberty.io/optimality-conditions-necessary-vs-sufficient.htmlFor a convex optimization problem, the first-order necessary condition says that at an optimum, the gradient is equal to zero. $\triangledown L(w*) = 0 $ The second-order sufficient condition ens......Sun, 30 Aug 2026 16:07:58 -0400https://liberty.io/what-does-9-with-a-line-above-the-9-mean.htmlhttps://liberty.io/what-does-9-with-a-line-above-the-9-mean.htmlWhat does this mean? $$\Large.\overline9 $$ I've never seen this notation before....Sun, 30 Aug 2026 15:55:10 -0400https://liberty.io/an-amazing-approximation-of-e.htmlhttps://liberty.io/an-amazing-approximation-of-e.htmlAs we can read in Wolfram Mathworld's article on approximations of $e$, the base of natural logarithm, An amazing pandigital approximation to e that is correct to $18457734525360901453873570$ de......Sun, 30 Aug 2026 15:48:53 -0400https://liberty.io/union-of-two-3x3-matrices.htmlhttps://liberty.io/union-of-two-3x3-matrices.htmlI'm trying to find the Union of two sets of matrices. The first set is all diagonal matrices, for example $$A = \begin{bmatrix} a & 0 & 0 \\ 0 & b & 0 \\ ......Sun, 30 Aug 2026 15:38:39 -0400https://liberty.io/what-is-the-difference-between-factoring-and-dividing-an-equation.htmlhttps://liberty.io/what-is-the-difference-between-factoring-and-dividing-an-equation.htmlNewbie question: Is pulling out factors and dividing the same? Please explain the difference in the examples below. Example: $$2x-4 \to 2(x-2)$$ Is it the same as $$4x^2 - 8x + 1 = 0$$ $$\fr......Sun, 30 Aug 2026 15:15:58 -0400https://liberty.io/solving-2-y-615-x-2-for-y.htmlhttps://liberty.io/solving-2-y-615-x-2-for-y.htmlHow can I transform the following equation: $$2^y = 615+x^2$$ to something of the form $$y = \cdots$$ SideQuest: What would the values of $x$ and $y$ that make the equality True, and how would you ......Sun, 30 Aug 2026 15:06:42 -0400https://liberty.io/proof-of-concavity-of-log-function.htmlhttps://liberty.io/proof-of-concavity-of-log-function.htmlDoes anybody have a proof of the concavity of the $\log{x}$ that does not use calculus?...Sun, 30 Aug 2026 15:06:27 -0400https://liberty.io/abc-is-a-triangle-such-that-bc-74-cm-and-point-d-is-on-bc-so-bd-14cm-i.htmlhttps://liberty.io/abc-is-a-triangle-such-that-bc-74-cm-and-point-d-is-on-bc-so-bd-14cm-i.htmlIn the figure below, $ABC$ is a triangle such that $\angle BAC = 150^\circ$ , $BC = 74 cm$ and point $D$ is on $BC$ such that $BD=14cm$.If $\angle ADB=60^\circ$,then what is the area, in $cm^2$, of ...Sun, 30 Aug 2026 15:04:05 -0400https://liberty.io/examples-with-not-periodic-functions.htmlhttps://liberty.io/examples-with-not-periodic-functions.htmlI'm looking for examples of non-periodic functions $f\in C^{\infty}$ satifying $|f^{(n)}(x)|\leq 1 \forall n\in \mathbb{N},\forall x\in\mathbb{R}$ I know only examples with periodic functions as......Sun, 30 Aug 2026 15:02:21 -0400https://liberty.io/is-zero-even-or-odd-duplicate.htmlhttps://liberty.io/is-zero-even-or-odd-duplicate.htmlThis question is already answered on here. head over to that link. I have been sitting here wondering if zero is even or odd and if it is even why is it even. I have no justification for why it is......Sun, 30 Aug 2026 15:00:27 -0400https://liberty.io/gram-schmidt-process-on-a-polynomial.htmlhttps://liberty.io/gram-schmidt-process-on-a-polynomial.htmlI'm currently learning Linear Algebra and I was asked to calculate the orthogonal projection of the vector $x^3$ on a subspace of $R_5[x]$ - $U = Sp\{1, x, x^2\}$ with the integral dot product in $......Sun, 30 Aug 2026 14:53:18 -0400https://liberty.io/calculus-related-rates-rectangle-area-problem-closed.htmlhttps://liberty.io/calculus-related-rates-rectangle-area-problem-closed.htmlQuestion: The length of a rectangle is increasing at a rate of $4$ inches per second while its width is decreasing at a rate of $3$ inches per second. At what rate, in square inches per second, is ......Sun, 30 Aug 2026 14:36:56 -0400https://liberty.io/natural-deduction-help-solving.htmlhttps://liberty.io/natural-deduction-help-solving.htmlI have this issue in natural deduction: B -> (A & ~A) ├ ~B I am having a really hard time understanding the rules in natural deduction - but this is what I got so far - can someone help me t......Sun, 30 Aug 2026 14:35:33 -0400https://liberty.io/evaluating-the-product-of-three-kronecker-delta.htmlhttps://liberty.io/evaluating-the-product-of-three-kronecker-delta.htmlI am trying to evaluate the following product of these 3 Kronecker delta: $\delta_{ij}\delta_{jk}\delta_{ki}$ I am not sure how to proceed. I understand that the Kronecker delta acts as a ...Sun, 30 Aug 2026 14:31:43 -0400https://liberty.io/the-least-upper-bound-and-the-greatest-lower-bound.htmlhttps://liberty.io/the-least-upper-bound-and-the-greatest-lower-bound.htmlI am taking math class, and I am not sure about LUB and GLB. I need someone to give this dummy a short explanation about them.... On interval (0,10), 0 is a lower bound and 10 is upper bound, but ......Sun, 30 Aug 2026 14:21:23 -0400https://liberty.io/trapezoidal-rule-for-system-of-ode.htmlhttps://liberty.io/trapezoidal-rule-for-system-of-ode.htmlI studied the trapezoidal rule for first-order differential equations. If the problem is $$y' = f(t, y)$$ the rule is given by the recursive formula $$y_{n + 1} = y_n + \frac12h(f(t_n, y_n) + f(t_{......Sun, 30 Aug 2026 14:20:59 -0400https://liberty.io/subgroups-of-s-n.htmlhttps://liberty.io/subgroups-of-s-n.htmlI am pretty confused right now with this. There should be a subgroup of $S_n$ with order x. So for example, for which n does $S_n$ contain a subgroup of order 60? What about if we are looking for a ...Sun, 30 Aug 2026 14:03:57 -0400https://liberty.io/why-are-natural-numbers-countable.htmlhttps://liberty.io/why-are-natural-numbers-countable.htmlI get how Cantor's diagonalization argument works for real numbers, but I don't see why you can't apply the same logic to natural numbers. I was reading this thread, but the explanations weren't ...Sun, 30 Aug 2026 14:02:46 -0400https://liberty.io/how-different-is-beta-computation-using-covariance-and-linear-regressi.htmlhttps://liberty.io/how-different-is-beta-computation-using-covariance-and-linear-regressi.htmlI wanted to compute Beta for a Stock against an Index (Say Stock X against S&P 500). I computed the daily returns for over one year applied the following logic : Beta = COVAR(X, S&P 500......Sun, 30 Aug 2026 13:47:41 -0400https://liberty.io/how-to-convert-radicals-to-decimals-without-a-calculator.htmlhttps://liberty.io/how-to-convert-radicals-to-decimals-without-a-calculator.htmlHow can one convert radicals to decimals(approximate value) when the number is not perfect such as $\sqrt2$, $\sqrt3$, $\sqrt5$, etc. Without the use of calculators....Sun, 30 Aug 2026 13:47:26 -0400https://liberty.io/proving-sequence-convergence.htmlhttps://liberty.io/proving-sequence-convergence.htmlI'm pretty confused...I understand bits and pieces but not how it all comes together...I would appreciate some help, either a written out example (you can make up one) and/or comments on how to fix......Sun, 30 Aug 2026 13:38:54 -0400https://liberty.io/understanding-related-rates-ladder-problem.htmlhttps://liberty.io/understanding-related-rates-ladder-problem.htmlI'm having some trouble really understanding this related rates problem. I will first state the problem and then point out where I'm confused. The top of a 25 foot ladder leaning against a vertica......Sun, 30 Aug 2026 13:34:08 -0400https://liberty.io/how-to-compute-the-de-rham-cohomology-of-the-punctured-plane-just-by-c.htmlhttps://liberty.io/how-to-compute-the-de-rham-cohomology-of-the-punctured-plane-just-by-c.htmlI have a classmate learning algebra.He ask me how to compute the de Rham cohomology of the punctured plane $\mathbb{R}^2\setminus\{0\}$ by an elementary way,without homotopy type,without Mayer-Vie......Sun, 30 Aug 2026 13:32:27 -0400https://liberty.io/how-to-find-if-the-points-fall-in-a-straight-line-or-not.htmlhttps://liberty.io/how-to-find-if-the-points-fall-in-a-straight-line-or-not.htmlThree points $(x_1,y_1), (x_2,y_2)$ and $(x_3,y_3)$ whether fall in a straight line or not. How do I do that?...Sun, 30 Aug 2026 13:28:07 -0400https://liberty.io/induction-with-negative-step.htmlhttps://liberty.io/induction-with-negative-step.htmlWe've learned that we can use induction to show that a statement holds for all natural numbers (or for all natural numbers above n). The steps are: prove that the statement holds for a base number b ...Sun, 30 Aug 2026 13:11:59 -0400https://liberty.io/solving-trigonometric-word-problems-with-limited-information.htmlhttps://liberty.io/solving-trigonometric-word-problems-with-limited-information.htmlThis is a diagram of this problem The word Problem: Andrew wants to construct a new stairway to the front door of his house. The door is 3 feet above the ground and at an angle of $30^{\circ}$ to ......Sun, 30 Aug 2026 13:10:41 -0400https://liberty.io/surface-parameterization-of-a-cylinder.htmlhttps://liberty.io/surface-parameterization-of-a-cylinder.htmlSuppose I wanted to parameterize the cylinder $x^2 + y^2 = R^2$ (for the purpose of computing a surface integral). Say $z$ is in range $-z_0 \le z \le z_0$. The standard parameterization I see ...Sun, 30 Aug 2026 13:06:18 -0400https://liberty.io/evaluate-integral-of-sin-ln-x-dx.htmlhttps://liberty.io/evaluate-integral-of-sin-ln-x-dx.htmlCan somebody verify this solution for me? Thanks! Evaluate $\displaystyle \int \sin(\ln(x))dx$. First, lets simplify things a bit by making the substitution $y=ln(x)$. Then $dy = \frac{1}{x}dx$ and......Sun, 30 Aug 2026 13:05:56 -0400https://liberty.io/what-is-the-name-of-a-game-that-cannot-be-won-until-it-is-over.htmlhttps://liberty.io/what-is-the-name-of-a-game-that-cannot-be-won-until-it-is-over.htmlConsider the following game: The game is to keep a friend's secret. If you ever tell the secret, you lose. As long as you don't you are winning. Clearly, it's a game that takes a lifetime to win. ...Sun, 30 Aug 2026 13:03:57 -0400https://liberty.io/1e-0-8-what-what-does-e-mean-duplicate.htmlhttps://liberty.io/1e-0-8-what-what-does-e-mean-duplicate.htmlHello I came across a math equation and I was wondering what did the "e" stand for? the equation is : y=47931x-1E+0.8 Can someone please help me by showing what the E stands for and the answer for......Sun, 30 Aug 2026 13:01:15 -0400https://liberty.io/why-is-the-number-e-so-important-in-mathematics.htmlhttps://liberty.io/why-is-the-number-e-so-important-in-mathematics.htmlI've heard a lot about this number $e$. Why is it so important? How does it fit into the 'bigger picture' of mathematics? How is it calculated and used?...Sun, 30 Aug 2026 13:00:13 -0400https://liberty.io/direction-and-magnitude-of-a-dog-running.htmlhttps://liberty.io/direction-and-magnitude-of-a-dog-running.htmlProblem: A dog in an open field runs 12.0m east and then 30.0m in a direction 54 degrees west of north. Part A: In what direction must the dog then run to end up 12.0m south of her original start......Sun, 30 Aug 2026 12:59:31 -0400https://liberty.io/finding-slope-of-a-curve-by-finding-the-limits-of-secant-slopes.htmlhttps://liberty.io/finding-slope-of-a-curve-by-finding-the-limits-of-secant-slopes.htmlFind the slope of the curve $y=x^2-4x-5$ at the point $P(3,-8)$ by finding the limit of the secant slopes through point $P$. My try: I picked another point $Q$ to get the secant $PQ$. Since $P......Sun, 30 Aug 2026 12:58:38 -0400https://liberty.io/power-series-expansion.htmlhttps://liberty.io/power-series-expansion.htmlI recently had a problem. I know how to evaluate power series but I cannot seem to find an expansion for $\sqrt{x+1}$. I've tried differentiating it, in order to bring it in reciprocal form but t......Sun, 30 Aug 2026 12:48:15 -0400https://liberty.io/what-is-an-example-of-an-involutory-matrix-which-is-not-normal.htmlhttps://liberty.io/what-is-an-example-of-an-involutory-matrix-which-is-not-normal.htmlIn another question, a user kindly explained that not all Involutory matrices are Normal. I thought they were 'very Normal': both Unitary and Hermitian, but clearly they are neither always Unitary ......Sun, 30 Aug 2026 12:47:01 -0400https://liberty.io/finding-dy-dx-by-implicit-differentiation-with-the-quotient-rule.htmlhttps://liberty.io/finding-dy-dx-by-implicit-differentiation-with-the-quotient-rule.htmlI've been stuck on a certain implicit differentiation problem that I've tried several times now. $$ \frac{x^2}{x+y} = y^2+6 $$ I know to take the derivatives of both sides and got: $$ \frac{(x+y)2x-\...Sun, 30 Aug 2026 12:39:03 -0400https://liberty.io/every-conservative-vector-field-is-irrotational.htmlhttps://liberty.io/every-conservative-vector-field-is-irrotational.htmlI have done an example where I needed to show that every conservative $C^2$ vector field is irrotational. However, there is something unclear in the solutions: Namely, I am uncertain what does the ...Sun, 30 Aug 2026 12:38:55 -0400https://liberty.io/phi-the-greek-letter-and-it-039-s-use-in-mathematics.htmlhttps://liberty.io/phi-the-greek-letter-and-it-039-s-use-in-mathematics.htmlIn the following inequality $$\frac{1+c}{a+b}\leq\varphi$$ how do I say it in English? I think that I should say: "The ratio of the sum of $1$ and $c$ to the sum of $a$ and $b$ is less than or e......Sun, 30 Aug 2026 12:34:50 -0400https://liberty.io/what-are-real-life-examples-where-a-linear-equation-would-end-up-being.htmlhttps://liberty.io/what-are-real-life-examples-where-a-linear-equation-would-end-up-being.htmlIn Algebra, linear equations in one variable can have either one solution (i.e. $x=3$), infinitely many solutions (i.e. $2x+6=3(x+2)-x$ ), or no solution (i.e. $x+3=x+5$). What would be some real-l......Sun, 30 Aug 2026 12:34:23 -0400https://liberty.io/formula-for-the-n-th-derivative-of-e-1-x-2.htmlhttps://liberty.io/formula-for-the-n-th-derivative-of-e-1-x-2.html$f(x) = \begin{cases} e^{-1/x^2} & \text{ if } x \ne 0 \\ 0 & \text{ if } x = 0 \end{cases}$ so $\displaystyle f'(0) = \lim_{x \to 0} \frac{f(x) - f(0)}{x - 0} = \lim_{x \to 0} \frac {e^{-1......Sun, 30 Aug 2026 12:30:03 -0400https://liberty.io/subtraction-when-second-number-is-bigger-than-first-number.htmlhttps://liberty.io/subtraction-when-second-number-is-bigger-than-first-number.htmlI'm a bit new to this. I'm trying to figure out how subtraction works pen and paper wise. I have a bit of a program where I can't seem to find any answers online. What I want to do is use the borr......Sun, 30 Aug 2026 12:29:03 -0400https://liberty.io/how-to-use-polyfit-function-in-matlab.htmlhttps://liberty.io/how-to-use-polyfit-function-in-matlab.htmlFirst of all i am new to Matlab, and not sure whether to use polyfit or something else for my problem. I plotted a graph with the following matlab code: t=transpose(linspace(-1,1,50)) y=1./(1+......Sun, 30 Aug 2026 12:27:13 -0400https://liberty.io/how-to-calculate-the-nth-term-and-the-sum-in-this-series-if-the-common.htmlhttps://liberty.io/how-to-calculate-the-nth-term-and-the-sum-in-this-series-if-the-common.htmlI have this series comprised of $$1,2,5,10,17,26,...$$, and so on. So far i have found that they add up in intervals being odd numbers. But I don't know how to find, let's say the 16th term and t......Sun, 30 Aug 2026 12:22:04 -0400https://liberty.io/area-of-a-rectangle-using-a-double-integral-in-polar-coordinates.htmlhttps://liberty.io/area-of-a-rectangle-using-a-double-integral-in-polar-coordinates.htmlI'm trying to find the area (which we know is equal to $ab$) of the rectangle bounded by $0\leq x\leq a$ and $0\leq y\leq b$ where $0<a<b$ using a double-integral in polar coordinates. The do......Sun, 30 Aug 2026 12:17:18 -0400https://liberty.io/how-to-represent-the-edges-of-a-complete-graph.htmlhttps://liberty.io/how-to-represent-the-edges-of-a-complete-graph.htmlLet say I have a graph $\mathcal{G}$. Denote this graph by $\mathcal{G}=(\mathcal{V}, \mathcal{E})$ where $\mathcal{V}$ is the set of vertices and $\mathcal{E}$ is the set of edges. My question is ...Sun, 30 Aug 2026 12:17:06 -0400