https://liberty.ioCurated star highlights with fresh energy.en-usMon, 17 Aug 2026 03:20:26 -0400https://liberty.io/distribution-and-absolute-value.htmlhttps://liberty.io/distribution-and-absolute-value.htmlI have a question about distribution and absolute values. I was solving a problem and was wondering if it would be okay to distribute a number into an absolute value with two terms. For example $3|......Mon, 17 Aug 2026 07:17:00 -0400https://liberty.io/what-is-the-definition-of-a-relation.htmlhttps://liberty.io/what-is-the-definition-of-a-relation.htmlLet $S$ be a set. An equivalence relation on $S$ is a subset $R\subseteq S\times S$ satisfying that $(s,s)\in R$ for all $s\in S$, $(s,t)\in R \implies (t,s)\in R$, $(s,t)\in R\; \land\; (t,u)\in R\...Mon, 17 Aug 2026 07:10:47 -0400https://liberty.io/should-we-use-parentheses-or-brackets-around-functions.htmlhttps://liberty.io/should-we-use-parentheses-or-brackets-around-functions.htmlShould we use parentheses or brackets around functions? Is there a widely accepted rule/style? For example, which of the following is the preferred way of writing formulas? $(f(x) + g(x)) * h(x)$......Mon, 17 Aug 2026 07:09:44 -0400https://liberty.io/weak-convergence-of-product-of-weakly-and-strongly-convergent-l-2-sequ.htmlhttps://liberty.io/weak-convergence-of-product-of-weakly-and-strongly-convergent-l-2-sequ.htmlthere is one question bothering me for quite a while now. Let $a_{n},b_{n}\in L^{2}:a_{n}\stackrel{L^{2}}{\rightharpoonup} a\in L^{2} $ weakly $ b_{n}\stackrel{L^{2}}{\rightarrow} b \in L^{2}$ str......Mon, 17 Aug 2026 06:55:50 -0400https://liberty.io/help-with-converting-the-derivative-of-cos-3-x.htmlhttps://liberty.io/help-with-converting-the-derivative-of-cos-3-x.htmlI needed to find the derivative of $\cos^3(x)$ and find it in a list of given derivatives, but I could not find $-3\sin(x)\cos^2(x)$, so I waited until I did more problems and the only one left was......Mon, 17 Aug 2026 06:44:55 -0400https://liberty.io/linear-algebra-notation-a-dagger.htmlhttps://liberty.io/linear-algebra-notation-a-dagger.htmlI'm a little confused as to some linear algebra notation and would appreciate some clarification. There exists an $n\times n$ diagonal matrix $Q$ where the entries along the diagonal are the eigen......Mon, 17 Aug 2026 06:38:18 -0400https://liberty.io/projecting-a-surface-segment-of-a-cone-onto-a-2d-plane.htmlhttps://liberty.io/projecting-a-surface-segment-of-a-cone-onto-a-2d-plane.htmlFirstly, I'd like to apologise - I do not know the correct terms for what I am asking. Assume that the top/bottom of the highlighted portion there is actually aligned with the base. To help expla......Mon, 17 Aug 2026 06:32:29 -0400https://liberty.io/extensions-and-kazhdan-039-s-property-t.htmlhttps://liberty.io/extensions-and-kazhdan-039-s-property-t.htmlIs Kazhdan's property (T) stable under extensions? i.e. if $G$ is an extension of a group with property (T) by a group with property (T), does it follow that $G$ has property (T)?...Mon, 17 Aug 2026 06:27:57 -0400https://liberty.io/the-langlands-program-for-beginners.htmlhttps://liberty.io/the-langlands-program-for-beginners.htmlAssuming that a person has taken standard undergraduate math courses (algebra, analysis, point-set topology), what other things must a person know before they can understand the Langlands program a......Mon, 17 Aug 2026 06:10:34 -0400https://liberty.io/proof-for-combination-formula-n-choose-k.htmlhttps://liberty.io/proof-for-combination-formula-n-choose-k.htmlI have been looking at this problem for a long time. Can anyone prove the combination formula using factorials N choose K? In case anyone does not know how to list all combinations in a set, you ......Mon, 17 Aug 2026 06:03:03 -0400https://liberty.io/understanding-the-ideal-generated-by-a-polynomial.htmlhttps://liberty.io/understanding-the-ideal-generated-by-a-polynomial.htmlSo my class on ring theory recently began, and I'm having a bit of trouble understanding ideals that are generated by polynomials. For an arbitrary ring, I know the definition of such an ideal, tha......Mon, 17 Aug 2026 06:00:50 -0400https://liberty.io/finding-the-inverse-of-a-number-under-a-certain-modulus.htmlhttps://liberty.io/finding-the-inverse-of-a-number-under-a-certain-modulus.htmlHow does one get the inverse of 7 modulo 11? I know the answer is supposed to be 8, but have no idea how to reach or calculate that figure. Likewise, I have the same problem finding the inverse o......Mon, 17 Aug 2026 05:49:54 -0400https://liberty.io/rational-numbers.htmlhttps://liberty.io/rational-numbers.htmlIt is a rule that: A rational number can be expressed in $\frac{a}{b}$ form and an irrational number cannot be expressed in such form. It is even said that (in order to promote the $\frac{a}{b}$......Mon, 17 Aug 2026 05:43:10 -0400https://liberty.io/what-is-the-integral-of-e-cos-x.htmlhttps://liberty.io/what-is-the-integral-of-e-cos-x.htmlQuestion: Find out $\displaystyle{\int e^{\cos x}~dx}$. My Attempt: Let $\cos x = y$. Hence $-\sin x\ dx = dy$ or $$dx = \displaystyle{\frac{-dy}{\sin x}=\frac{-dy}{\sqrt{1-\cos^2x}}=\frac{-dy}{\...Mon, 17 Aug 2026 05:28:54 -0400https://liberty.io/pontryagin-classes-of-a-product-manifold.htmlhttps://liberty.io/pontryagin-classes-of-a-product-manifold.htmlI'm imagining there's a way to relate the pontryagin classes of $T(M\times N)$ to the pontryagin classes of $M$ and those of $N$, but I haven't been able to find a helpful reference. Could someone ...Mon, 17 Aug 2026 05:21:37 -0400https://liberty.io/the-hat-function-closed.htmlhttps://liberty.io/the-hat-function-closed.htmlA hat function in 2D Cartesian is defined as $$f(x)= \begin{cases} x, & x\geq 0\\ -x, & x<0 \end{cases}$$ Could any one help me how the hat function is defined in 3D Cartesian. Note, ......Mon, 17 Aug 2026 05:12:40 -0400https://liberty.io/finding-the-scale-factor-and-rotation-angle-of-a-matrix.htmlhttps://liberty.io/finding-the-scale-factor-and-rotation-angle-of-a-matrix.htmlI was given the following question without the material appearing first in the book (I am learning independently). I would gladly look into the matter, but I am not sure where to look - because I d......Mon, 17 Aug 2026 05:09:06 -0400https://liberty.io/calculate-the-value-of-x-y-and-z-coordinates.htmlhttps://liberty.io/calculate-the-value-of-x-y-and-z-coordinates.htmlSystem of the equations: Equation 1: $$-360 = -6x + x^2 - 40y + y^2 - 20z + z^2$$ Equation 2: $$-600 = -40x + x^2 - 30y + y^2 - 100z + z^2$$ Equation 3: $$59.85 = -10x + x^2 - 4y + y^2 - 10z ......Mon, 17 Aug 2026 04:44:00 -0400https://liberty.io/how-to-solve-ln-e-2.htmlhttps://liberty.io/how-to-solve-ln-e-2.htmlI know that $$\ln e^2=2$$ But what about this? $$(\ln e)^2$$ A calculator gave 1. I'm really confused....Mon, 17 Aug 2026 04:33:12 -0400https://liberty.io/why-was-o-gabber-not-an-author-of-the-1983-039-s-faisceaux-pervers.htmlhttps://liberty.io/why-was-o-gabber-not-an-author-of-the-1983-039-s-faisceaux-pervers.htmlEssentially the title. I noticed that in the 2018 re-printed version of Faisceaux Pervers, O. Gabber is an author (making the work BBDG) as opposed to the classic Faisceaux Pervers (BBD) of 1983 in......Mon, 17 Aug 2026 04:19:44 -0400https://liberty.io/b-splines-of-degree-1-2-and-3.htmlhttps://liberty.io/b-splines-of-degree-1-2-and-3.htmlThe basis functions for linear B-splines are: \begin{equation*} \begin{aligned} & B_0(u) = (1-u)\\ & B_1(u) = u. \end{aligned} \end{equation*} For quadratic B-plines: \b......Mon, 17 Aug 2026 04:04:18 -0400https://liberty.io/finding-delta-algebraically-for-a-given-epsilon.htmlhttps://liberty.io/finding-delta-algebraically-for-a-given-epsilon.htmlFor the limit $$\lim_{x\to 5}\sqrt{x-1}=2$$ find a $\delta>0$ that works for $\epsilon=1$. In another words, find a $\delta>0$ such that for all $x$, $$0<|x-5|<\delta \implies |\sqrt......Mon, 17 Aug 2026 04:01:57 -0400https://liberty.io/square-with-perpendicular-lines-drawn-prove-that-they-are-equal.htmlhttps://liberty.io/square-with-perpendicular-lines-drawn-prove-that-they-are-equal.htmlABCD is a square. C' is a point on BA and B' is a point on AD such that BB' and CC' are perpendicular. Show that BB' = CC' I don't know where to start. Use similarities?...Mon, 17 Aug 2026 04:01:51 -0400https://liberty.io/verification-of-stokes-039-theorem-for-a-paraboloid-and-a-given-field.htmlhttps://liberty.io/verification-of-stokes-039-theorem-for-a-paraboloid-and-a-given-field.htmlI've been trying different methods but keep getting the same answer for the following question: Verify that Stokes’ Theorem is true for the given vector field F and surface S. $14.\; \vec F(x, y,......Mon, 17 Aug 2026 03:56:50 -0400https://liberty.io/technique-for-simplifying-e-g-sqrt-8-4-sqrt-3-to-sqrt-6-sqrt-2.htmlhttps://liberty.io/technique-for-simplifying-e-g-sqrt-8-4-sqrt-3-to-sqrt-6-sqrt-2.htmlHow to find the square root of an irrational expression, to simplify that root. e.g.: $$ \sqrt{ 8 - 4\sqrt{3} } = \sqrt{6} - \sqrt{2} $$ Easy to verify: \begin{align} (\sqrt{6} - \sqrt{2})^2 = 6......Mon, 17 Aug 2026 03:53:27 -0400https://liberty.io/how-do-i-simplify-a-fraction-with-a-square-root-on-the-numerator-close.htmlhttps://liberty.io/how-do-i-simplify-a-fraction-with-a-square-root-on-the-numerator-close.htmlHow do I simplify this fraction : $$\frac{(3x^2+4x)\sqrt {x+1}}{2(x+1)^2}$$ The final result is $\frac{3x^2+4x} {2(x+1)^{3/2}}$....Mon, 17 Aug 2026 03:44:39 -0400https://liberty.io/sum-of-the-series-sum-n-1-infty-n-2e-n.htmlhttps://liberty.io/sum-of-the-series-sum-n-1-infty-n-2e-n.htmlThe question is to find out the sum of the series $$\sum_{n=1}^\infty n^2 e^{-n}$$ I tried to bring the summation in some form of telescoping series but failed. I then tried approximating the sum ......Mon, 17 Aug 2026 03:42:53 -0400https://liberty.io/show-that-9-5-sqrt-3-2-sqrt-2-is-not-a-square-in-mathbb-q-sqrt-2-sqrt-.htmlhttps://liberty.io/show-that-9-5-sqrt-3-2-sqrt-2-is-not-a-square-in-mathbb-q-sqrt-2-sqrt-.htmlShow that $(9-5\sqrt 3)(2-\sqrt 2) $ is not a square in $\mathbb Q (\sqrt 2, \sqrt 3) $; equivalently, prove that $\mathbb Q (\sqrt 2, \sqrt 3, u)$ is of degree $2$ over $\mathbb Q (\sqrt 2, \sqrt ......Mon, 17 Aug 2026 03:41:00 -0400https://liberty.io/find-a-system-of-linear-equations-for-the-given-solution-set.htmlhttps://liberty.io/find-a-system-of-linear-equations-for-the-given-solution-set.html$$\{ (-1, 1, -2, 2) + \alpha(1, 2, -3, 4) + \beta(1, -2, 3, -4) \mid \alpha, \beta \in \mathbb{R}\}$$ The given solution set suggests the system of equations will have 2 pivots and 2 free variable......Mon, 17 Aug 2026 03:39:22 -0400https://liberty.io/what-039-s-the-difference-between-direction-sense-and-orientation.htmlhttps://liberty.io/what-039-s-the-difference-between-direction-sense-and-orientation.htmlI'm trying to understand the difference between the sense, orientation, and direction of a vector. According to this, sense is specified by two points on a line parallel to a vector. Orientation is ...Mon, 17 Aug 2026 03:24:52 -0400https://liberty.io/evaluate-the-integral-int-cot-x-cos-x-dx.htmlhttps://liberty.io/evaluate-the-integral-int-cot-x-cos-x-dx.htmlI am asked to integrate $$ \int \cot(x) \cos(x) \ dx $$ How ca I do that? Using the trig substitution $$ \cos^2(x) = \frac{1}{2} \left( 1+\cos(2x) \right) $$ doesn't get me any further. Thank ......Mon, 17 Aug 2026 03:18:17 -0400https://liberty.io/expectation-and-variance-of-the-pareto-distribution.htmlhttps://liberty.io/expectation-and-variance-of-the-pareto-distribution.htmlGiven the distribution funciton of the r.v. $X$ for $\alpha, \beta >0$ $$ F(x) = 1-\Big( \frac{\beta}{\beta +x}\Big)^{\alpha} $$ for $x \geq 0$ and $0$ elsewhere What is the expectation and vari......Mon, 17 Aug 2026 03:15:25 -0400https://liberty.io/how-to-solve-x-2-ln-x-0.htmlhttps://liberty.io/how-to-solve-x-2-ln-x-0.htmlI was just investigating $y = f(x) = e^{-x^2}$ and then went ahead to plot $x=f(y), y=-f(x), and x=-f(y)$, and what I got was interest rounded square shape, and I think we can calculate this area u......Mon, 17 Aug 2026 03:14:36 -0400https://liberty.io/help-finding-coordinates-of-centre-and-radius-of-circle.htmlhttps://liberty.io/help-finding-coordinates-of-centre-and-radius-of-circle.htmlI'm having trouble with some questions relating to finding the centre coordinates and radius of a circle when given the equation. I understand how to find it in the form: (x-p)^2 + (y-q)^2 = r^2 I.e ...Mon, 17 Aug 2026 02:54:33 -0400https://liberty.io/create-context-free-grammar-for-w-mid-w-is-odd-with-a-0-in-the-middle.htmlhttps://liberty.io/create-context-free-grammar-for-w-mid-w-is-odd-with-a-0-in-the-middle.htmlI need to find a CFG where the word length $|w|$ is odd. Plus there must be a $0$ in the middle. In a previous exercise I had to specify a CFG only for odd word length. I chose the following: $G ......Mon, 17 Aug 2026 02:47:11 -0400https://liberty.io/to-prove-cayley-hamilton-theorem-why-can-039-t-we-substitute-a-for-lam.htmlhttps://liberty.io/to-prove-cayley-hamilton-theorem-why-can-039-t-we-substitute-a-for-lam.htmlCayley Hamilton Theorem states that if $A$ is an $n \times n$ matrix over the field $F$ then $p(A) = 0$. We note that $p(\lambda) = \det(\lambda I - A)$. Hence, why can't we just substitute $\lamb......Mon, 17 Aug 2026 02:41:39 -0400https://liberty.io/find-the-second-derivative-d-2-y-over-d-x-2-in-terms-of-t-when-x-3-2-t.htmlhttps://liberty.io/find-the-second-derivative-d-2-y-over-d-x-2-in-terms-of-t-when-x-3-2-t.htmlThis is my attempt: $\eqalign{ & x = 3 - 2{t^2} \cr & y = {1 \over t} \cr & {{dx} \over {dt}} = - 4t \cr & {{dy} \over {dt}} = - {t^{ - 2}} = {{ - 1} \over {{t^2}}}......Mon, 17 Aug 2026 02:40:18 -0400https://liberty.io/expansion-of-cube-of-four-terms-closed.htmlhttps://liberty.io/expansion-of-cube-of-four-terms-closed.htmlHow do I get to know the pattern that appears when we open the cube of four terms ? For example, how do I get to know this pattern? $(a+b+c+d)^3=\sum a^3 $ +6 $\sum abc$ + 3$\sum b a^2$...Mon, 17 Aug 2026 02:36:46 -0400https://liberty.io/inverse-of-a-singular-matrix.htmlhttps://liberty.io/inverse-of-a-singular-matrix.htmlMy question is why we have to restrict over selves to find inverse for non singular matrix? We can define inverse of a square matrix as follows: A matrix $B$ is said to be inverse of $A$ if $BA =......Mon, 17 Aug 2026 02:36:04 -0400https://liberty.io/why-are-representations-of-quivers-such-a-big-deal.htmlhttps://liberty.io/why-are-representations-of-quivers-such-a-big-deal.htmlQuivers are directed graphs where loops and multi-arrows are allowed. And we can talk about representations of quivers by assigning each vertex a vector space and each arrow a homomorphism. Moreover, ...Mon, 17 Aug 2026 02:24:57 -0400https://liberty.io/how-to-solve-an-equation-of-the-form-ax-2-by-2-cx-dy-e-0.htmlhttps://liberty.io/how-to-solve-an-equation-of-the-form-ax-2-by-2-cx-dy-e-0.htmlI am trying to find out how to solve $ax^2 - by^2 + cx - dy + e = 0$ to get integer solutions, failing this the rational solutions. Thanks!...Mon, 17 Aug 2026 02:24:31 -0400https://liberty.io/mathematics-behind-this-card-trick.htmlhttps://liberty.io/mathematics-behind-this-card-trick.htmlSuppose I have $21$ playing cards. I distribute them in $3$ columns and tell you to choose mentally a card. Then just indicate in which column the card is. I pick up one of the columns which doe......Mon, 17 Aug 2026 02:04:55 -0400https://liberty.io/converting-an-expression-to-a-trigonometric-form.htmlhttps://liberty.io/converting-an-expression-to-a-trigonometric-form.htmlI have this expression to convert in trig form: $z=-\cos\frac{\pi}{7}+ i\sin\frac{\pi}{7}$ Tried to move the minus sine inside the $\cos$ function, but that just wouldn't work, then with the help ......Mon, 17 Aug 2026 02:00:27 -0400https://liberty.io/why-does-it-matter-if-a-graph-is-a-function.htmlhttps://liberty.io/why-does-it-matter-if-a-graph-is-a-function.htmlWe know an equation when plotted on a graph is a representation of a function if the graph passes the vertical line test. Consider $x=y^2$. Its graph is a parabola and it fails the vertical line......Mon, 17 Aug 2026 01:52:22 -0400https://liberty.io/for-real-numbers-a-and-b-when-is-the-equation-a-b-a-b-true.htmlhttps://liberty.io/for-real-numbers-a-and-b-when-is-the-equation-a-b-a-b-true.htmlI put that the statement was true only when a = 0 and b = 0 but the correct answer was that it only held true for a = 0 OR b = 0. With 'and' I figured |0 + 0| = 0 and |0 - 0| = 0. Could someone exp......Mon, 17 Aug 2026 01:41:11 -0400https://liberty.io/terminology-facet-versus-face-in-polytope.htmlhttps://liberty.io/terminology-facet-versus-face-in-polytope.htmlIn a polytope, what are the difference and relation between facet and face? How are they defined respectively? Thanks and regards!...Mon, 17 Aug 2026 01:36:08 -0400https://liberty.io/how-to-change-integral-bounds.htmlhttps://liberty.io/how-to-change-integral-bounds.htmlIn the following integral I want to change the bounds from $(0, 2)$ to $(-1, 1)$: $\displaystyle{\int_{0}^{2}(1+x)^3 dx}$ How do I change them? I know that a variable changing is needed, but don'......Mon, 17 Aug 2026 01:35:48 -0400https://liberty.io/solving-a-logarithmic-equation-log-2-2-x-1-x-log-4-144.htmlhttps://liberty.io/solving-a-logarithmic-equation-log-2-2-x-1-x-log-4-144.htmlI need to solve this: $$\log_2 (2^x-1)+x=\log_4 (144)$$ I can work out that $x=\log_2 (2^x)$ and $\log_4 (144)=log_2(12)$ but I'm stuck after that....Mon, 17 Aug 2026 01:22:56 -0400https://liberty.io/trichotomy-implies-totality-of-partial-order.htmlhttps://liberty.io/trichotomy-implies-totality-of-partial-order.htmlTheorem: A partially ordered set is totally ordered if it obeys the law of trichotomy. Things I know: A relation on some set $A$ is said to be a partially ordered set if the relation is reflexive, ...Mon, 17 Aug 2026 01:21:17 -0400https://liberty.io/prove-the-randomness-of-elements-in-list.htmlhttps://liberty.io/prove-the-randomness-of-elements-in-list.htmlI wrote a function that generates random numbers (but not with libraries) I could use numpy or statistic library in python, But generate them from scratch. Now I have a list of numbers, But the pro......Mon, 17 Aug 2026 01:18:48 -0400https://liberty.io/definition-of-an-ordered-field.htmlhttps://liberty.io/definition-of-an-ordered-field.htmlA text I'm looking at has the following definition of an ordered field: DEFINITION A field ($F$, $+$, $\cdot$) is ordered iff there is a relation $\lt$ on $F$ such that for all $\quad\quad\qua......Mon, 17 Aug 2026 01:00:02 -0400https://liberty.io/smallest-possible-dimensions-of-a-piece-of-paper-after-one-fold.htmlhttps://liberty.io/smallest-possible-dimensions-of-a-piece-of-paper-after-one-fold.htmlSo I've been thinking, can someone explain mathematically if I started with a square of paper dimensions 1x1, what the smallest single side dimension could be reached after one fold. My gut instinc......Mon, 17 Aug 2026 00:56:08 -0400https://liberty.io/how-to-express-a-sum-of-two-series-in-terms-of-known-sums-closed.htmlhttps://liberty.io/how-to-express-a-sum-of-two-series-in-terms-of-known-sums-closed.htmlIf i have $\sum_{n=1}^\infty a_n=A$ and $\sum_{n=1}^\infty b_n=B$, can i write $\sum_{n=1}^\infty a_n\cdot b_n$ in function of $A$ and $B$ ?...Mon, 17 Aug 2026 00:35:20 -0400https://liberty.io/multiplying-radicals-by-variables.htmlhttps://liberty.io/multiplying-radicals-by-variables.htmlLet's say I am to multiply $3x^2$ by ${\sqrt 8}$. Which answer is true and why? $3 {\sqrt 8x^2}$ or $3x^2 {\sqrt 8}$...Mon, 17 Aug 2026 00:30:17 -0400https://liberty.io/you-roll-a-6-sided-die-what-is-p-6-or-even-simplified-your-answer-and-.htmlhttps://liberty.io/you-roll-a-6-sided-die-what-is-p-6-or-even-simplified-your-answer-and-.htmlYou roll a 6 sided die. What is p(6 or even)? Simplified your answer and write it as a fraction or whole number. You roll a 6-sided die. I am working on this. Thanks for having me. ibcletter.pdf...Mon, 17 Aug 2026 00:30:12 -0400https://liberty.io/what-is-the-difference-between-continuous-derivative-and-derivative.htmlhttps://liberty.io/what-is-the-difference-between-continuous-derivative-and-derivative.htmlWhat is the difference between continuous derivative and derivative? According to my teachers solution to the assignment,it seems there exits difference between continuous derivative and derivative. ...Mon, 17 Aug 2026 00:27:22 -0400https://liberty.io/division-of-a-convex-shape-results-in-a-convex-shape.htmlhttps://liberty.io/division-of-a-convex-shape-results-in-a-convex-shape.htmlLet $P$ be almost a convex polygon in the plane. By almost, we mean that some edges could be arcs and not just line segments, as long as $P$ remains convex. Let $L$ be a line going through $P$, div......Mon, 17 Aug 2026 00:24:37 -0400https://liberty.io/eigenvalues-for-4-times-4-matrix.htmlhttps://liberty.io/eigenvalues-for-4-times-4-matrix.htmlShow that $0,2,4$ are the eigenvalues for the matrix $A$: $$A=\pmatrix{ 2 & -1 & -1 & 0 \\ -1 & 3 & -1 & -1 \\ -1 & -1 & 3 & -1 \\ 0 & -1 &amp......Mon, 17 Aug 2026 00:21:21 -0400https://liberty.io/finding-center-of-circle.htmlhttps://liberty.io/finding-center-of-circle.htmlHow can I calculate center of a circle $x,y$? I have 2 points on the circumference of the circle and the angle between them. The 2 points on the circle are $P_1(x_1,y_1)$ and $P_2(x_2,y_2)$. The a......Mon, 17 Aug 2026 00:12:03 -0400https://liberty.io/what-does-it-mean-to-solve-a-math-problem-analytically.htmlhttps://liberty.io/what-does-it-mean-to-solve-a-math-problem-analytically.htmlI'm reading a Calculus book for my own edification and at the beginning the pre-calculus introduction has the problem, $3x+y=7$ They talk about solving the problem graphically, analytically, and ...Mon, 17 Aug 2026 00:04:16 -0400https://liberty.io/does-l-039-hopitals-rule-hold-for-second-derivative-third-derivative-e.htmlhttps://liberty.io/does-l-039-hopitals-rule-hold-for-second-derivative-third-derivative-e.htmlDoes L'Hopitals Rule hold for second derivative, third derivative, etc...? Assuming the function is differential up to the $k$th derivative, is the following true? $$\lim_{x\to c} \frac{f(x)}{g(x)......Mon, 17 Aug 2026 00:03:35 -0400https://liberty.io/how-do-i-simplify-nested-summations-also-how-does-it-work-like-a-loop.htmlhttps://liberty.io/how-do-i-simplify-nested-summations-also-how-does-it-work-like-a-loop.html$$ \sum^{n}_{j=1}\sum^{n}_{k=1} jk $$ How do I simplify this? Also can someone explain how this works? its gonna be like this right? or am I misunderstanding it? $$ \sum^{n}_{j=1}(j+2j+3j+...+nj) $$...Sun, 16 Aug 2026 23:54:37 -0400https://liberty.io/how-to-prove-tan-a-sec-a-frac-1-sec-a-tan-a.htmlhttps://liberty.io/how-to-prove-tan-a-sec-a-frac-1-sec-a-tan-a.htmlhow to prove $\tan A+\sec A=\frac{1}{(\sec A-\tan A)}$ ? I already tried: $$\begin{align} \sin A/\cos A+1/\cos A&=1/(\sec A-\tan A)\\ \sin A+1/\cos A&=1/(\sec A-\tan A)\\ \end{align}$$...Sun, 16 Aug 2026 23:39:58 -0400https://liberty.io/the-area-of-a-truncated-pyramid-with-irregular-top-and-bottom-surface-.htmlhttps://liberty.io/the-area-of-a-truncated-pyramid-with-irregular-top-and-bottom-surface-.htmlThis question is inspired by the question The volume for truncated pyramid with irregular base. Given that we have the top and bottom surface area ($A_1$, $A_2$) of a pyramid, ad the height of the ...Sun, 16 Aug 2026 23:36:36 -0400https://liberty.io/how-to-find-the-p-value-in-the-chi-square-test.htmlhttps://liberty.io/how-to-find-the-p-value-in-the-chi-square-test.htmlI'm reading Probability and Statistical Inference, ninth edition by Hogg and on page 436, he gives an example using the chi-square goodness-of-fit test. How did he find this p-value?...Sun, 16 Aug 2026 23:30:47 -0400https://liberty.io/how-to-convert-r-cos-theta-sin-theta-to-rectangular-form.htmlhttps://liberty.io/how-to-convert-r-cos-theta-sin-theta-to-rectangular-form.htmlWe are studying polar and rectangular equations in math. The last question on the homework was: Write $r = \cos\theta + \sin\theta$ in rectangular form. I have tried a few ways with no luck. I wo......Sun, 16 Aug 2026 23:29:50 -0400https://liberty.io/what-is-kink.htmlhttps://liberty.io/what-is-kink.htmlPleas tell me that what a "Kink" is and what this sentence means: Distance functions have a kink at the interface where $d = 0$ is a local minimum....Sun, 16 Aug 2026 23:21:47 -0400https://liberty.io/is-it-possible-to-draw-this-picture-without-lifting-the-pen.htmlhttps://liberty.io/is-it-possible-to-draw-this-picture-without-lifting-the-pen.htmlSome days ago, our math teacher said that he would give a good grade to the first one that will manage to draw this: To draw this without lifting the pen and without tracing the same line more tha......Sun, 16 Aug 2026 23:20:02 -0400https://liberty.io/what-is-a-regular-submanifold.htmlhttps://liberty.io/what-is-a-regular-submanifold.htmlDefinition: Let $M$ be a non-empty subset of $\mathbb{R}^n$ and consider $m\in\mathbb{N}$ such that $1\leqslant m\leqslant n$. M is named a regular sub manifold of $\mathbb{R}^n$ of dimension $m$ ......Sun, 16 Aug 2026 23:17:14 -0400https://liberty.io/calculating-a-limit-with-floor-function.htmlhttps://liberty.io/calculating-a-limit-with-floor-function.htmlCan someone help me understand how to calculate the limit: $\lim_{n \to \infty} n [\sqrt{n+4}-\sqrt{n} ] $ ? How can I det rid the floor function ?! (Multiplying by $ \sqrt{n+4}+\sqrt{n} $ gives me ...Sun, 16 Aug 2026 23:16:26 -0400https://liberty.io/method-of-least-squares-why-is-it-preferred-duplicate.htmlhttps://liberty.io/method-of-least-squares-why-is-it-preferred-duplicate.htmlPossible Duplicate: Why do we use a Least Squares fit? To find the normal equations for derivation of the regression line we use the method of least squares, We want to make the error smallest......Sun, 16 Aug 2026 23:15:38 -0400https://liberty.io/why-multiplication-of-zero-by-any-number-is-zero-closed.htmlhttps://liberty.io/why-multiplication-of-zero-by-any-number-is-zero-closed.htmlIf I multiply $a$ by $0$, it means $a$ is added $0$ times, but I am not able to visualize $0$ times!...Sun, 16 Aug 2026 23:13:33 -0400https://liberty.io/plotting-f-x-sin-x-cos-x-by-converting-it-to-another-form.htmlhttps://liberty.io/plotting-f-x-sin-x-cos-x-by-converting-it-to-another-form.htmlAre there any trigonometric identity that can make $f(x) = \sin(x)+\cos(x)$ easier to plot? I have no idea how it becomes a sin graph shape in the end....Sun, 16 Aug 2026 23:06:14 -0400https://liberty.io/questions-tagged-multivariable-calculus.htmlhttps://liberty.io/questions-tagged-multivariable-calculus.htmlQ&A for people studying math at any level and professionals in related fields...Sun, 16 Aug 2026 23:02:51 -0400https://liberty.io/tangent-the-trig-ratio-vs-tangent-the-geometric-concept.htmlhttps://liberty.io/tangent-the-trig-ratio-vs-tangent-the-geometric-concept.htmlDo the terms "tangent" (the trigonometric ratio), which is defined as "opposite-side / adjacent-side", and "tangent" (of the curve), which is a line that touches a curve at a point, have the SAME m......Sun, 16 Aug 2026 23:02:35 -0400https://liberty.io/ratio-test-vs-cauchy-hadamard.htmlhttps://liberty.io/ratio-test-vs-cauchy-hadamard.htmlI'm sorry if this had been asked before, but there are many other questions that make it difficult to find an answer to my own. I know that the radius of convergence of a power series can be deter......Sun, 16 Aug 2026 22:56:22 -0400https://liberty.io/how-to-explain-if-a-quantity-is-a-pivot-quantity.htmlhttps://liberty.io/how-to-explain-if-a-quantity-is-a-pivot-quantity.htmlfrom what i have gathered . I know that pivots are functions that give approximate or exact assumption of CI . I'm not entirely sure what is the quantity the question is referring to. does the quan......Sun, 16 Aug 2026 22:56:05 -0400https://liberty.io/classificating-the-isometries-of-the-hyperbolic-plane.htmlhttps://liberty.io/classificating-the-isometries-of-the-hyperbolic-plane.htmlI just did the classification of the isometries of the Euclidean Plane using some results of linear algebra and euclidean geometry, guiding myself through this article: http://repositorio.ul.pt/...Sun, 16 Aug 2026 22:47:47 -0400https://liberty.io/find-the-equation-of-the-cone-z-sqrt-x-2-y-2-in-spherical-coordinates.htmlhttps://liberty.io/find-the-equation-of-the-cone-z-sqrt-x-2-y-2-in-spherical-coordinates.htmlI have the following... $$z=\sqrt{x^2+y^2}$$ I need to write this as an equation in spherical coordinates. I know that $p^2 = x^2+y^2+z^2$ and that... $$x = p\sin\phi \cos\theta$$ $$y=p\sin\phi......Sun, 16 Aug 2026 22:46:36 -0400https://liberty.io/equivalence-between-middle-excluded-law-and-double-negation-eliminatio.htmlhttps://liberty.io/equivalence-between-middle-excluded-law-and-double-negation-eliminatio.htmlIt's well-know that in intuitionistic logic, middle excluded law and double negation elimination are equivalent. For example, in Johnstone - Topo theory, I read that, in a Heyting algebra, $p\vee\......Sun, 16 Aug 2026 22:46:01 -0400https://liberty.io/show-v-x-0-is-the-only-solution-for-y-039-039-cy-0-v-0-0-v-039-0-0-c-g.htmlhttps://liberty.io/show-v-x-0-is-the-only-solution-for-y-039-039-cy-0-v-0-0-v-039-0-0-c-g.htmlShow $v(x) = 0$ is the only solution for $y'' + cy=0, v(0) = 0, v'(0)=0, c\gt 0$ I have this question as the only practice problem I can't do for my exam in thirty minutes. I can see that it is t......Sun, 16 Aug 2026 22:17:30 -0400https://liberty.io/the-pattern-in-mathematical-induction-proofs.htmlhttps://liberty.io/the-pattern-in-mathematical-induction-proofs.htmlWhen given a statement to be proven by mathmatical induction the statement tends to look like this $1 + 2 + 3 + \dots + n = \frac{n(n+1)}{2}$ so going about the proof. 1) Prove the base case $\......Sun, 16 Aug 2026 22:12:24 -0400https://liberty.io/for-compass-and-straightedge-problems-are-you-allowed-to-use-the-compa.htmlhttps://liberty.io/for-compass-and-straightedge-problems-are-you-allowed-to-use-the-compa.htmlFor compass and straightedge problems, you could have a line between two points A and B, and want to make a line the same size between C and line DE. If you placed the two points of the compass be......Sun, 16 Aug 2026 22:09:36 -0400https://liberty.io/principal-value-of-complex-number.htmlhttps://liberty.io/principal-value-of-complex-number.htmlLet $z=-1-i$, find the principal value ? Here $x=-1,y=-1$ therefore $\arg(z)=\tan \alpha=|\frac{y}{x}|=|\frac{-1}{-1}|$ Therefore, $\alpha =\tan^{-1}=\frac{\pi}{4}$ which lies between $0$ and $\f......Sun, 16 Aug 2026 22:06:15 -0400https://liberty.io/regular-and-irregular-points.htmlhttps://liberty.io/regular-and-irregular-points.htmlIn the following 2 examples, I am trying to find all the singular points of the given equations and determine whether each one is regular or irregular I can determine the singular points of the ...Sun, 16 Aug 2026 22:04:55 -0400https://liberty.io/do-carmo-differential-geometry-exercise-4-4-17.htmlhttps://liberty.io/do-carmo-differential-geometry-exercise-4-4-17.htmlLet $\alpha:I\to\Bbb R^3$ be a curve parametrized by arc length $s$, with nonzero curvature and torsion. Consider the parametrized surface $$\textbf{x}(s,v) = \alpha(s)+vb(s),\ s\in I,-\epsilon<......Sun, 16 Aug 2026 22:03:26 -0400https://liberty.io/why-do-we-need-the-commutative-axiom-of-multiplication.htmlhttps://liberty.io/why-do-we-need-the-commutative-axiom-of-multiplication.htmlI think I can prove that $ab = ba$ by other axioms. Am I wrong? Why? Edit: proof updated with notes showing where I actually used the commutative property without noticing. Proof. For any numb......Sun, 16 Aug 2026 22:01:34 -0400https://liberty.io/how-many-balls-does-he-put-into-each-green-bag.htmlhttps://liberty.io/how-many-balls-does-he-put-into-each-green-bag.htmlThis is a homework problem my son brought home -- 5th grade. Raul has 56 bouncy balls. He puts three times as many balls into red gift bags as he puts into green gift bags. If he puts the same ...Sun, 16 Aug 2026 21:54:06 -0400https://liberty.io/find-all-the-primitive-roots-of-13.htmlhttps://liberty.io/find-all-the-primitive-roots-of-13.htmlFind all the primitive roots of $13$ My attempt: Since that $13$ is a prime I need to look for $g$ such that $g^{13-1}\equiv 1\pmod{13}$ There are $\phi(12)=4$ classes modulo $12$ how can I fin......Sun, 16 Aug 2026 21:48:41 -0400https://liberty.io/integration-shortcut-to-avoid-tedious-integration-by-parts-closed.htmlhttps://liberty.io/integration-shortcut-to-avoid-tedious-integration-by-parts-closed.htmlIn addition to the common integration shortcut of $\int x \cdot e^{ax}=\frac{xe^{ax}}{a}-\frac {e^{ax}}{a^2}+c$ Is there supplemental shortcut that can help me to avoid tedious repetitive integr......Sun, 16 Aug 2026 21:36:34 -0400https://liberty.io/why-do-graphing-calculators-plot-this-function-wrong-wolfram-mathemati.htmlhttps://liberty.io/why-do-graphing-calculators-plot-this-function-wrong-wolfram-mathemati.htmlWrong graph of the function f(x)=-1^x The function in question is $f(x)=-1^x$ and the plot example is from Wolfram Mathematica. The graph is supposed to show a zig-zag otherwise known as the trian......Sun, 16 Aug 2026 21:24:34 -0400https://liberty.io/good-function-039-s.htmlhttps://liberty.io/good-function-039-s.htmlI'm trying to solve the following question: Let $(X,d)$ be a metric space. We call a continuous function $f:X\to \mathbb R$ "good function" if for every continuous function $g:X\to \mathbb R$,......Sun, 16 Aug 2026 21:19:06 -0400https://liberty.io/what-is-the-growth-rate-and-continuous-growth-rate.htmlhttps://liberty.io/what-is-the-growth-rate-and-continuous-growth-rate.htmlThe problem: $\;f(t) = 4 \cdot 2^{\,t/5}$ I know continuous is e^k, but this problem doesn't seem to work for that. Is the initial value, 4, able to be used to find the growth rates?...Sun, 16 Aug 2026 21:11:29 -0400https://liberty.io/using-logarithmic-differentiation-with-trig-functions.htmlhttps://liberty.io/using-logarithmic-differentiation-with-trig-functions.htmlMy first intuition to derive the following function - $f(x) = \sec(x^x)$ - was to take the natural log of both sides and simplify to: $\ln(f(x)) = x(\ln(\sec(x))$. However, this does not lead me to......Sun, 16 Aug 2026 20:47:36 -0400https://liberty.io/how-to-get-all-the-factors-of-a-number-using-its-prime-factorization.htmlhttps://liberty.io/how-to-get-all-the-factors-of-a-number-using-its-prime-factorization.htmlFor example, I have the number $420$. This can be broken down into its prime factorization of $$2^2 \times3^1\times5^1\times7^1 = 420 $$ Using $$\prod_{i=1}^r (a_r + 1)$$ where $a$ is the magnitud......Sun, 16 Aug 2026 20:36:49 -0400https://liberty.io/should-this-definition-of-set-equality-be-an-axiom.htmlhttps://liberty.io/should-this-definition-of-set-equality-be-an-axiom.htmlThis question is about set equality, as it is presented in Chapter 3, Set Theory, Section 1, Fundamentals, of Terence Tao's textbook, Analysis I. However, I think it is prudent to begin earlier (...Sun, 16 Aug 2026 20:23:11 -0400https://liberty.io/does-the-series-sum-n-1-infty-1-n-ln-n-converge-or-diverge.htmlhttps://liberty.io/does-the-series-sum-n-1-infty-1-n-ln-n-converge-or-diverge.htmlSuppose you have the series: $$\sum_{n=1}^\infty (-1)^n\ln(n)$$ Does it converge or diverge? You cannot apply the alternating series test since $b_n$ is not decreasing. Similarly, you cannot apply ......Sun, 16 Aug 2026 20:18:44 -0400https://liberty.io/existence-and-uniqueness-theorems-for-first-order-ode-039-s.htmlhttps://liberty.io/existence-and-uniqueness-theorems-for-first-order-ode-039-s.htmlThere is a theorem regarding the existence and uniqueness of solutions to first-order ODE's: Thm.: Consider the initial value problem $y'=f(t,y)$, $y(t_0)=y_0$. Suppose $f$ and $\frac{∂f}{∂y}......Sun, 16 Aug 2026 20:04:46 -0400https://liberty.io/what-is-the-formula-for-finding-the-summation-of-an-exponential-functi.htmlhttps://liberty.io/what-is-the-formula-for-finding-the-summation-of-an-exponential-functi.htmlI am having a hard time researching how to handle summations of functions with exponential growth or decay. I know that simple summations can be calculated as follows: $$\sum_{i=1}^{n} i = \frac{n(......Sun, 16 Aug 2026 20:03:50 -0400https://liberty.io/a-pattern-appearing-in-the-powers-of-phi.htmlhttps://liberty.io/a-pattern-appearing-in-the-powers-of-phi.html\begin{align} \phi^5 &= 11,\underline{0}901699\cdots\\ \phi^6 &= 17,\underline{9}44271\cdots\\ \phi^7 &= 29,\underline{6}34441\cdots\\ \phi^8 &= 46,\underline{9}7871\cdots\\ \phi^9 ......Sun, 16 Aug 2026 20:00:37 -0400