https://track.liberty.ioFresh celebrity buzz with instant attention value.en-usSun, 06 Sep 2026 09:18:46 -0400https://track.liberty.io/calculating-christoffel-symbol-of-the-second-kind-in-coordinates-where.htmlhttps://track.liberty.io/calculating-christoffel-symbol-of-the-second-kind-in-coordinates-where.htmlExercise 86,pg-73 of Pavel Grinfeld's Tensor Calculus Book: Derive the Christoffel symbols in spherical and cylindrical coordinates by the equation:$$\Gamma_{ij}^k= \frac12 Z^{km} \left[ \frac{\par......Sun, 06 Sep 2026 13:14:55 -0400https://track.liberty.io/how-to-find-time-period-of-a-discrete-time-signal.htmlhttps://track.liberty.io/how-to-find-time-period-of-a-discrete-time-signal.html𝑥[𝑛] = cos (𝑛/8− 𝜋) Getting the value of cos(0)=1. For that we get n=8𝜋 and for again getting cos(x)=1, we can put n=24𝜋. So the value is repeating itself in the interval of 16𝜋. How do i s......Sun, 06 Sep 2026 13:10:55 -0400https://track.liberty.io/implication-and-interpretation-of-banach-tarski.htmlhttps://track.liberty.io/implication-and-interpretation-of-banach-tarski.htmlAs I understand, the Banach-Tarski paradox says a ball in 3-space may be decomposed into finitely many pieces and reassembled into two balls each of the same size as the original. Despite being cal......Sun, 06 Sep 2026 13:07:42 -0400https://track.liberty.io/big-oh-notation-of-7x-2-confused.htmlhttps://track.liberty.io/big-oh-notation-of-7x-2-confused.htmlI'm supposed to figure out the Big-Oh notation of $7x^2$. Take a look at this. Now this says: Show that $7x^2$ is $O(x^3)$ When $x>7, 7x^2<x^3$, So let $C=1$ and $k=7$, we see $7x^2$ is......Sun, 06 Sep 2026 13:07:25 -0400https://track.liberty.io/solve-limit-containing-1-0-using-l-039-hopital-039-s-rule.htmlhttps://track.liberty.io/solve-limit-containing-1-0-using-l-039-hopital-039-s-rule.htmlI was assigned a problem in my calculus 2 course that I can't seem to solve. We're going over indeterminate forms solved by L'Hopital's Rule. $$ \lim_{x\to \pi^-} \left( \frac{\csc(x)+x}{\tan(\fr......Sun, 06 Sep 2026 13:06:41 -0400https://track.liberty.io/what-set-does-mathbb-w-denote.htmlhttps://track.liberty.io/what-set-does-mathbb-w-denote.htmlWhat set does $\mathbb W$ denote? I know this may horribly lack context, but I've seen multiple times on M.SE $\mathbb W$ used in some fairly elementary context I think....Sun, 06 Sep 2026 13:04:06 -0400https://track.liberty.io/question-about-right-triangle-and-sin-2theta.htmlhttps://track.liberty.io/question-about-right-triangle-and-sin-2theta.htmlThis is a pretty basic question but I just wanted clarification. I know that sin(theta) is opposite/hypotenuse regarding right triangles. But what would sin(2theta) be? would it be (opposite/hypote......Sun, 06 Sep 2026 12:57:20 -0400https://track.liberty.io/lattice-points-in-x-y-plane.htmlhttps://track.liberty.io/lattice-points-in-x-y-plane.htmlWhat are Lattice Points? Which points in x-y planes are Lattice Points? Is (m,n) a lattice point where m,n are any integers?...Sun, 06 Sep 2026 12:53:20 -0400https://track.liberty.io/why-is-l-infty-not-separable.htmlhttps://track.liberty.io/why-is-l-infty-not-separable.htmlMy functional analysis textbook says "The metric space $l^\infty$ is not separable." The metric defined between two sequences $\{a_1,a_2,a_3\dots\}$ and $\{b_1,b_2,b_3,\dots\}$ is $\sup\limits_......Sun, 06 Sep 2026 12:48:19 -0400https://track.liberty.io/why-do-people-interchange-between-int-and-sum-so-easily.htmlhttps://track.liberty.io/why-do-people-interchange-between-int-and-sum-so-easily.htmlOne of the things I found curious in many texts is how in certain cases interchange the $\sum$ operator with $\int$. What are the "terms" for such a swap? I understand that integration in the early......Sun, 06 Sep 2026 12:48:15 -0400https://track.liberty.io/n-points-can-be-equidistant-from-each-other-only-in-dimensions-ge-n-1.htmlhttps://track.liberty.io/n-points-can-be-equidistant-from-each-other-only-in-dimensions-ge-n-1.html2 points are from equal distance to each other in dimensions 1,2,3,... 3 points can be equidistant from each other in 2,3,... dimensions 4 points can be equidistant from each other only in dimen......Sun, 06 Sep 2026 12:47:11 -0400https://track.liberty.io/proving-when-gram-039-s-determinant-is-equal-to-zero-duplicate.htmlhttps://track.liberty.io/proving-when-gram-039-s-determinant-is-equal-to-zero-duplicate.htmlProve that Gram's determinant $G(x_1,\dots, x_n)=0$ if and only if $x_1, \dots, x_k$ are linearly dependent. So I know that $G(x_1,\dots, x_n)=\det \begin{vmatrix} \xi( x_1,x_1) & \xi( x_1,......Sun, 06 Sep 2026 12:35:03 -0400https://track.liberty.io/how-to-solve-function-equations.htmlhttps://track.liberty.io/how-to-solve-function-equations.htmlConsider: $$g(x)=x^{1/3}$$ $$f(g(x))=2x+3$$ Find $f(x)$ It should be fairly obvious that a solution is $f(x)=2x^3+3$ But is this solution unique? And what method should I use to solve such an ...Sun, 06 Sep 2026 12:34:53 -0400https://track.liberty.io/where-can-i-find-a-review-of-discrete-math.htmlhttps://track.liberty.io/where-can-i-find-a-review-of-discrete-math.htmlI'm looking for course notes and assignments and hopefully some example exams for Discrete Math, I'm taking a placement exam in the subject after having taken it 4 years ago....Sun, 06 Sep 2026 12:05:36 -0400https://track.liberty.io/why-historically-do-we-multiply-matrices-as-we-do.htmlhttps://track.liberty.io/why-historically-do-we-multiply-matrices-as-we-do.htmlMultiplication of matrices — taking the dot product of the $i$th row of the first matrix and the $j$th column of the second to yield the $ij$th entry of the product — is not a very intu......Sun, 06 Sep 2026 12:04:12 -0400https://track.liberty.io/proof-of-the-chain-rule-for-calculus.htmlhttps://track.liberty.io/proof-of-the-chain-rule-for-calculus.htmlI was comparing my attempt to prove the chain rule by my own and the proof given in Spivak's book but they seems to be rather different. Please tell me if I'm wrong or if I'm missing something. I r......Sun, 06 Sep 2026 12:00:57 -0400https://track.liberty.io/definition-of-compact-support.htmlhttps://track.liberty.io/definition-of-compact-support.htmlAccording to http://mathworld.wolfram.com/CompactSupport.html , A function has compact support if it is zero outside of a compact set. Alternatively, one can say that a function has compact suppo......Sun, 06 Sep 2026 12:00:56 -0400https://track.liberty.io/question-39-in-folland-039-s-real-analysis-chapter-3.htmlhttps://track.liberty.io/question-39-in-folland-039-s-real-analysis-chapter-3.htmlThe question "If {$F_j$} is a sequence of nonnegative increasing functions on $[a,b]$ such that $F(x)= \sum_1^\infty F_j(x) < \infty$" for all $x \in [a,b]$, then $F\prime(x)=\sum_1^\infty F\pri......Sun, 06 Sep 2026 11:49:45 -0400https://track.liberty.io/method-of-finding-a-p-adic-expansion-to-a-rational-number.htmlhttps://track.liberty.io/method-of-finding-a-p-adic-expansion-to-a-rational-number.htmlCould someone go though the method of finding a p-adic expansion of say $-\frac{1}{6}$ in $\mathbb{Z}_7?$...Sun, 06 Sep 2026 11:45:36 -0400https://track.liberty.io/what-does-mathbb-r-n-to-mathbb-r-m-mean-and-what-is-mathbb-r-n.htmlhttps://track.liberty.io/what-does-mathbb-r-n-to-mathbb-r-m-mean-and-what-is-mathbb-r-n.htmlWhat the does $\mathbb{R^n}$ mean? For example if something says that it is a transformation $T:\mathbb{R}^2 \rightarrow \mathbb{R}^3$. Does that mean that $\mathbb{R}^2 = 2 \times 2$ matrix? and t......Sun, 06 Sep 2026 11:43:23 -0400https://track.liberty.io/the-set-of-real-numbers-is-a-subset-of-the-set-of-complex-numbers.htmlhttps://track.liberty.io/the-set-of-real-numbers-is-a-subset-of-the-set-of-complex-numbers.htmlSo, I was taught that $\mathbb{Z}\subseteq\mathbb{Q}\subseteq\mathbb{R}$. But since the set of complex numbers is by definition $$\mathbb{C}=\{a+bi:a,b\in\mathbb{R}\},$$ doesn't this mean $\mathbb......Sun, 06 Sep 2026 11:36:23 -0400https://track.liberty.io/finding-the-equation-of-the-ellipse-by-completing-the-square.htmlhttps://track.liberty.io/finding-the-equation-of-the-ellipse-by-completing-the-square.htmlI have the following equation of an ellipse: $$5x^2+9y^2+40x=100$$ I need to put it in the form: $$\frac{(x-h)^2}{m^2} + \frac{(y-k)^2}{n^2} =1 $$ I was trying to complete the square with the ...Sun, 06 Sep 2026 11:33:35 -0400https://track.liberty.io/how-can-i-apply-absorption-law-in-this-expression.htmlhttps://track.liberty.io/how-can-i-apply-absorption-law-in-this-expression.htmlI have this expression: A + A * B * C' I know that i can use the Absorption law in this and the answer is A, but how can i apply that? Thank you....Sun, 06 Sep 2026 11:14:44 -0400https://track.liberty.io/why-measure-theory-for-probability.htmlhttps://track.liberty.io/why-measure-theory-for-probability.htmlI studied elementary probability theory. For that, density functions were enough. What is a practical necessity to develop measure theory? What is a problem that cannot be solved using elementary d......Sun, 06 Sep 2026 11:13:04 -0400https://track.liberty.io/convergence-of-the-series-involving-log-function.htmlhttps://track.liberty.io/convergence-of-the-series-involving-log-function.htmlI was asked about convergence of summation of $\log(n)/n^2$ and $\log(1+(1/n))/n^2$. I want to use only p series test. Is there any approximation for $\log(x)$?, Like sterling's approximation for ...Sun, 06 Sep 2026 11:05:53 -0400https://track.liberty.io/understanding-a-limiting-procedure.htmlhttps://track.liberty.io/understanding-a-limiting-procedure.htmlSay I have a function $f(x,y)$ that is defined for all $x \leq 0$ except at a single point $x=x_0 \neq 0.$ There exists one representation of the function in the domain $x < x_0$ and another ...Sun, 06 Sep 2026 11:02:58 -0400https://track.liberty.io/calculating-deposit-based-on-interest-and-required-withdrawal-in-futur.htmlhttps://track.liberty.io/calculating-deposit-based-on-interest-and-required-withdrawal-in-futur.htmlI am really stuck with example 2-5 and 2-6. I don't really understand example 2-6 and example 2-5 I just can't figure out...I was able to do example 2-4 which was easy... For example 2-4 I did F=P......Sun, 06 Sep 2026 10:59:04 -0400https://track.liberty.io/two-different-notations-for-inner-product.htmlhttps://track.liberty.io/two-different-notations-for-inner-product.htmlIn my linear algebra course on the faculty of mathematics, we used the following notation for the inner product: $$\langle v, w \rangle$$ On the other hand, my friends from the faculty of physics r......Sun, 06 Sep 2026 10:53:03 -0400https://track.liberty.io/give-a-big-o-estimate-for-a-function-f-x-use-a-simple-function-g-of-th.htmlhttps://track.liberty.io/give-a-big-o-estimate-for-a-function-f-x-use-a-simple-function-g-of-th.htmlTask Find a function $g(x)$ that is Big-O of $f(x)$ $f(x)$ is $\mathcal{O}(g(x))$ $f(x)=n\cdot log(n^2+1)+n^2\cdot log(n)$ I tried to follow/copy one of the examples in my book Discrete Mathematic......Sun, 06 Sep 2026 10:46:59 -0400https://track.liberty.io/solve-surface-integral-of-a-scalar-function-using-divergence-theorem.htmlhttps://track.liberty.io/solve-surface-integral-of-a-scalar-function-using-divergence-theorem.htmlUse the Divergence Theorem to evaluate: $$\iint_S (2x+2y+z^2) dS$$ where $S$ is the sphere $$x^2+y^2+z^2=1.$$ Most of the examples on the book that needs to be solved using divergence theorem are ...Sun, 06 Sep 2026 10:29:24 -0400https://track.liberty.io/how-can-i-find-the-basis-solutions-of-homogeneous-linear-ode.htmlhttps://track.liberty.io/how-can-i-find-the-basis-solutions-of-homogeneous-linear-ode.htmlSecond order linear differential equation is given below. $y''+\frac{2}{x}y'+k^2y=0,$ where $k$ is constant and $x\neq 0$ I already know that the basis are $y_1=\frac{e^{-ikx}}{x}$ and $y_2=\fra......Sun, 06 Sep 2026 10:28:46 -0400https://track.liberty.io/laplace-transform-for-solving-differential-equation.htmlhttps://track.liberty.io/laplace-transform-for-solving-differential-equation.htmlI solved the following task, but since I am new in this field I need to check if it is correct or if there is anything I am missing or doing wrong. Task : Solve differential equation using Laplace ...Sun, 06 Sep 2026 10:26:44 -0400https://track.liberty.io/user-allen-knutson-mathematics-stack-exchange.htmlhttps://track.liberty.io/user-allen-knutson-mathematics-stack-exchange.htmlQ&A for people studying math at any level and professionals in related fields...Sun, 06 Sep 2026 10:26:08 -0400https://track.liberty.io/finding-value-of-k-for-which-fg-x-k-has-equal-roots.htmlhttps://track.liberty.io/finding-value-of-k-for-which-fg-x-k-has-equal-roots.htmlI've been going through this community and I Find this really helpful. About me(I know I should be precise but ya), I'm just a highschool student who can't afford any coachings/schools. Self school......Sun, 06 Sep 2026 10:22:33 -0400https://track.liberty.io/how-do-i-graph-2-30-on-a-polar-grid-closed.htmlhttps://track.liberty.io/how-do-i-graph-2-30-on-a-polar-grid-closed.htmlI tried to graph the point $(-2,30)$ on a polar graph and I think I got it wrong, how should it be graphed?...Sun, 06 Sep 2026 10:12:26 -0400https://track.liberty.io/number-of-lines-to-connect-n-times-n-dots.htmlhttps://track.liberty.io/number-of-lines-to-connect-n-times-n-dots.htmlThe following is a popular riddle: Draw a $3 \times 3$ grid, and connect all the dots using only $4$ straight consecutive lines. The solution is to think outside of the box and do the followin......Sun, 06 Sep 2026 10:10:54 -0400https://track.liberty.io/uniqueness-of-hamiltonian-cycle.htmlhttps://track.liberty.io/uniqueness-of-hamiltonian-cycle.htmlLet G be a graph on $n$ vertices, where $n \geq 8$ and with minimum degree at least $n/2$. Does $G$ have at least 2 distinct Hamilton cycles? (Two Hamilton cycles are said to be distinct if they di......Sun, 06 Sep 2026 09:55:05 -0400https://track.liberty.io/solution-of-tanx-x.htmlhttps://track.liberty.io/solution-of-tanx-x.htmlHow do I find the solutions of tanx = x upto any number of decimals? (Of course, there is the graphical method but it just helps in finding the approximate value...)...Sun, 06 Sep 2026 09:53:29 -0400https://track.liberty.io/derivative-of-cost-function-for-logistic-regression.htmlhttps://track.liberty.io/derivative-of-cost-function-for-logistic-regression.htmlI am going over the lectures on Machine Learning at Coursera. I am struggling with the following. How can the partial derivative of $$J(\theta)=-\frac{1}{m}\sum_{i=1}^{m}y^{i}\log(h_\theta(x^{i}))+......Sun, 06 Sep 2026 09:50:06 -0400https://track.liberty.io/rotate-triangle-abc-around-the-origin.htmlhttps://track.liberty.io/rotate-triangle-abc-around-the-origin.htmlHow do you get from triangle ABC (blue) to triangle ABC (red) The instructions are to rotate it $270$ degrees I am trying to help a friend but forgot how to do this? Is it using a formula?...Sun, 06 Sep 2026 09:48:54 -0400https://track.liberty.io/what-is-the-opposite-of-a-cross-term.htmlhttps://track.liberty.io/what-is-the-opposite-of-a-cross-term.htmlWhen we multiply out $(x + y)(x + y)$, we refer to the two $xy$ terms as "cross terms". Is there a corresponding term for the $x^2$ and $y^2$ terms?...Sun, 06 Sep 2026 09:46:02 -0400https://track.liberty.io/what-is-the-difference-between-a-sound-argument-and-a-valid-argument.htmlhttps://track.liberty.io/what-is-the-difference-between-a-sound-argument-and-a-valid-argument.htmlIn my notes, these are the definitions of a valid argument An argument form is valid if and only if whenever the premises are all true, then conclusion is true. An argument is valid if its arg......Sun, 06 Sep 2026 09:44:17 -0400https://track.liberty.io/how-do-i-write-e-x-2-as-a-summation.htmlhttps://track.liberty.io/how-do-i-write-e-x-2-as-a-summation.htmlI am new to power series. I know how to write $e^x$ as a summation, but i do not know how that helps me....Sun, 06 Sep 2026 09:35:22 -0400https://track.liberty.io/how-to-calculate-critical-value-in-hypothesis-testing.htmlhttps://track.liberty.io/how-to-calculate-critical-value-in-hypothesis-testing.htmlYou wish to test the following claim ($H_a$) at a significance level of alpha=0.10. $H_o: \mu=70.4$ $H_a: \mu<70.4$ You believe the population is normally distributed and you know the standard ...Sun, 06 Sep 2026 09:31:48 -0400https://track.liberty.io/elementary-product-rule-for-ito-formula.htmlhttps://track.liberty.io/elementary-product-rule-for-ito-formula.htmlIn "Baxter, Rennie" book, there is an explanation of product rule for Ito formula. They apply Ito formula (without any details) to $$\frac{1}{2}((X_t + Y_t)^2 - X_t^2 - Y_t^2) = X_tY_t$$ and ob......Sun, 06 Sep 2026 09:14:39 -0400https://track.liberty.io/integrate-int-sqrt-frac-x-3-3-x-11-dx-closed.htmlhttps://track.liberty.io/integrate-int-sqrt-frac-x-3-3-x-11-dx-closed.html$$ \int \sqrt{\frac{x^3-3}{x^{11}}}\ dx $$ It seems like substitution could not do it. Is there another way?...Sun, 06 Sep 2026 09:14:37 -0400https://track.liberty.io/condition-of-the-mean-value-theorem.htmlhttps://track.liberty.io/condition-of-the-mean-value-theorem.htmlThe usual formulation of the mean value theorem in a real analysis course is something like this: Let $f\colon [a,b] \to \mathbb{R}$ be continous on $[a,b]$ and differentiable on $]a,b[$. Then the......Sun, 06 Sep 2026 09:12:43 -0400https://track.liberty.io/how-to-find-the-limit-of-cos-2x-x-as-x-approaches-zero-duplicate.htmlhttps://track.liberty.io/how-to-find-the-limit-of-cos-2x-x-as-x-approaches-zero-duplicate.htmlBasicly the title. I know it's easy but I can't figure it out. [Edit] Not a duplicate has a 2x instead of x....Sun, 06 Sep 2026 09:11:28 -0400https://track.liberty.io/find-from-first-principle-the-derivative-of.htmlhttps://track.liberty.io/find-from-first-principle-the-derivative-of.htmlFind, from first principle, the derivative of: $$\log (ax+b)$$ My Attempt: $$f(x)=\log (ax+b)$$ $$f(x+\Delta x)=\log (ax+a\Delta x+b)$$ Now, $$f'(x)=\lim_{\Delta x\to 0} \dfrac {f(x+\Delta x)-f(x)}{\...Sun, 06 Sep 2026 09:10:59 -0400https://track.liberty.io/why-two-line-coincide-in-the-product-of-reflections.htmlhttps://track.liberty.io/why-two-line-coincide-in-the-product-of-reflections.htmlExample. Let $l$ and $m$ be two different lines intersecting at a point $P$. Show that $R_m R_l =R_{p, \theta}$ where $B$ is twice the directed angle $\alpha$ from $l$ to $m$. Solution. Let $h$ be......Sun, 06 Sep 2026 09:07:16 -0400https://track.liberty.io/how-do-you-divide-an-inequality-by-another-inequality.htmlhttps://track.liberty.io/how-do-you-divide-an-inequality-by-another-inequality.htmlThe question I'm having trouble with is follows: Suppose you have already proved the proposition that, “If $a$ and $b$ are nonnegative real numbers, then $\frac{a + b}{2} ≥ \sqrt{ab}$.” a. Explai......Sun, 06 Sep 2026 09:01:14 -0400https://track.liberty.io/the-rotation-group-so-3-may-be-mapped-to-a-2-sphere-by-sending-a-rotat.htmlhttps://track.liberty.io/the-rotation-group-so-3-may-be-mapped-to-a-2-sphere-by-sending-a-rotat.htmlThe rotation group $SO(3)$ may be mapped to a $2$-sphere by sending a rotation matrix to its first column. How can I describe the fibres of the map?...Sun, 06 Sep 2026 09:00:41 -0400https://track.liberty.io/how-do-you-prove-a-number-is-prime.htmlhttps://track.liberty.io/how-do-you-prove-a-number-is-prime.htmlI am a software engineer but try to keep up with maths as I really enjoyed the subject at school. I just saw a great TED talk: Why I fell in love with monster prime numbers The talk states that the ...Sun, 06 Sep 2026 08:30:55 -0400https://track.liberty.io/proving-that-the-composition-of-relations-are-equal-to-each-other.htmlhttps://track.liberty.io/proving-that-the-composition-of-relations-are-equal-to-each-other.htmlPlease excuse my English if it's not understandable, exercises are translated so I don't know all the English terms in math. So I'm doing exercises regarding relations and compositions, and one of ......Sun, 06 Sep 2026 08:21:13 -0400https://track.liberty.io/finding-the-error-of-a-taylor-series.htmlhttps://track.liberty.io/finding-the-error-of-a-taylor-series.htmlWhen calculating the error of a Taylor series, the formula is as follows: $$R_n(x)=\frac{f^{(n+1)}(z)(x-c)^{(n+1)}}{(n+1)!}$$ $z$ is the maximum value of the expression on the interval between $x$......Sun, 06 Sep 2026 08:15:00 -0400https://track.liberty.io/two-sides-and-angle-between-them-triangle-question.htmlhttps://track.liberty.io/two-sides-and-angle-between-them-triangle-question.htmlis it possible to find the third side of a triangle if you know the lengths of the other two and the angle between the known sides? the triangle is not equilateral. we're using the kinect camera a......Sun, 06 Sep 2026 08:14:25 -0400https://track.liberty.io/what-is-a-coordinate-function-x-i-of-a-manifold-given-a-chart-u-x.htmlhttps://track.liberty.io/what-is-a-coordinate-function-x-i-of-a-manifold-given-a-chart-u-x.htmlI am trying to understand the notes here: http://unapologetic.wordpress.com/2011/04/13/cotangent-vectors-differentials-and-the-cotangent-bundle/. Specifically, this sentence: If we have local ...Sun, 06 Sep 2026 08:13:23 -0400https://track.liberty.io/if-e-euler-039-s-constant-1-frac-1-n-n-as-n-approaches-infinity-why-is.htmlhttps://track.liberty.io/if-e-euler-039-s-constant-1-frac-1-n-n-as-n-approaches-infinity-why-is.htmlIf $e$ (Euler's constant) $= (1+\frac{1}{n})^n$ as $n$ approaches infinity, why is $e^x$ not equal to $e$ if x equals any number, real or not real. Let r = any real number I think that $\left(1+\......Sun, 06 Sep 2026 08:05:53 -0400https://track.liberty.io/local-max-min-critical-points-of-integral.htmlhttps://track.liberty.io/local-max-min-critical-points-of-integral.htmlGiven $$f(t) = \int_0^t \frac{x^2+14x+45}{1+\cos^2(x)}dx $$ I need to find the local max of f(t). Well here using the fundamental theorem of calculus, I know I can just replace the $x$ with $t$. B......Sun, 06 Sep 2026 08:02:00 -0400https://track.liberty.io/show-that-mathbb-r-when-equipped-with-the-standard-topology-is-not-com.htmlhttps://track.liberty.io/show-that-mathbb-r-when-equipped-with-the-standard-topology-is-not-com.htmlProof: Let $\mathscr{U}$ := {$U_n$}$_{n \geq 1}$ = {$(n-1,n+1)$}$_{n \geq 1}$ be an open cover of $\mathbb{R}$. Consider finitely many of them, say: $U_{n_1} \ldots U_{n_k}$. Setting $N := \max(n_1 \...Sun, 06 Sep 2026 07:55:05 -0400https://track.liberty.io/is-there-a-correct-mathematical-term-for-the-inverse-of-the-slope.htmlhttps://track.liberty.io/is-there-a-correct-mathematical-term-for-the-inverse-of-the-slope.htmlLinear graphs all have a slope that can be calculated by deviding the progress on the y-axis by the progress on the x-axis. Is there a correct term to refer to the inverse of the slope, that means ......Sun, 06 Sep 2026 07:48:16 -0400https://track.liberty.io/what-is-the-affine-connection-and-what-is-the-intuition-behind-for-aff.htmlhttps://track.liberty.io/what-is-the-affine-connection-and-what-is-the-intuition-behind-for-aff.htmlHere is the definition of affine connection, as appears in Milnor's book Morse Theory. DEFINITION. An affine connection at a point $p \in \text{M}$ is a function which assigns to each tangent vect......Sun, 06 Sep 2026 07:46:51 -0400https://track.liberty.io/polynomial-division-modulo-5-gcd-of-two-polynomials.htmlhttps://track.liberty.io/polynomial-division-modulo-5-gcd-of-two-polynomials.htmlI have to find the gcd $h$ of $f$ and $g$ in $\mathbb{F}_5[X]$. $f=X^9+X^8+X^7+X^6+X^5+X^4+X^3+X^2+X+1,$ $g=X^4+X-2.$ Here is my polynomial division: My solution is $3X^3-X^2+3X+1$ but this is ......Sun, 06 Sep 2026 07:42:25 -0400https://track.liberty.io/prove-that-an-isometry-preserves-straight-lines-using-intuitive-geomet.htmlhttps://track.liberty.io/prove-that-an-isometry-preserves-straight-lines-using-intuitive-geomet.htmlThis is an exercise of the book "Basic Mathematics" by Serge Lang, p.145. I've been working on this proof for a few days and I can't seem to make it more coherent than this. Would appreciate some h......Sun, 06 Sep 2026 07:38:03 -0400https://track.liberty.io/what-makes-the-cauchy-principal-value-the-correct-value-for-a-integral.htmlhttps://track.liberty.io/what-makes-the-cauchy-principal-value-the-correct-value-for-a-integral.htmlI haven't been able to find a good answer to this searching around online. There is a related old question here, but it never received much attention. Suppose I have some physical property that I ...Sun, 06 Sep 2026 07:36:54 -0400https://track.liberty.io/calculating-expected-value-and-variance-of-a-probability-density-funct.htmlhttps://track.liberty.io/calculating-expected-value-and-variance-of-a-probability-density-funct.htmlIf I was given a probability density function: $$f(y) = \left\{\begin{array}{ll}\frac{3y^2(4-y)}{64} & \textrm{for } 0 \leq y \leq 4\\ 0 & \textrm{elsewhere} \end{array}\right.......Sun, 06 Sep 2026 07:26:38 -0400https://track.liberty.io/prove-taylor-expansion-with-mean-value-theorem.htmlhttps://track.liberty.io/prove-taylor-expansion-with-mean-value-theorem.htmlhttp://vapour-trail.blogspot.com/2006/03/brief-explanation-of-taylor-series-via.html provides an intuitive derivation of Taylor expansions from the mean value theorem that confuses me. The derivat......Sun, 06 Sep 2026 07:18:28 -0400https://track.liberty.io/how-to-find-the-exact-value-of-the-cosine-of-50-degree-angle.htmlhttps://track.liberty.io/how-to-find-the-exact-value-of-the-cosine-of-50-degree-angle.htmlI want to know the exact value of $\cos 50^\circ$. Actually I have already tried lot of times to solve but I cannot find the exact value of $\cos 50^\circ$....Sun, 06 Sep 2026 07:07:44 -0400https://track.liberty.io/calculating-the-coordinates-of-end-terminal-point-of-an-arc-from-known.htmlhttps://track.liberty.io/calculating-the-coordinates-of-end-terminal-point-of-an-arc-from-known.htmlI have searched the site and found similar questions of mine but not exaclty I needed. I have an arc. I know the coordinates of the arc center (P0), r (radius), coordinates of starting point of th......Sun, 06 Sep 2026 06:55:05 -0400https://track.liberty.io/more-comprehensive-general-topology-039-s-book-than-engelking.htmlhttps://track.liberty.io/more-comprehensive-general-topology-039-s-book-than-engelking.htmlOther than Engelking General Topology, I also come across other graduate general topology text such as Dugundji and Kelley, which I also find them interesting. However, I find Engelking's book more ...Sun, 06 Sep 2026 06:54:52 -0400https://track.liberty.io/solve-for-x-in-matrix-equation.htmlhttps://track.liberty.io/solve-for-x-in-matrix-equation.htmlHow can I solve for $X$ in this matrix equation? $$\begin{bmatrix}-3&-8\\-9&5\end{bmatrix} X + \begin{bmatrix}4&-7\\3&-2\end{bmatrix} = \begin{bmatrix}5&8\\-1&-1\end{bmatri......Sun, 06 Sep 2026 06:53:21 -0400https://track.liberty.io/why-is-this-answer-wrong-point-of-intersection-between-parabola-and-li.htmlhttps://track.liberty.io/why-is-this-answer-wrong-point-of-intersection-between-parabola-and-li.htmlQuestion: Use the discriminant to determine the number of points of intersection of the line $y=3x+5$ and the quadratic functions $f(x)=3x^2-2x-4$. Solve to find the points of intersection. U......Sun, 06 Sep 2026 06:49:43 -0400https://track.liberty.io/how-to-solve-5x5-grid-with-16-diagonals.htmlhttps://track.liberty.io/how-to-solve-5x5-grid-with-16-diagonals.htmlI have a grid 5x5 and I have to fill 16 little squares with a diagonal Rules You cannot place more than 1 diagonal in each square The diagonals cannot touch each other (example bellow) Click her......Sun, 06 Sep 2026 06:44:30 -0400https://track.liberty.io/how-to-calculate-the-transformation-matrix-if-the-coordinate-system-tr.htmlhttps://track.liberty.io/how-to-calculate-the-transformation-matrix-if-the-coordinate-system-tr.htmlI was trying to calculate the new coordinates of the 4 corners of the rectangle in the image, but I think I only formed the rotation matrix. what is the correct transformation matrix for this opera......Sun, 06 Sep 2026 06:44:03 -0400https://track.liberty.io/solutions-manual-for-analysis-on-manifolds.htmlhttps://track.liberty.io/solutions-manual-for-analysis-on-manifolds.htmlA few months ago,I wanted to learn something fundmental about manifolds. From highly recommend , I decided to choice Analysis on Manifolds by James R.Munkres as my self-learning textbook.Until now ,I ...Sun, 06 Sep 2026 06:33:15 -0400https://track.liberty.io/properties-of-a-non-invertible-square-matrix.htmlhttps://track.liberty.io/properties-of-a-non-invertible-square-matrix.htmlI am wondering what properties does $A$, an $n\times n$ non invertible matrix, have. An obvious one is $\det(A)=0$ But, I am not sure about other intuitions that I am having. For example: $rank(......Sun, 06 Sep 2026 06:31:30 -0400https://track.liberty.io/how-can-i-determine-the-value-of-angle-x-in-this-figure.htmlhttps://track.liberty.io/how-can-i-determine-the-value-of-angle-x-in-this-figure.html[![I used geometry but Did not find the answer. ][1]][1] **I used Geometry but not avail ** [1]: https://i.stack.imgur.com/Ejxoz.jpg N:B The fiqure is a square. Here is what I tred - When using ...Sun, 06 Sep 2026 06:30:43 -0400https://track.liberty.io/find-if-three-points-in-3-dimensional-space-are-collinear.htmlhttps://track.liberty.io/find-if-three-points-in-3-dimensional-space-are-collinear.htmlFind if the points joining $A=(6,7,1), B=(2,-3,1)$ and $C=(4,-5,0)$ are collinear. How to determine collinearity in three dimensions? In two dimensions, one can compare the slopes of segments $AB$......Sun, 06 Sep 2026 06:29:44 -0400https://track.liberty.io/geometry-proof-convex-quadrilateral.htmlhttps://track.liberty.io/geometry-proof-convex-quadrilateral.htmlA quadrilateral $ABCD$ is formed from four distinct points (called the vertices), no three of which are collinear, and from the segments $AB$, $CB$, $CD$, and $DA$ (called the sides), which have no ...Sun, 06 Sep 2026 06:29:22 -0400https://track.liberty.io/how-do-you-validate-a-pmf-closed.htmlhttps://track.liberty.io/how-do-you-validate-a-pmf-closed.html$$P\left(X=x\right)=\left\{ \begin{array}{rl} 0.02 & X=0\\ 0.03 & X=1\\ 0.2 & X\in\left\{ 2,\,5\right\} \\ 0.25 & X\in\left\{ 3,\,4\right\} \\ c & X=6 \end{array}\right.$$ I ha......Sun, 06 Sep 2026 06:28:34 -0400https://track.liberty.io/why-there-is-no-sign-of-logic-symbols-in-mathematical-texts.htmlhttps://track.liberty.io/why-there-is-no-sign-of-logic-symbols-in-mathematical-texts.htmlEither in undergraduate or graduate textbooks on Mathematics (Real/Complex Analysis, General Topology, Differential Geometry, ...), I never saw symbols $\Rightarrow$, $\iff$, $\forall$, $\exists$, ......Sun, 06 Sep 2026 06:28:10 -0400https://track.liberty.io/maximising-largest-integer-on-a-list-given-mean-median-and-mode.htmlhttps://track.liberty.io/maximising-largest-integer-on-a-list-given-mean-median-and-mode.htmlA list of $10$ positive integers has a mean of $11$, a median of $10$ and a unique mode of $7$. What is the largest possible value of an integer in the list? In an ascending (albeit not strictly ...Sun, 06 Sep 2026 06:24:24 -0400https://track.liberty.io/find-the-first-three-non-zero-terms-of-the-taylor-series-of-f.htmlhttps://track.liberty.io/find-the-first-three-non-zero-terms-of-the-taylor-series-of-f.htmlConsider the function: $$f(x) = \frac{1}{x^2+10}$$ Find the first three non-zero terms of the Taylor series of...Sun, 06 Sep 2026 06:22:43 -0400https://track.liberty.io/how-to-calculator-plus-minus-quadratic.htmlhttps://track.liberty.io/how-to-calculator-plus-minus-quadratic.htmlThe book says there should be a quadratic formula, but I don't understand how to solve it. I have: $\pm 6.0 = \sqrt(\frac{36 - 4(4.0)( \pm 7.04)}{14.2})$ I don't understand how to input this into a ...Sun, 06 Sep 2026 06:00:47 -0400https://track.liberty.io/how-to-calculate-sin-65-without-a-calculator.htmlhttps://track.liberty.io/how-to-calculate-sin-65-without-a-calculator.htmlI know about the sum and difference formula but I can't think of two values which will be able to use for sin(65). Therefore, I come to the question: How to calculate sin(65) without a calculator....Sun, 06 Sep 2026 06:00:03 -0400https://track.liberty.io/q-quadratic-division-how-to-divide-two-quadratics.htmlhttps://track.liberty.io/q-quadratic-division-how-to-divide-two-quadratics.htmlMy studies into graphs and models following examples from Khan Academy has helped me on my goal to learn how to chart and model via the quadratic formula. However, while I have been successful in ...Sun, 06 Sep 2026 05:59:14 -0400https://track.liberty.io/what-039-s-the-difference-between-a-proof-and-a-derivation.htmlhttps://track.liberty.io/what-039-s-the-difference-between-a-proof-and-a-derivation.htmlI did a BSc in Theoretical Physics, meaning that a lot of my time was spent deriving equations, making hand-wavy arguments, and arriving at solutions with a distinct lack of rigour. I'm now doing......Sun, 06 Sep 2026 05:57:23 -0400https://track.liberty.io/why-does-r-cos-theta-produce-a-circle.htmlhttps://track.liberty.io/why-does-r-cos-theta-produce-a-circle.htmlI am trying to do a double integral over the following region in polar coordinates: I know that the limits of integration are: $$\theta = -\frac{\pi}{2} \quad \to \quad \theta = \frac{\pi}{2} \\ ......Sun, 06 Sep 2026 05:50:16 -0400https://track.liberty.io/finding-all-solutions-to-an-inequality-equation.htmlhttps://track.liberty.io/finding-all-solutions-to-an-inequality-equation.htmlI have the following inequality that I need to find all solutions of: $2x^3-8x > 5x^2-20$ My guess is that you would have to turn this into a polynomial equation and let the right hand side ......Sun, 06 Sep 2026 05:49:20 -0400https://track.liberty.io/what-is-the-domain-of-a-division-of-functions.htmlhttps://track.liberty.io/what-is-the-domain-of-a-division-of-functions.htmlThis question is about real functions of real variables. I think that, in general, if the domain of some function $f(x)$ is A, and the domain of another function $g(x)$ is B, then the domain of $(......Sun, 06 Sep 2026 05:32:59 -0400https://track.liberty.io/finding-deceleration-and-initial-speed-given-time-and-distance.htmlhttps://track.liberty.io/finding-deceleration-and-initial-speed-given-time-and-distance.html(I'm sorry if this is a bad way to frame a math question) Let's say a car is moving at 40m/s and a brake is applied, the car decelerates at 10m/s thus stops after 4 seconds, traveled 100m after the ...Sun, 06 Sep 2026 05:23:42 -0400https://track.liberty.io/lim-x-to0-sin-2x-2x-x-3.htmlhttps://track.liberty.io/lim-x-to0-sin-2x-2x-x-3.htmlEvaluate $$ \lim_{x\to0} \frac{\sin(2x)-2x}{x^3} $$ I thought that I could use L'Hopital's rule to get to an answer of $-1$ but according to the answer manual that isn't correct. What am I doing w......Sun, 06 Sep 2026 05:21:26 -0400https://track.liberty.io/is-there-any-difference-between-empty-graph-and-null-graph.htmlhttps://track.liberty.io/is-there-any-difference-between-empty-graph-and-null-graph.htmlI was going through one of the links Is the empty graph connected? here I found that it is mentioned that empty graph is a graph with no edges while null graph is a graph with no vertices but I ...Sun, 06 Sep 2026 05:18:27 -0400https://track.liberty.io/how-to-prove-operatorname-tr-ab-operatorname-tr-ba.htmlhttps://track.liberty.io/how-to-prove-operatorname-tr-ab-operatorname-tr-ba.htmlthere is a similar thread here Coordinate-free proof of $\operatorname{Tr}(AB)=\operatorname{Tr}(BA)$?, but I'm only looking for a simple linear algebra proof....Sun, 06 Sep 2026 05:08:57 -0400https://track.liberty.io/evaluating-limits-algebraically.htmlhttps://track.liberty.io/evaluating-limits-algebraically.htmlI bought the eighth edition of Stewart Calculus (metric version) and I'm up to the section about limits. It's been pretty easy so far, but I've come across a class of limit problems that don't seem ...Sun, 06 Sep 2026 04:52:52 -0400https://track.liberty.io/what-is-the-definition-of-tensor-contraction.htmlhttps://track.liberty.io/what-is-the-definition-of-tensor-contraction.htmlAccording to Wikipedia's page on tensor contraction: In general, a tensor of type $(m,n)$ (with $m \geq 1$ and $n \geq 1$) is an element of the vector space $V \otimes \ldots \otimes V \otimes V......Sun, 06 Sep 2026 04:41:54 -0400https://track.liberty.io/eigenfunction-fourier-series-relationship.htmlhttps://track.liberty.io/eigenfunction-fourier-series-relationship.htmlI'm having difficulty formulating a proof detailing that every eigenfunction of $D^2$ is either a constant or of the form $a$cos$(nx)$ + $b$sin($nx$) for some value $n$. I understand that the Fou......Sun, 06 Sep 2026 04:35:26 -0400https://track.liberty.io/property-of-dirac-delta-function-in-mathbb-r-n.htmlhttps://track.liberty.io/property-of-dirac-delta-function-in-mathbb-r-n.htmlHow does one prove the following identity? $$\int _Vf(\pmb{r})\delta (g(\pmb{r})) \, d\pmb{r}=\int_S \frac{f(\pmb{r})}{\left|\operatorname{grad} g(\pmb{r})\right|} \, d\sigma$$ where $S$ is the sur......Sun, 06 Sep 2026 04:34:30 -0400https://track.liberty.io/is-there-a-way-to-find-all-roots-of-a-polynomial-equation.htmlhttps://track.liberty.io/is-there-a-way-to-find-all-roots-of-a-polynomial-equation.htmlIs there a way to find all roots of a polynomial equation? Lets say $$x^5+ax^4+bx^3+cx^2+dx+e=0$$ how to find its all roots?...Sun, 06 Sep 2026 04:30:35 -0400https://track.liberty.io/what-is-the-mathematical-definition-of-index-set.htmlhttps://track.liberty.io/what-is-the-mathematical-definition-of-index-set.htmlI find some descriptions http://en.wikipedia.org/wiki/Index_set and http://mathworld.wolfram.com/IndexSet.html . But can't find any definition....Sun, 06 Sep 2026 04:27:20 -0400