https://liberty.ioCurated star highlights with fresh energy.en-usFri, 28 Aug 2026 19:18:16 -0400https://liberty.io/find-the-relative-maximum-and-relative-minimum-of-the-function-f-x-49x.htmlhttps://liberty.io/find-the-relative-maximum-and-relative-minimum-of-the-function-f-x-49x.htmlFind the relative maximum and relative minimum of the function $f(x) = 49x + \frac{4}{x}=49x+4x^{-1}$ Solution: Step 1: Find the values of $x$ where $f'(x)=0$ and $f'(x)$ DNE. $f'(x)=49-4x^{-2}=......Fri, 28 Aug 2026 23:16:34 -0400https://liberty.io/numerically-solving-a-poisson-equation-with-neumann-boundary-condition.htmlhttps://liberty.io/numerically-solving-a-poisson-equation-with-neumann-boundary-condition.htmlThe Problem Suppose I have an equation of the form $\nabla^2 \phi(x) = f(x)$ on the interval $A \le x \le B$, where $f(x)$ is known and $\phi(x)$ is unknown. I have Neumann-type boundary condition......Fri, 28 Aug 2026 23:13:52 -0400https://liberty.io/understanding-how-to-prove-when-a-subset-is-a-subgroup.htmlhttps://liberty.io/understanding-how-to-prove-when-a-subset-is-a-subgroup.htmlLemma 3.4. Let $(G ,*)$ be a group. A nonempty subset $H$ of $G$ is a subgroup of $(G,*)$, iff, for every $a, b\in H$, $a*b^{-1}\in H$. Proof. First, suppose $H$ is a subgroup. If $b\in H$, then $b......Fri, 28 Aug 2026 23:04:53 -0400https://liberty.io/graph-isomorphisms-on-6-vertices-with-degree-3.htmlhttps://liberty.io/graph-isomorphisms-on-6-vertices-with-degree-3.htmlI want to find another graph that has 6 vertices and each has degree $3$ that is not isomorphic to these two graphs below. I know that these two graphs are isomorphic. They will all have the same d......Fri, 28 Aug 2026 22:54:33 -0400https://liberty.io/necessary-to-prove-the-inverse-is-invertible.htmlhttps://liberty.io/necessary-to-prove-the-inverse-is-invertible.htmlI am just starting out on linear algebra and I have come to a section the book that confuses me somewhat. The authour defines an invertible matrix A as: "A square matrix A is said to be invertible......Fri, 28 Aug 2026 22:53:13 -0400https://liberty.io/what-039-s-the-proof-that-the-euler-totient-function-is-multiplicative.htmlhttps://liberty.io/what-039-s-the-proof-that-the-euler-totient-function-is-multiplicative.htmlThat is, why is $\varphi(AB)=\varphi(A)\varphi(B)$, if $A$ and $B$ are two coprime positive integers? It's not just a technical trouble—I can't see why this should be, intuitively: I bellyfeel ......Fri, 28 Aug 2026 22:47:34 -0400https://liberty.io/maximum-modulus-principle-and-open-mapping-theorem.htmlhttps://liberty.io/maximum-modulus-principle-and-open-mapping-theorem.htmlThe Maximum Modulus Principal states: let $f$ be a function holomorphic on some connected open subset $D$ of the complex plane $\mathbb{C}$ and taking complex values. If $z_{0}$ is a point in $D$ s......Fri, 28 Aug 2026 22:31:18 -0400https://liberty.io/pursuit-curves-solution.htmlhttps://liberty.io/pursuit-curves-solution.htmlFor our math class we have to do some calculations with respect to pursuit curves. The chased object starts at point $(p,0)$. Chaser starts at $(0,0)$.(x,y) Speed of the chased object is $u$. Speed ...Fri, 28 Aug 2026 22:27:27 -0400https://liberty.io/find-the-limit-as-x-approaches-infinity-of-e-x-lnx-and-sin2x-x.htmlhttps://liberty.io/find-the-limit-as-x-approaches-infinity-of-e-x-lnx-and-sin2x-x.htmlI'm not sure how to evaluate the limit as $x \rightarrow \infty $ of the following: $$ \bullet \lim_{x\to\infty}e^{-x}\cdot \log x $$ $$\bullet\lim_{x\to\infty} \frac{\sin(2x)}{x}$$...Fri, 28 Aug 2026 22:22:40 -0400https://liberty.io/what-are-the-ways-of-proving-that-the-cantor-set-is-uncountable-apart-.htmlhttps://liberty.io/what-are-the-ways-of-proving-that-the-cantor-set-is-uncountable-apart-.htmlWhat are the ways of proving that the Cantor set is uncountable apart from Cantor diagonalization? Are there any based on dynamical systems?...Fri, 28 Aug 2026 22:19:54 -0400https://liberty.io/limit-of-sin-x-as-x-approaches-zero-from-the-left.htmlhttps://liberty.io/limit-of-sin-x-as-x-approaches-zero-from-the-left.htmlI'm trying to find a proof that: $$ \lim_{x\to0^-}\sin(x)=0$$ I'd like to be able to do the proof without reference to advanced theorems (mean value theorem, series, etc). I have a geometric approa......Fri, 28 Aug 2026 22:14:14 -0400https://liberty.io/what-is-the-difference-between-square-of-sum-and-sum-of-square.htmlhttps://liberty.io/what-is-the-difference-between-square-of-sum-and-sum-of-square.htmlWhat is difference between square of sum $(\sum_{i=1}^{n}x_i)^2$ and sum of square $\sum_{i=1}^{n}x_i^2$? I think square of sum is bigger than sum of square but i can not find a relation between t......Fri, 28 Aug 2026 22:13:48 -0400https://liberty.io/on-laplace-transform-of-periodic-functions.htmlhttps://liberty.io/on-laplace-transform-of-periodic-functions.htmlI recently bumped into this theorem regarding the Laplace transform of periodic function: Given a periodic function $f(t)=f(t+p)$, where $p$ is its period, then its Laplace transform is given by: ......Fri, 28 Aug 2026 22:08:09 -0400https://liberty.io/least-squares-estimate-b1-formula.htmlhttps://liberty.io/least-squares-estimate-b1-formula.htmlMy regression textbook textbook says that $\sum_{i=1}^{n} (x_i - \overline{x}) = 0$. I know that: $\sum_{i=1}^{n} (x_i - \overline{x})(y_i - \overline{y})$ $= \sum_{i=1}^{n} (x_i - \overline......Fri, 28 Aug 2026 21:58:20 -0400https://liberty.io/linear-ode-roots-of-characteristic-equation-having-multiplicity-1.htmlhttps://liberty.io/linear-ode-roots-of-characteristic-equation-having-multiplicity-1.htmlI know the method. For a linear homogeneous ordinary differential equation with a root of the characteristic polynomial in $\alpha$ has multiplicity $k$, then $y=x^me^{\alpha x}$ with $m=0,1,\cdot......Fri, 28 Aug 2026 21:55:12 -0400https://liberty.io/explicit-formula-for-floor-x.htmlhttps://liberty.io/explicit-formula-for-floor-x.htmlIn number theory we have so-called explicit formula's in terms of the Riemann zeta zero's. For instance to count the sum of the logarithms of the primes below some given integer. (second Chebyshev ...Fri, 28 Aug 2026 21:51:07 -0400https://liberty.io/how-to-rotate-a-polar-equation.htmlhttps://liberty.io/how-to-rotate-a-polar-equation.htmlTake a simple polar equation like r = θ/2 that graphs out to: But, how would I achieve a rotation of the light-grey plot in this image (roughly 135 degrees)? Is there a way to easily shift the plot?...Fri, 28 Aug 2026 21:47:34 -0400https://liberty.io/finding-eigenpairs-of-matrix-a.htmlhttps://liberty.io/finding-eigenpairs-of-matrix-a.htmlHow do you find the eigenpairs of this? Given two matrices $A=\begin{bmatrix} a& b\\b&-a\end{bmatrix}$ and $B=\begin{bmatrix} a& -b\\b&a\end{bmatrix}$ where $b \neq 0$, find all ...Fri, 28 Aug 2026 21:44:19 -0400https://liberty.io/if-a-is-not-relatively-prime-to-n-prove-modulo-property.htmlhttps://liberty.io/if-a-is-not-relatively-prime-to-n-prove-modulo-property.htmlIf $n>1$ is integer and $1\le a \le n$ is integer such that $(a,n)\neq 1$ then prove there exist integer $1 \le b <n $ such that $ab \equiv 0(mod \; n)$ I have tried everything from going to ...Fri, 28 Aug 2026 21:42:37 -0400https://liberty.io/expression-for-the-current-as-a-function-of-time.htmlhttps://liberty.io/expression-for-the-current-as-a-function-of-time.htmlWhen the electromotive force (emf) is removed from a circuit containing inductance and resistance but no capacitors, the rate of decrease of current is proportional to the current. If the initial ...Fri, 28 Aug 2026 21:33:31 -0400https://liberty.io/evaluate-double-integral-of-triangular-region.htmlhttps://liberty.io/evaluate-double-integral-of-triangular-region.htmlEvaluate $$ \int\int_D e^{y^2} dA $$ where D is the triangular region with vertices (0,0), (0,1) and (2,1) My attempts: $$ \int^{2}_0 \int^{1}_\frac{x}{2} e^{y^2} dy dx $$ or $$ \int^{0}_1\int^{......Fri, 28 Aug 2026 21:32:22 -0400https://liberty.io/for-which-values-of-b-c-is-the-matrix-invertible.htmlhttps://liberty.io/for-which-values-of-b-c-is-the-matrix-invertible.html$A =\left(\begin{array}{ccc} 0 & 1 & b \\ -1 & 0 & c \\ -b & -c & 0 \end{array}\right)$ I tried to come up with the RREF($A$), but in the final step I have the matrix: $A ......Fri, 28 Aug 2026 21:27:37 -0400https://liberty.io/is-there-any-deep-reason-that-23456789-is-prime-closed.htmlhttps://liberty.io/is-there-any-deep-reason-that-23456789-is-prime-closed.htmlI was recently coding a prime factorization function and wanted to test that it could factor large numbers reasonably well. Arbitrarily, I slid my finger across the number pad, and got 23456789 as ......Fri, 28 Aug 2026 21:26:47 -0400https://liberty.io/recurrent-and-transient-states.htmlhttps://liberty.io/recurrent-and-transient-states.htmlI know which are the recurrent states {2,4,5} and It's clear to see it drawing a diagram but how can I prove that {2,4,5} are the recurrent states? In a formal way? Also the transient states are ......Fri, 28 Aug 2026 21:22:08 -0400https://liberty.io/ambiguity-in-rule-for-determining-an-outlier.htmlhttps://liberty.io/ambiguity-in-rule-for-determining-an-outlier.htmlI have the following set of numbers, in which I need to determine if there are any outliers: 11, 14, 21, 26, 29, 33, 61. I graphed them in Desmos (see above), and visually, it appears to me that 6......Fri, 28 Aug 2026 21:12:57 -0400https://liberty.io/method-of-characteristics-for-burgers-039-equation-with-rectangular-da.htmlhttps://liberty.io/method-of-characteristics-for-burgers-039-equation-with-rectangular-da.htmlUsing the method of characteristics, find a solution to Burgers' equation \begin{cases} u_t+\left(\frac{u^2}{2} \right)_x =0 & \text{in }\mathbb{R}\times(0,\infty) \\ \qquad \qquad \, \, u=g......Fri, 28 Aug 2026 20:57:09 -0400https://liberty.io/computing-the-iterated-logarithm-log-star-by-hand.htmlhttps://liberty.io/computing-the-iterated-logarithm-log-star-by-hand.htmlI'm having trouble figuring out how to compute the iterated logarithm (log-star) by hand without using a calculator or programming language. I wrote out the following program in Java to check my an......Fri, 28 Aug 2026 20:54:31 -0400https://liberty.io/how-to-choose-between-real-and-complex-coefficients-of-fourier-series.htmlhttps://liberty.io/how-to-choose-between-real-and-complex-coefficients-of-fourier-series.htmlSometimes in exercises we are asked to calculate the fourier series of a function. But there are two ways to do that. If $f:\Bbb R\mapsto\Bbb R$ is $T$-periodic over $\Bbb R$ then what conditions......Fri, 28 Aug 2026 20:50:02 -0400https://liberty.io/why-does-two-terms-immediately-adjacent-mean-multiply.htmlhttps://liberty.io/why-does-two-terms-immediately-adjacent-mean-multiply.htmlI am currently teaching a GED math class. While learning about the order of operations, the students asked why does a number next to a parentheses mean multiplication? I understand the rule that two ...Fri, 28 Aug 2026 20:40:35 -0400https://liberty.io/what-does-the-plus-sign-contained-in-a-circle-oplus-mean-in-this-case.htmlhttps://liberty.io/what-does-the-plus-sign-contained-in-a-circle-oplus-mean-in-this-case.htmlThe fundamental group of the torus is isomorphic to $\mathbb{Z}\oplus\mathbb{Z}$. I know that the $\oplus$ symbol is the exclusive or symbol but I don't understand how two of the same sets are XOR......Fri, 28 Aug 2026 20:18:57 -0400https://liberty.io/remainder-when-2-108-is-divided-by-11-closed.htmlhttps://liberty.io/remainder-when-2-108-is-divided-by-11-closed.htmlWhat is the remainder obtained when $2^{108}$ is divided by $11$? I tried bringing in $11$ in the given no. such as in $(11-3)^{36}$ and then using binomial expansion...but its not helping. Any ...Fri, 28 Aug 2026 20:08:18 -0400https://liberty.io/complexity-analysis-of-alpha-beta-pruning-of-a-full-tree.htmlhttps://liberty.io/complexity-analysis-of-alpha-beta-pruning-of-a-full-tree.htmlI am trying to understand the derivation of a time complexity for an alpha-beta pruning algorithm but up till now have not found any reasonable recourse. Many recourses claim that if you take a full ...Fri, 28 Aug 2026 20:04:37 -0400https://liberty.io/relationship-between-groups-and-permutations.htmlhttps://liberty.io/relationship-between-groups-and-permutations.htmlWhat is the relationship between groups and permutations? The way I understand it, a permutation is a set of numbers that you can do things with. A group is a set of elements equipped with an opera......Fri, 28 Aug 2026 20:03:35 -0400https://liberty.io/combinations-probability-dividing-9-people-into-3-groups.htmlhttps://liberty.io/combinations-probability-dividing-9-people-into-3-groups.htmlI'm struggling to understand the procedure to solve the following problem: The group consists of 9 people, 3 men and 6 women. They are divided randomly into 3 groups (3 persons each). What's the ...Fri, 28 Aug 2026 19:53:05 -0400https://liberty.io/how-do-i-calculate-the-value-of-this-series.htmlhttps://liberty.io/how-do-i-calculate-the-value-of-this-series.htmlI want to find the value to which this series converges $$\sum_{n=0}^\infty \frac{(-1)^n}{n^2+1}$$ I tried looking at the sequence of partial sums $$S_k = \sum_{n=0}^k \frac{(-1)^n}{n^2+1}$$ and I ...Fri, 28 Aug 2026 19:42:27 -0400https://liberty.io/riemannian-ricci-curvature-for-round-n-sphere.htmlhttps://liberty.io/riemannian-ricci-curvature-for-round-n-sphere.htmlWhat is the best way to see that the Ricci scalar curvature of $(S^n(r),g_{round})$ is a constant $n(n-1)/r^2$ ? I essentially only see this value stated in the literature, but no computation assoc......Fri, 28 Aug 2026 19:41:20 -0400https://liberty.io/do-non-convex-platonic-solids-exist.htmlhttps://liberty.io/do-non-convex-platonic-solids-exist.htmlConsider a solid with the following properties - It is composed of congruent, regular polygons. At each vertex, the same number of edges and faces meet. This is the same as the requirement for the ...Fri, 28 Aug 2026 19:34:14 -0400https://liberty.io/motivation-behind-point-set-topology-closed.htmlhttps://liberty.io/motivation-behind-point-set-topology-closed.htmlWhy should I study point-set topology? What initially interested me in topology was the pop-sci rubber sheet stuff or coffee cup-donut stuff or proving fundamental theorem of algebra using curves ......Fri, 28 Aug 2026 19:28:58 -0400https://liberty.io/binary-decimal-primes.htmlhttps://liberty.io/binary-decimal-primes.htmlConverting an integer $N$ into binary and then reading the result as a base-10 number $N_2$, the prime $N=3$ gives $N_2=11$ (which is also prime) and $N=5$ gives $N_2=101$. Are $3$ and $5$ the only......Fri, 28 Aug 2026 19:19:44 -0400https://liberty.io/proving-that-f-n-to-z-is-a-bijection-closed.htmlhttps://liberty.io/proving-that-f-n-to-z-is-a-bijection-closed.htmlProve that $f: \mathbb{N} \to \mathbb{Z} $ is a bijection where the relation between $\mathbb{N}$ and $\mathbb{Z}$ is as follows: $f(n)= \frac n2$ if n is even, and $-\frac {n+1}{2}$ if n is odd ...Fri, 28 Aug 2026 19:10:59 -0400https://liberty.io/what-exactly-is-a-number.htmlhttps://liberty.io/what-exactly-is-a-number.htmlWe've just been learning about complex numbers in class, and I don't really see why they're called numbers. Originally, a number used to be a means of counting (natural numbers). Then we extend t......Fri, 28 Aug 2026 19:09:35 -0400https://liberty.io/chain-rule-f-t-e-t-sin2t.htmlhttps://liberty.io/chain-rule-f-t-e-t-sin2t.htmlI'm missing a step somewhere on this: $$f(t)=e^{t\sin2t}.$$ I know I have to use the chain rule but I'm getting tripped up. here are my steps: $\frac d{dx}(e^{t\sin2t})$ $e^{t\sin2t} \frac d{dx}(t\...Fri, 28 Aug 2026 18:58:04 -0400https://liberty.io/what-exactly-is-calculus.htmlhttps://liberty.io/what-exactly-is-calculus.htmlI've researched this topic a lot, but couldn't find a proper answer to this, and I can't wait a year to learn it at school, so my question is: What exactly is calculus? I know who invented it,......Fri, 28 Aug 2026 18:42:47 -0400https://liberty.io/prove-left-frac-2-5-right-frac-2-5-ln-2.htmlhttps://liberty.io/prove-left-frac-2-5-right-frac-2-5-ln-2.htmlInadvertently, I find this interesting inequality. But this problem have nice solution? prove that $$\ln{2}>(\dfrac{2}{5})^{\frac{2}{5}}$$ This problem have nice solution? Thank you. ago,I find ...Fri, 28 Aug 2026 18:38:03 -0400https://liberty.io/radius-arc-length-what-angle-from-c-p.htmlhttps://liberty.io/radius-arc-length-what-angle-from-c-p.htmlI know the radius and length of an arc. how do I figure out the angle from center point to the ends of the arc? ( I'm trying to figure out the length of the arc if I offset/increase radius by $2$" ......Fri, 28 Aug 2026 18:37:01 -0400https://liberty.io/definition-of-continuous-map.htmlhttps://liberty.io/definition-of-continuous-map.htmlLet $X,Y$ be topological spaces, let $f:X\longrightarrow Y$ a function. There are several (equivalent) ways to define continuity of $f$, one of these says: $f$ is continuous if $$(1)\qquad \forall A\...Fri, 28 Aug 2026 18:29:07 -0400https://liberty.io/finding-the-upper-and-lower-bounds-positive-and-negative-each-if-appli.htmlhttps://liberty.io/finding-the-upper-and-lower-bounds-positive-and-negative-each-if-appli.htmlGiven $|z^4 -8 + 3|, |z|=2 $ I need to find the positive upper and lower bounds of the above expression. My attempt: By triangle inequality, we have $||z^4|+|-8z|+3|\geq|z^{4} -8z + 3| \leq ||z^4 ......Fri, 28 Aug 2026 18:22:45 -0400https://liberty.io/how-to-find-the-laplace-transform-of-f-t-t-e-t.htmlhttps://liberty.io/how-to-find-the-laplace-transform-of-f-t-t-e-t.htmlHow to find the Laplace transform of the following function: $$f(t)=t e^t$$ $$F(s)=\int_0^\infty (te^t e^{-st})dt$$ What method do I use to find the integral?...Fri, 28 Aug 2026 18:22:28 -0400https://liberty.io/what-039-s-the-point-of-trigonometric-proofs-identities-in-introductor.htmlhttps://liberty.io/what-039-s-the-point-of-trigonometric-proofs-identities-in-introductor.htmlI remember back in high school at some point delving into worksheet after worksheet of trigonometric "identities", the vast majority of which are basically restatements of $\sin^2(x) + \cos^2(x) = ......Fri, 28 Aug 2026 18:20:11 -0400https://liberty.io/how-to-correctly-find-csc-of-an-angle.htmlhttps://liberty.io/how-to-correctly-find-csc-of-an-angle.htmlAlright, so one of my questions is CSC (angle) -5. When I plug CSC in my calculator, it says "math error." I'm using a Casio fx-300 MS, and using shift + cos, then putting an angle, such as 90....Fri, 28 Aug 2026 18:17:35 -0400https://liberty.io/volume-of-an-irregular-octahedron-from-edge-lengths.htmlhttps://liberty.io/volume-of-an-irregular-octahedron-from-edge-lengths.htmlDoes anyone know how to calculate the volume of an irregular octahedron from the lengths of the edges? The octahedron has triangular faces, but the only information are the edge lengths. Alternat......Fri, 28 Aug 2026 17:57:38 -0400https://liberty.io/equation-or-equality.htmlhttps://liberty.io/equation-or-equality.htmlIs a numerical equality (like $1+2 = 3$) an algebraic equation? In google search we have: "equation ɪˈkweɪʒ(ə)n/ noun noun: equation; plural noun: equations 1. Mathematics a statement that the va......Fri, 28 Aug 2026 17:56:47 -0400https://liberty.io/how-to-solve-this-simple-2d-markov-chain.htmlhttps://liberty.io/how-to-solve-this-simple-2d-markov-chain.htmlI have 2D Markov chain like below : I have already solved for $p_{b,0}$ and $p_{0,d}$ like this : $$p_{b,0} = \rho^bp_{0,0}$$ $$p_{0,d} = \rho^dp_{0,0}$$ But I don't know how to get $p_{b,d}$.......Fri, 28 Aug 2026 17:54:23 -0400https://liberty.io/what-does-039-express-in-terms-of-x-039-mean.htmlhttps://liberty.io/what-does-039-express-in-terms-of-x-039-mean.htmlFor the following question : $f(x) = 2x^2 + 4x $ It asks me to express the following in terms of $x$: $f(-2x)$ What does the question mean by this? Does it mean make $x$ the subject?...Fri, 28 Aug 2026 17:45:10 -0400https://liberty.io/approximate-the-sum-of-each-alternating-series.htmlhttps://liberty.io/approximate-the-sum-of-each-alternating-series.htmlApproximate the sum of each series to three decimal places. $\sum {(-1)^n {1\over {n^3}}}$ From alternating series test, this series convergence. $S\approx a_3+S_2$ $S\approx {1\over {27}} + {7\ov......Fri, 28 Aug 2026 17:30:51 -0400https://liberty.io/whether-multiplication-by-a-constant-matrix-alone-can-represent-every-.htmlhttps://liberty.io/whether-multiplication-by-a-constant-matrix-alone-can-represent-every-.htmlI read recently that every function $f : \mathbb{R}^m \to \mathbb{R}^n$ can be written as $f(\mathbf{x}) = \mathbf{Ax}$ where $\mathbf{A}$ is a matrix of constants. On the one hand this is intuitive ...Fri, 28 Aug 2026 17:23:47 -0400https://liberty.io/determine-relationship-between-vectors.htmlhttps://liberty.io/determine-relationship-between-vectors.htmlUsing the product i would like to understand if it is possible to determine relationship between two vectors,let us consider the following problem: Vectors $u$ and $v$ have length $1$. Which of......Fri, 28 Aug 2026 17:22:17 -0400https://liberty.io/relationship-between-smooth-manifold-and-differentiable-manifold.htmlhttps://liberty.io/relationship-between-smooth-manifold-and-differentiable-manifold.htmlDefine a smooth manifold to be a ringed space (a Hausdorff, second countable, $n$-manifold) that is locally isomorphic (as a ringed space) to the sheaf of smooth real valued functions on some open ......Fri, 28 Aug 2026 17:22:07 -0400https://liberty.io/finding-the-range-of-a-function-without-graph.htmlhttps://liberty.io/finding-the-range-of-a-function-without-graph.htmlI'm asked to find the range of the function : $ (-x^2 , +1) $ Taking R as a real number which represents a quantity along a line and graphing : then the range for this function from viewing the ...Fri, 28 Aug 2026 17:20:40 -0400https://liberty.io/book-recommendation-hard-problems-for-multivariable-calculus-w-solutio.htmlhttps://liberty.io/book-recommendation-hard-problems-for-multivariable-calculus-w-solutio.htmlI'm looking for a text that covers roughly what's sometimes called "Calculus III" or multivariable calculus.* But this text must satisfy certain additional criteria: (1) It must be more in-depth (......Fri, 28 Aug 2026 17:19:05 -0400https://liberty.io/gompertz-function-as-a-finite-product.htmlhttps://liberty.io/gompertz-function-as-a-finite-product.htmlI have a growth model such as: $$G(n)= (1-1/N)(1-2/N)...(1-n/N) \text{, with } n<N$$ As analytical approximation of $G(n)$, I numerically found that $1-\exp(-\exp(1.71 (1-1.05 n/\sqrt{N}))$ w......Fri, 28 Aug 2026 17:18:13 -0400https://liberty.io/how-many-10-letter-words-can-be-formed-using-the-26-letters-of-the-alp.htmlhttps://liberty.io/how-many-10-letter-words-can-be-formed-using-the-26-letters-of-the-alp.htmlHow many $10$-letter words can be formed using the $26$ letters of the alphabet if: a) Repetition is allowed b) Repetition is not allowed c) Repetition is allowed, but letters are listed ...Fri, 28 Aug 2026 17:06:15 -0400https://liberty.io/the-points-where-function-is-discontinuous-are-those-points-counted-co.htmlhttps://liberty.io/the-points-where-function-is-discontinuous-are-those-points-counted-co.htmlThe points where function is discontinuous,are those points counted/considered in the domain of the function. $(1)[x]$,greatest integer function is discontinuous at all integer points but integers ......Fri, 28 Aug 2026 17:02:43 -0400https://liberty.io/a-little-question-about-clairaut-039-s-theorem.htmlhttps://liberty.io/a-little-question-about-clairaut-039-s-theorem.htmlClairaut's Theorem Let $f$ be a function of two variables,let$(x_{0},y_{0})$ be a point and let $U$ be an open disk with center $(x_{0},y_{0})$.Assume that $f$ is defined on $U$ and its partial ...Fri, 28 Aug 2026 16:35:45 -0400https://liberty.io/what-is-a-jordan-cell.htmlhttps://liberty.io/what-is-a-jordan-cell.htmlGoogle has been surprisingly unhelpful for me. A homework problem from my algebra class asks me to Calculate p(A) where A is a Jordan cell and p is a polynomial. I'd like to attempt the problem ......Fri, 28 Aug 2026 16:32:04 -0400https://liberty.io/what-does-sin-theta-0-mean-here-if-tan-theta-frac-8-15-and-sin-theta-0.htmlhttps://liberty.io/what-does-sin-theta-0-mean-here-if-tan-theta-frac-8-15-and-sin-theta-0.htmlThe question is: If $\tan(\theta) = -\frac{8}{15}$, and $\sin(\theta) > 0$, then find $\cos(\theta)$. What I did was draw a triangle on the unit circle with sides $8, 15$ and therefore hypot......Fri, 28 Aug 2026 16:30:13 -0400https://liberty.io/finding-x-y-z-of-a-triangle.htmlhttps://liberty.io/finding-x-y-z-of-a-triangle.htmlMy younger brother was asked to do this problem and was given the problem exactly as the problem was given with no additional information. Basically, we are trying to find the values of x,y,and z ......Fri, 28 Aug 2026 16:28:19 -0400https://liberty.io/how-do-you-factor-x-3-8-0.htmlhttps://liberty.io/how-do-you-factor-x-3-8-0.htmlI factored $x^3 - 8 = 0$ and I only got $x = 2$, but the answer said it's $x=2$ and $x=-1 \pm \sqrt{3}i$? How do you get $x=-1 \pm\sqrt{3}i$?...Fri, 28 Aug 2026 16:25:39 -0400https://liberty.io/how-can-the-surd-sqrt-2-sqrt-3-be-expressed.htmlhttps://liberty.io/how-can-the-surd-sqrt-2-sqrt-3-be-expressed.htmlI was wondering how $\sqrt{2-\sqrt{3}}$ could be expressed in terms of $\frac{\sqrt{3}-1}{\sqrt{2}}$. I did try to solve both the expressions separately but none of them seemed to match. I would ...Fri, 28 Aug 2026 16:19:51 -0400https://liberty.io/integrate-frac-x-ln-x-sqrt-x-2-1-sqrt-x-2-1-closed.htmlhttps://liberty.io/integrate-frac-x-ln-x-sqrt-x-2-1-sqrt-x-2-1-closed.html$$\int \frac{x\ln\left(x+\sqrt{x^{2}+1}\right)}{\sqrt{x^{2}+1}}dx$$ I honestly have been struggling with this integral for 2 hours straight and none of my methods (including substitution and integ......Fri, 28 Aug 2026 16:19:48 -0400https://liberty.io/what-039-s-wrong-with-my-solution-for-equation-x-1-2-3x-1-2-10x-3-2.htmlhttps://liberty.io/what-039-s-wrong-with-my-solution-for-equation-x-1-2-3x-1-2-10x-3-2.htmlThe solution given in the website: Here's my solution: What am I doing wrong?...Fri, 28 Aug 2026 16:19:39 -0400https://liberty.io/how-to-solve-equations-when-logarithm-is-the-exponent.htmlhttps://liberty.io/how-to-solve-equations-when-logarithm-is-the-exponent.htmlHere is an example problem where the logarithm is expressed as an exponent. Please help me understand this concept its not properly covered in my textbook. $$ 3^{\log_3 (2k)} = 9 $$...Fri, 28 Aug 2026 16:01:27 -0400https://liberty.io/taking-the-area-of-isosceles-cross-sections-of-an-ellipse.htmlhttps://liberty.io/taking-the-area-of-isosceles-cross-sections-of-an-ellipse.htmlThe base of S is an elliptical region with boundary curve $25x^2 + 4y^2 = 100$. Cross-sections perpendicular to the x-axis are isosceles right triangles with hypotenuse in the base. I have gotten ......Fri, 28 Aug 2026 15:49:18 -0400https://liberty.io/prove-that-arcsin-3-5-pi-is-irrational.htmlhttps://liberty.io/prove-that-arcsin-3-5-pi-is-irrational.htmlI was asked to prove that on the unit circle, the rational points are dense. My idea is to first show that the point (3/5,4/5) corresponds to an irrational angle (more precisely, its ratio with res......Fri, 28 Aug 2026 15:34:35 -0400https://liberty.io/how-can-i-use-qr-in-matlab-to-compute-lq-decomposition.htmlhttps://liberty.io/how-can-i-use-qr-in-matlab-to-compute-lq-decomposition.htmlI want to compute this: $$\begin{bmatrix} U\\ Y \end{bmatrix} = \begin{bmatrix} L_{11} &0 \\ L_{21} & L_{22} \end{bmatrix}\begin{bmatrix} Q_1\\ Q_2 \end{bmatrix}$$ Is this matlab command ...Fri, 28 Aug 2026 15:23:55 -0400https://liberty.io/is-there-anything-special-about-intersection-points-of-vesica-pisces-c.htmlhttps://liberty.io/is-there-anything-special-about-intersection-points-of-vesica-pisces-c.htmlTake a triangle $\triangle ABC$ On each pair of vertices of the triangle, place a pair of vesica pisces circles with the properties that each circle has the same radius as the other circle of th......Fri, 28 Aug 2026 15:09:15 -0400https://liberty.io/how-to-find-the-dimensions-of-a-rectangle-given-a-the-area-and-the-dia.htmlhttps://liberty.io/how-to-find-the-dimensions-of-a-rectangle-given-a-the-area-and-the-dia.htmlI need to write concise formulas of these two:$\space \space \space$ $(1)$ finding the dimensions of a rectangle given the Area and the diagonal $\space \space \space$$(2)$ finding the dimensions o......Fri, 28 Aug 2026 15:01:12 -0400https://liberty.io/how-to-find-the-length-of-one-of-the-sides-of-a-triangle-given-the-are.htmlhttps://liberty.io/how-to-find-the-length-of-one-of-the-sides-of-a-triangle-given-the-are.htmlThe triangles are drawn to scale. The first triangle has side lengths of 17, 17, 16 while the second triangle has side lengths of 17,17,$k$. The triangles have the same area. Find the value of $k$ ...Fri, 28 Aug 2026 15:00:27 -0400https://liberty.io/what-039-s-the-intuition-behind-definition-of-chaotic-function.htmlhttps://liberty.io/what-039-s-the-intuition-behind-definition-of-chaotic-function.htmlI read books A First Course in Discrete Dynamical Systems by Richard A. Holmgren and An Introduction to Chaotic Dynamical Systems by Robert L. Devaney. I want to understand which concepts of "chaos......Fri, 28 Aug 2026 14:54:26 -0400https://liberty.io/normalizer-of-the-normalizer-of-the-sylow-p-subgroup.htmlhttps://liberty.io/normalizer-of-the-normalizer-of-the-sylow-p-subgroup.htmlIf $P$ is a Sylow $p$-subgroup of $G$, how do I prove that normalizer of the normalizer $P$ is same as the normalizer of $P$ ?...Fri, 28 Aug 2026 14:53:47 -0400https://liberty.io/can-a-5th-degree-equation-with-repeated-roots-be-solved-by-radicials.htmlhttps://liberty.io/can-a-5th-degree-equation-with-repeated-roots-be-solved-by-radicials.htmlI realized that generally a 5th degree equation cannot be solved by radicals, but what if we have a 5th degree equation with repeated roots, how about its solvability by radicals? Intuitively, it w......Fri, 28 Aug 2026 14:50:29 -0400https://liberty.io/how-to-draw-a-phase-portrait-of-a-two-dimensional-ode.htmlhttps://liberty.io/how-to-draw-a-phase-portrait-of-a-two-dimensional-ode.htmlIf we're given: $$\dot{x}=-x+y$$ $$\dot{y}=xy-1$$ How do I draw a phase portrait of this system? I don't understand which direction the arrows are supposed to point. This is what I got so far: I ...Fri, 28 Aug 2026 14:46:14 -0400https://liberty.io/finding-the-largest-singular-value-easily.htmlhttps://liberty.io/finding-the-largest-singular-value-easily.htmlI'm only interested in finding the largest singular value of an $m\times n$ real matrix $A$. I don't need the corresponding (left and right) singular vectors. Is there a way to do so without perf......Fri, 28 Aug 2026 14:43:04 -0400https://liberty.io/classification-of-pde-into-linear-nonlinear.htmlhttps://liberty.io/classification-of-pde-into-linear-nonlinear.htmlConsider the following second order PDEs \begin{align} u_{t} + v_{t} + x^{4}u_{xx} + v_{yy} &=0\,\, (1)\\ u_{t} + v_{t} + xu_{xx} + v_{yy} + x^{2}v_{y}&=0\,\,(2) \end{align} I want to ...Fri, 28 Aug 2026 14:41:03 -0400https://liberty.io/finding-the-order-of-an-element-in-the-group-of-integers-modulo-n-with.htmlhttps://liberty.io/finding-the-order-of-an-element-in-the-group-of-integers-modulo-n-with.htmlI'm a bit new to abstract algebra and while learning it, I've come across a somewhat tedious problem. I'm curious as to whether or not there is some modular congruency trick/number theory that let......Fri, 28 Aug 2026 14:33:08 -0400https://liberty.io/convert-angle-radians-to-a-heading-vector.htmlhttps://liberty.io/convert-angle-radians-to-a-heading-vector.htmlI have been looking everywhere trying to find out how to convert an angle in radians (expressed as -Pi to Pi) to a heading vector. The only [x,y] answer I have found is, [cos(angle), sin(angle)] , ...Fri, 28 Aug 2026 14:24:35 -0400https://liberty.io/user-aretor-mathematics-stack-exchange.htmlhttps://liberty.io/user-aretor-mathematics-stack-exchange.htmlQ&A for people studying math at any level and professionals in related fields...Fri, 28 Aug 2026 14:23:41 -0400https://liberty.io/proving-the-so-called-well-ordering-principle.htmlhttps://liberty.io/proving-the-so-called-well-ordering-principle.htmlIs there anything wrong with the following proof? Theorem. Every non-empty subset $B \subset\mathbb{N}$ has a least member. Proof. Assume not. Then, of necessity, we'd have to have $B=\varnothing......Fri, 28 Aug 2026 14:18:20 -0400https://liberty.io/sample-standard-deviation-given-population-standard-deviation.htmlhttps://liberty.io/sample-standard-deviation-given-population-standard-deviation.htmlHow do you find the sample standard deviation when given the population standard deviation? What formula do you use? If you can make up an example that would be great....Fri, 28 Aug 2026 14:08:47 -0400https://liberty.io/why-are-3d-transformation-matrices-4-times-4-instead-of-3-times-3.htmlhttps://liberty.io/why-are-3d-transformation-matrices-4-times-4-instead-of-3-times-3.htmlBackground: Many (if not all) of the transformation matrices used in $3D$ computer graphics are $4\times 4$, including the three values for $x$, $y$ and $z$, plus an additional term which usually h......Fri, 28 Aug 2026 14:07:52 -0400https://liberty.io/let-t-be-the-centroid-of-a-triangle-abc-and-let-p-be-the-midpoint-of-t.htmlhttps://liberty.io/let-t-be-the-centroid-of-a-triangle-abc-and-let-p-be-the-midpoint-of-t.htmlQuestion : Let $T$ be the centroid of a triangle $ABC$, and let $P$ be the midpoint of the side $\overline{AC}$. The line containing the point $T$ parallel to the line $BC$ intersects the side $\ov......Fri, 28 Aug 2026 14:00:53 -0400https://liberty.io/how-to-express-logical-equivalence-arrow-using-pierce-039-s-arrow.htmlhttps://liberty.io/how-to-express-logical-equivalence-arrow-using-pierce-039-s-arrow.htmlI am really confused about this and not sure how to show $P\iff Q$ with the ↓ arrow and only the ↓ arrow. I understand that $P \iff Q$ is $P\implies Q$ and $Q\implies P$. I also know that $P\impl......Fri, 28 Aug 2026 13:45:44 -0400https://liberty.io/why-are-the-trig-functions-defined-by-the-counterclockwise-path-of-a-c.htmlhttps://liberty.io/why-are-the-trig-functions-defined-by-the-counterclockwise-path-of-a-c.htmlMy understanding is that $\cos$ is defined by the value of $x$ as you trace the graph of a circle counterclockwise, starting at the point $(1, 0)$. Similarly, $\sin$ traces the $y$ value. I underst......Fri, 28 Aug 2026 13:32:13 -0400https://liberty.io/find-y-function-of-cos-y-x.htmlhttps://liberty.io/find-y-function-of-cos-y-x.htmlLet's say that there is no limit to $x$, then I believe $y = \arccos(x) + \pi n $ because $\cos(y) = x$ for $x + \pi n , n\in \mathbb{Z}$. However, in the last step of the following calculation, it......Fri, 28 Aug 2026 13:28:50 -0400https://liberty.io/prove-series-frac-1-x-ln-x-diverges-duplicate.htmlhttps://liberty.io/prove-series-frac-1-x-ln-x-diverges-duplicate.htmlProve series $$\sum\limits_{n=2}^{\infty}\frac{1}{n\ln n}$$ doesn't converge without integral test. I really don't know any hint for $\ln n$. So I understand I have to show $$\frac{1}{n\ln n} > ......Fri, 28 Aug 2026 13:22:32 -0400https://liberty.io/lagrange-multipliers-with-multiple-constraints.htmlhttps://liberty.io/lagrange-multipliers-with-multiple-constraints.htmlI want to maximize $f(x,y,z)=x+y+z$, according to the constraints $g_1(x,y,z)=x^2-y^2-1=0$ and $g_2(x,y,z)=2x+z-1=0$ . So I get $5$ equations using lagrange multipliers solving $\nabla f = \lambda......Fri, 28 Aug 2026 13:15:05 -0400https://liberty.io/optimization-related-rate-searchlight.htmlhttps://liberty.io/optimization-related-rate-searchlight.htmlI can't seem to figure out how to set up this problem. I'm a bit confused on the relationship between the variables here; I've seen a similar problem involving a searchlight but the angle of $\thet......Fri, 28 Aug 2026 13:13:11 -0400https://liberty.io/center-of-mass-of-a-circle.htmlhttps://liberty.io/center-of-mass-of-a-circle.htmlHow would one find the center of mass of a circle? The center of mass of a rod is given by: $$\frac{1}{M}\int^{L}_{0}\rho x dx$$ So, for a sphere, it would be an area integral, such as: $$\frac{......Fri, 28 Aug 2026 13:11:14 -0400https://liberty.io/normalizing-eigenvectors-from-pauli-matrices.htmlhttps://liberty.io/normalizing-eigenvectors-from-pauli-matrices.htmlFor this example of a Pauli matrix, \begin{bmatrix} 0 & -i \\ i & 0 \end{bmatrix} I found that one of its eigenvectors (for $\lambda = 1$) is \begin{bmatrix} -i \\ 1 \end{bmat......Fri, 28 Aug 2026 13:00:04 -0400https://liberty.io/intuition-normal-subgroup-test-fraleigh-p-141-theorem-14-13.htmlhttps://liberty.io/intuition-normal-subgroup-test-fraleigh-p-141-theorem-14-13.html(1.) Not querying proofs or formality. I do this in my other question. Normal Subgroup Test says H is normal in G $\iff gH{g}^{-1}\subseteq H$ for all $g \in G$. What's the intuition of this su......Fri, 28 Aug 2026 12:59:08 -0400