https://pop.com.sgExplosive pop stories with broad entertainment appeal.en-usSun, 06 Sep 2026 13:23:26 -0400https://pop.com.sg/history-behind-the-choice-of-letters-h-and-k-for-the-vertex-of-a-parab.htmlhttps://pop.com.sg/history-behind-the-choice-of-letters-h-and-k-for-the-vertex-of-a-parab.htmlAfter failing to find a historical explanation for usage of letters $h$ and $k$ for the vertex of a parabola in most relatively recent textbooks in anglosphere, I turn to math.SE. Is there any ...Sun, 06 Sep 2026 17:20:26 -0400https://pop.com.sg/prove-n-prime-ideal-of-mathbb-z-iff-n-is-prime-or-zero.htmlhttps://pop.com.sg/prove-n-prime-ideal-of-mathbb-z-iff-n-is-prime-or-zero.htmlprove $(n)$ prime ideal of $\mathbb{Z}$ iff $n$ is prime or zero Defintions Def of prime Ideal (n) $$ ab\in (n) \implies a\in(n) \vee b\in(n) $$ Def 1] integer n is prime if $n \neq 0,\pm 1 $ and ...Sun, 06 Sep 2026 17:01:44 -0400https://pop.com.sg/is-the-classifying-space-a-fully-faithful-functor.htmlhttps://pop.com.sg/is-the-classifying-space-a-fully-faithful-functor.htmlGiven a topological group $G$, we can form its classifying space $BG$; suppose we have chosen some specific construction, say the bar construction. $B$ is a functor - given any homomorphism $G \to ......Sun, 06 Sep 2026 16:44:31 -0400https://pop.com.sg/finding-factors-of-a-complex-polynomial.htmlhttps://pop.com.sg/finding-factors-of-a-complex-polynomial.htmlLet $P(z) = z^4-z^3+z^2+2=0$. Express $P(z)$ as a product of two real quadratic factors and hence find the other two zeros. I am given that $P(1+i)=0$, but I don't know to begin finding the other ...Sun, 06 Sep 2026 16:43:13 -0400https://pop.com.sg/oracles-for-tqbf.htmlhttps://pop.com.sg/oracles-for-tqbf.htmlI've seen this question somewhere and I've been thinking about it a lot but couldnt think of an answer. Say you have oracles A and B for the TQFB (True Quantified Boolean Formula) decision problem,......Sun, 06 Sep 2026 16:36:46 -0400https://pop.com.sg/using-l-039-hopital-rule-on-a-limit-that-does-not-exist.htmlhttps://pop.com.sg/using-l-039-hopital-rule-on-a-limit-that-does-not-exist.htmlI was trying to solve an improper integral and had to evaluate the expression below with the upper limit as infinity (limit is 0) and lower limit as 0 (the limit I'm asking about here) Here's the ...Sun, 06 Sep 2026 16:36:31 -0400https://pop.com.sg/why-are-quadrants-defined-the-way-they-are.htmlhttps://pop.com.sg/why-are-quadrants-defined-the-way-they-are.htmlI was thinking about planes and things, and suddenly wondered why quadrants are defined the way they are, the first on the top-right, and so on. I wonder if this gives us any benefit, or if any ...Sun, 06 Sep 2026 16:26:51 -0400https://pop.com.sg/modulo-calculation-with-multiple-exponents-via-crt.htmlhttps://pop.com.sg/modulo-calculation-with-multiple-exponents-via-crt.htmlI'm aware that there are already a few questions like this but unfortunately I wasn't able to find an answer yet. $$ (14^{2014)^{2014}} \pmod {60} $$ So I started off by putting the modular in : ......Sun, 06 Sep 2026 16:15:44 -0400https://pop.com.sg/how-do-i-find-the-angular-acceleration-of-a-pulley-when-there-one-obje.htmlhttps://pop.com.sg/how-do-i-find-the-angular-acceleration-of-a-pulley-when-there-one-obje.htmlThe problem is as follows: The figure from below shows two blocks joined by a rope on each end goind through a pulley. The masses of each block are as follows $m_1=5\,kg$, $m_2=4\,kg$. The ...Sun, 06 Sep 2026 16:15:30 -0400https://pop.com.sg/a-tough-differential-calculus-problem.htmlhttps://pop.com.sg/a-tough-differential-calculus-problem.htmlThis is a question I've had a lot of trouble with. I HAVE solved it, however, with a lot of trouble and with an extremely ugly calculation. So I want to ask you guys (who are probably more '...Sun, 06 Sep 2026 16:13:12 -0400https://pop.com.sg/constant-of-integration-positive-or-negative.htmlhttps://pop.com.sg/constant-of-integration-positive-or-negative.htmlI was solving some equation related to mechanics ... anyways I reached this step: $\int dx=-10\int 1+\frac{20}{v-20}dv$ It's correct according to the answer key of the book. So the next step is: ......Sun, 06 Sep 2026 16:03:35 -0400https://pop.com.sg/deriving-the-normal-equation-used-to-solve-least-squares-problems.htmlhttps://pop.com.sg/deriving-the-normal-equation-used-to-solve-least-squares-problems.htmlI am studying for my linear algebra final and it was suggested to me that I learn how to derive the normal equation used in solving least squares problems. I have been looking in various places onl......Sun, 06 Sep 2026 15:59:48 -0400https://pop.com.sg/prove-that-the-gradient-of-a-unit-vector-equals-2-magnitude-of-the-vec.htmlhttps://pop.com.sg/prove-that-the-gradient-of-a-unit-vector-equals-2-magnitude-of-the-vec.htmlLet $\vec r=(x,y,z)$ Firstly find $\vec \nabla (\frac 1 r)$ where r is the magnitude of $\vec r$. I think I've done this correctly to get $-x(x^2+y^2+z^2)^{-\frac32} \hat i-y(x^2+y^2+z^2)^{-\frac32......Sun, 06 Sep 2026 15:48:56 -0400https://pop.com.sg/line-ellipsoid-intersection.htmlhttps://pop.com.sg/line-ellipsoid-intersection.htmlI am trying to determine whether a line and an ellipsoid (in general, inc. rotations) intersect. Following a similar argument to the Line-sphere intersection using vector notation I have construc......Sun, 06 Sep 2026 15:39:29 -0400https://pop.com.sg/problem-with-solving-equation-with-hyperbolic-functions-2-cosh-2x-sinh.htmlhttps://pop.com.sg/problem-with-solving-equation-with-hyperbolic-functions-2-cosh-2x-sinh.htmlI want to solve the following equation for real values of $x$, by substituting the exponential forms of the hyperbolic functions. \begin{equation} 2\cosh(2x) - \sinh(2x) = 2 \end{equation} If someone ...Sun, 06 Sep 2026 15:29:15 -0400https://pop.com.sg/is-there-any-inequality-between-2-norm-condition-number-and-frobenius-.htmlhttps://pop.com.sg/is-there-any-inequality-between-2-norm-condition-number-and-frobenius-.htmlWhat I have found in [1] Condition number inequality between Frobenius norm and 2-norm for square matrix, Consider a full rank matrix $X \in \mathbb{C}^{n \times m}$, $m=n$, then we can have, $$......Sun, 06 Sep 2026 15:11:41 -0400https://pop.com.sg/deriving-the-kinematic-equation-v-2-v-0-2-2a-x-x-0.htmlhttps://pop.com.sg/deriving-the-kinematic-equation-v-2-v-0-2-2a-x-x-0.htmlI have a question on deriving the kinematic equation $v^2=v_{0}^2+2a(x-x_{0})$ from first principles and the known kinematic equations. Is this simply differentiating, but with respect to time($t$)......Sun, 06 Sep 2026 14:59:22 -0400https://pop.com.sg/gradient-of-a-matrix.htmlhttps://pop.com.sg/gradient-of-a-matrix.htmlI was following Stephen Boyd's convex optimisation course and came across the following slide: Can somebody explain to me how the gradient was calculated for the quadratic and least-squares object......Sun, 06 Sep 2026 14:59:03 -0400https://pop.com.sg/the-volume-of-sphere-using-integrals.htmlhttps://pop.com.sg/the-volume-of-sphere-using-integrals.htmlIn spherical coordinate system I have the volume element $$dV=r^{2}\sin(\theta)\ d\theta\ d\varphi\ dr$$ I want to calculate the volume for the radius equal to $R$. I calculate the integral: $$\int......Sun, 06 Sep 2026 14:55:28 -0400https://pop.com.sg/infinimum-of-a-set.htmlhttps://pop.com.sg/infinimum-of-a-set.htmlGiven the following conditions: $x \in \Bbb R$ and $y\in (0,1)$ I was asked to prove that inf $ |x-y |=0 $ My Attempt: By the elementary properties of the modulus function , we know that $ ......Sun, 06 Sep 2026 14:47:34 -0400https://pop.com.sg/does-sin-2-x-cos-2-x-1-2-cos-2-x.htmlhttps://pop.com.sg/does-sin-2-x-cos-2-x-1-2-cos-2-x.htmlI am finishing a proof. It seems like I can use $\cos^2 + \sin^2 = 1$ to figure this out, but I just can't see how it works. So I've got two questions. Does $\sin^2 x - \cos^2 x = 1-2\cos^2 x$......Sun, 06 Sep 2026 14:44:13 -0400https://pop.com.sg/differentiating-an-inner-product.htmlhttps://pop.com.sg/differentiating-an-inner-product.htmlIf $(V, \langle \cdot, \cdot \rangle)$ is a finite-dimensional inner product space and $f,g : \mathbb{R} \longrightarrow V$ are differentiable functions, a straightforward calculation with components ...Sun, 06 Sep 2026 14:42:16 -0400https://pop.com.sg/find-the-arc-length-of-the-parametric-curve.htmlhttps://pop.com.sg/find-the-arc-length-of-the-parametric-curve.html$$x=6t-6sint$$ $$y=6-6cost$$ Find the arc length of the parametric curve $$Arc length = \int_{0}^{2\pi} \sqrt{(6-6cost)^2+(6sint)^2}dt\\ =\int_{0}^{2\pi} \sqrt{36-72cost+36cos^2t+36sin^2t}dt \\ =\......Sun, 06 Sep 2026 14:41:56 -0400https://pop.com.sg/write-an-equation-for-a-rational-function-with.htmlhttps://pop.com.sg/write-an-equation-for-a-rational-function-with.htmlWrite an equation for a rational function with: Vertical asymptotes at $x = -5$ and $x = 5$ $x$ intercepts at $x = -2$ and $x = -6$ Horizontal asymptote at $y = 6$ $y =$ ? I have $$\frac{(x+2)......Sun, 06 Sep 2026 14:33:10 -0400https://pop.com.sg/minimum-spanning-tree-edge-count.htmlhttps://pop.com.sg/minimum-spanning-tree-edge-count.htmlGiven is a weighted complete graph where every weigth is a positive ineger. Let n be the amount of vertices. I have to prove that the number of edges of a minimum spanning tree of that graph is eq......Sun, 06 Sep 2026 14:25:50 -0400https://pop.com.sg/prove-that-unit-impulse-function-integral-is-equal-to-one-duplicate.htmlhttps://pop.com.sg/prove-that-unit-impulse-function-integral-is-equal-to-one-duplicate.htmlUnit impulse function is one of the special functions which is widely used in the field of signal processing. It has nice properties that helps in some situations specially its sifting property. Bu......Sun, 06 Sep 2026 14:24:34 -0400https://pop.com.sg/order-on-eigenvalues-on-diagonal-matrix.htmlhttps://pop.com.sg/order-on-eigenvalues-on-diagonal-matrix.htmlIf the eigenvalues are say $-1$, $-1$ and $2$ for a $3$ x $3$ matrix, then when comes to the diagonal matrix, is it (from top left, to bottom right) $-1$ $-1$ $2$ or $2$ $-1$ $-1$ or $-1$ $2$ $-1$?......Sun, 06 Sep 2026 14:21:09 -0400https://pop.com.sg/example-of-a-not-recursively-enumerable-set-a-subseteq-mathbb-n.htmlhttps://pop.com.sg/example-of-a-not-recursively-enumerable-set-a-subseteq-mathbb-n.htmlCan someone give me an example if a not recursively enumerable set $A \subseteq \mathbb{N}$ ? I came up with this question, when trying to show, that there exist partial functions $f: \mathbb{N} \...Sun, 06 Sep 2026 14:18:59 -0400https://pop.com.sg/how-can-i-finish-integrating-int-sqrt-x-2-49-over-x-using-trig-substit.htmlhttps://pop.com.sg/how-can-i-finish-integrating-int-sqrt-x-2-49-over-x-using-trig-substit.html$$\int {\sqrt{x^2-49} \over x}\,dx $$ $$ x = 7\sec\theta$$ $$ dx = 7\tan\theta \sec\theta \,d\theta$$ $$\int {\sqrt{7^2\sec^2\theta - 7^2} \over 7\sec\theta}\left(7\tan\theta \sec\theta \,d\theta\r......Sun, 06 Sep 2026 14:12:17 -0400https://pop.com.sg/what-is-a-residue.htmlhttps://pop.com.sg/what-is-a-residue.htmlI've heard of residues in complex analysis, contour integration, etc. but all I really know it to be is the $c_{-1}$ term in the Laurent series for a function. Is there some sort of intuition on wh......Sun, 06 Sep 2026 13:58:47 -0400https://pop.com.sg/difference-between-parallel-and-equal-lines.htmlhttps://pop.com.sg/difference-between-parallel-and-equal-lines.htmlBasically I want to know that when did a pair of parallel lines become equal. And with the above please tell me the difference between the two...Sun, 06 Sep 2026 13:58:03 -0400https://pop.com.sg/why-can-039-t-i-use-product-rule-to-derive-x-ln-3.htmlhttps://pop.com.sg/why-can-039-t-i-use-product-rule-to-derive-x-ln-3.htmlThe product rule is defined as $$(f \cdot g)' = f' \cdot g + g' \cdot f.$$ I have the following function $u(x) = x\cdot \ln(3)$. I understand that you can derive it by implicit differentiation and......Sun, 06 Sep 2026 13:56:08 -0400https://pop.com.sg/volume-of-solid-rotated-about-y-axis.htmlhttps://pop.com.sg/volume-of-solid-rotated-about-y-axis.htmlFind the volume $V$ of the solid obtained by rotating the region bounded by the given curves about the specified line. $y = \ln{7x}$; $y = 3$ $y = 4$ $x = 0$ Rotated about the $Y$-Axis. How ......Sun, 06 Sep 2026 13:56:06 -0400https://pop.com.sg/understanding-the-musical-isomorphisms-in-vector-spaces.htmlhttps://pop.com.sg/understanding-the-musical-isomorphisms-in-vector-spaces.htmlI am trying to solidify my understanding of the muscial isomorphisms in the context of vector spaces. I believe I understand the definitions but would appreciate corrections if my understanding is ......Sun, 06 Sep 2026 13:47:18 -0400https://pop.com.sg/power-series-and-intervals-of-convergence.htmlhttps://pop.com.sg/power-series-and-intervals-of-convergence.htmlWhile teaching power series in my AP Calculus class, an interesting question came up. I gave an answer that I believe is correct, but I'm not certain why it is correct except "by definition", but ......Sun, 06 Sep 2026 13:45:42 -0400https://pop.com.sg/cross-product-of-two-sums-of-three-vectors.htmlhttps://pop.com.sg/cross-product-of-two-sums-of-three-vectors.htmlThis seems like a simple question but I couldn't find anywhere to verify Is it true that: $$(\mathbf{a}+\mathbf{b}+\mathbf{c})\times (\mathbf{d}+\mathbf{e}+\mathbf{f}) = (\mathbf{a}\times \mathbf......Sun, 06 Sep 2026 13:44:07 -0400https://pop.com.sg/solving-an-8th-degree-polynomial.htmlhttps://pop.com.sg/solving-an-8th-degree-polynomial.htmlI know that through the Abel Ruffini Theorem the general solution to a polynomial of degree five or more cannot be found explicitly. But are there are any other ways to find the roots of such a ...Sun, 06 Sep 2026 13:35:38 -0400https://pop.com.sg/how-to-find-a-basis-for-r-3-which-contains-a-basis-of-im-c.htmlhttps://pop.com.sg/how-to-find-a-basis-for-r-3-which-contains-a-basis-of-im-c.htmlFind a basis for $R^3$ which contains a basis of $im(C)$ (image of C), where $$C=\begin{pmatrix}1 & 2 & 3&4\\\ 2 & -4 & 6& -2\\ -1 & 2 & -3 &1 \end{pmatrix}$$......Sun, 06 Sep 2026 13:35:18 -0400https://pop.com.sg/how-many-faces-of-a-solid-can-one-see.htmlhttps://pop.com.sg/how-many-faces-of-a-solid-can-one-see.htmlWhat is the maximum number of faces of totally convex solid that one can "see" from a point? ...and, more importantly, how can I ask this question better? (I'm a college student with little exper......Sun, 06 Sep 2026 13:34:11 -0400https://pop.com.sg/second-order-linear-odes-and-analytic-coefficients.htmlhttps://pop.com.sg/second-order-linear-odes-and-analytic-coefficients.htmlA second order, linear, homogeneous ODE can be expressed as: $$a(x)y''(x) + b(x)y'(x) + c(x)y(x) = 0$$ I have only a basic knowledge about differential equations. However, all the texts state tha......Sun, 06 Sep 2026 13:32:52 -0400https://pop.com.sg/how-can-we-minimize-a-function-of-two-variables.htmlhttps://pop.com.sg/how-can-we-minimize-a-function-of-two-variables.htmlHere we are more interested in the method to minimize the function, rather than what the actual result is. The function I currently have, which I may need to change, is: $$b / d + (1/6 \log{(b)})......Sun, 06 Sep 2026 13:25:01 -0400https://pop.com.sg/measure-theory-semifinite-measure-folland-problem-1-14-duplicate.htmlhttps://pop.com.sg/measure-theory-semifinite-measure-folland-problem-1-14-duplicate.htmlLet $\mu$ be a semi-finite (and countably-additive) measure and $\mu(E)=\infty$. Then for any $C>0$, there exists a measurable set $F$ such that $F \subseteq E$ and $C<\mu(F)<\infty$. By ...Sun, 06 Sep 2026 13:21:05 -0400https://pop.com.sg/mathbb-z-sqrt-11-is-norm-euclidean.htmlhttps://pop.com.sg/mathbb-z-sqrt-11-is-norm-euclidean.htmlI'm trying to show that $\mathbb{Z}[\sqrt{11}]$ is Euclidean with respect to the function $a+b\sqrt{11} \mapsto|N(a+b\sqrt{11})| = | a^2 -11b^2|$ By multiplicativity, it suffices to show that $\fo......Sun, 06 Sep 2026 13:18:38 -0400https://pop.com.sg/finding-area-bounded-by-2-lines-curves-by-integration.htmlhttps://pop.com.sg/finding-area-bounded-by-2-lines-curves-by-integration.htmlWhen finding area bounded by 2 lines/curves by integration, I believe I 1st need to find the intersection point eg. $y=1-x$, $y=\sqrt{1+x}$ $1-x = \sqrt{1+x}$ ... $x(x+3)=0$ So $x=0 \text{ o......Sun, 06 Sep 2026 13:16:18 -0400https://pop.com.sg/a-good-book-for-introduction-to-mathematical-statistics.htmlhttps://pop.com.sg/a-good-book-for-introduction-to-mathematical-statistics.htmlI am trying to read this book (Introduction to Mathematical Statistics by Robert,Joseph, Allen,7th edition) I find it hard to follow. So, can anyone please recommend another book has similar conten......Sun, 06 Sep 2026 13:16:17 -0400https://pop.com.sg/the-gamma-function-and-the-pi-function.htmlhttps://pop.com.sg/the-gamma-function-and-the-pi-function.htmlI have been studying differential equation, in particular special functions. Euler's Gamma function, and Gauss's Pi function are essentially the same, differing only by an offset of one unit. for......Sun, 06 Sep 2026 13:14:02 -0400https://pop.com.sg/helmholtz-theorem.htmlhttps://pop.com.sg/helmholtz-theorem.htmlI have been told that the Helmholtz decomposition theorem says that every smooth vector field $\boldsymbol{F}$ [where I am not sure what precise assumptions are needed on $\boldsymbol{F}$] on an ...Sun, 06 Sep 2026 13:11:24 -0400https://pop.com.sg/x-coordinates-of-points-where-tangent-line-is-horizontal-or-vertical-u.htmlhttps://pop.com.sg/x-coordinates-of-points-where-tangent-line-is-horizontal-or-vertical-u.htmlFor the implicit equation $x^3 + y^3 - xy^2 = 8$ Determine the exact x-coordinates of all points where the tangent line is horizontal or vertical I figured out $\dfrac{dy}{dx} =\dfrac{y^2 - 3x^......Sun, 06 Sep 2026 13:07:58 -0400https://pop.com.sg/what-does-principal-mean-in-principal-unit-normal-vector.htmlhttps://pop.com.sg/what-does-principal-mean-in-principal-unit-normal-vector.htmlWhen working with space curves, why is it called the "Principal Unit Normal Vector"? I know there are two. So how do we know which one is the principal one? Also, is there a principal unit binor......Sun, 06 Sep 2026 12:59:01 -0400https://pop.com.sg/how-to-resolve-a-trigonometric-equation-involving-contangent.htmlhttps://pop.com.sg/how-to-resolve-a-trigonometric-equation-involving-contangent.htmlI am resolving some trigonometric equation systems. A typical system (the following one is invented) could be: $\sin(x) + \cos(y) = 1.5$ $2\sin(x) + 3\cos(y) = 1$ After getting the values, you j......Sun, 06 Sep 2026 12:38:35 -0400https://pop.com.sg/find-laplace-transform-using-unit-step-function-given-graph-of-a-perio.htmlhttps://pop.com.sg/find-laplace-transform-using-unit-step-function-given-graph-of-a-perio.htmlPlease correct my work. The textbook answer which is expressed exactly like this $1/s(1+e^{-s})$ does not match my own. Find the Laplace Transform for one period of the perpetual periodic function......Sun, 06 Sep 2026 12:33:09 -0400https://pop.com.sg/volume-of-truncated-prism.htmlhttps://pop.com.sg/volume-of-truncated-prism.htmlI needed to find the volume of what Wikipedia calls a truncated prism, which is a prism (with triangle base) that is intersected with a halfspace such that the boundary of the halfspace intersects......Sun, 06 Sep 2026 12:26:14 -0400https://pop.com.sg/proving-pascal-039-s-rule-n-choose-r-n-1-choose-r-1-n-1-choose-r-when-.htmlhttps://pop.com.sg/proving-pascal-039-s-rule-n-choose-r-n-1-choose-r-1-n-1-choose-r-when-.htmlI'm trying to prove that ${n \choose r}$ is equal to ${{n-1} \choose {r-1}}+{{n-1} \choose r}$ when $1\leq r\leq n$. I suppose I could use the counting rules in probability, perhaps combination= $......Sun, 06 Sep 2026 12:12:54 -0400https://pop.com.sg/notation-and-the-name-choice-for-meet-and-join-in-order-theory.htmlhttps://pop.com.sg/notation-and-the-name-choice-for-meet-and-join-in-order-theory.htmlI have two simple questions: From where do the names meet and join come from? I don't see any intuition between those names in context of order theory. From where does the notation come? I have to......Sun, 06 Sep 2026 12:12:00 -0400https://pop.com.sg/frames-and-co-frames.htmlhttps://pop.com.sg/frames-and-co-frames.htmlI'm trying to write a precise definition of a frame and co-frame. I have that a frame is a basis for a tangent space $T_pM$ and the co-frame is the dual basis (basis for the co-tangent space $T^{\a......Sun, 06 Sep 2026 11:58:45 -0400https://pop.com.sg/list-of-all-256-binary-combinations-of-8-digits-closed.htmlhttps://pop.com.sg/list-of-all-256-binary-combinations-of-8-digits-closed.htmli don't have any coding/java etc software. how can i view all the 256 binary combinatios from 8 digits? can someone tell me where can i find them in a list (any site/online generator etc) or copy t......Sun, 06 Sep 2026 11:51:41 -0400https://pop.com.sg/properties-of-subtraction-and-division.htmlhttps://pop.com.sg/properties-of-subtraction-and-division.htmlI'm (re)learning math and was looking for the properties of the most basic math operations. For addition and multiplication these are very easy to find (e.g. https://www.aaamath.com/g74a_px1.htm and ...Sun, 06 Sep 2026 11:50:35 -0400https://pop.com.sg/why-do-two-logs-with-negative-values-not-multiply-to-make-a-positive-l.htmlhttps://pop.com.sg/why-do-two-logs-with-negative-values-not-multiply-to-make-a-positive-l.htmlI was pondering a question while exploring logarithms, and logarithmic functions. $$ \log(-x) + \log(-x) = \log x^2 $$ Why is this considered incorrect? Seeing as: $$ \log(3) + \log(3) = \log3^2......Sun, 06 Sep 2026 11:38:29 -0400https://pop.com.sg/total-differential-of-a-function-f-x-y.htmlhttps://pop.com.sg/total-differential-of-a-function-f-x-y.htmlI found this definition of total variation of a function $F(x, y)$ \begin{eqnarray} dF(x, y) &\equiv& F(x+dx, y+dy) -F(x, y)\\ & = & [F(x+dx, y+dy) - F(x, y+dy)] + [F(x, y+dy) - F(x......Sun, 06 Sep 2026 11:32:29 -0400https://pop.com.sg/how-do-you-explain-what-a-hyperbolic-pde-is.htmlhttps://pop.com.sg/how-do-you-explain-what-a-hyperbolic-pde-is.htmlA friend of mine who just finished an introductory calculus class saw my books on fluid mechanics and asked me what a different PDEs look like. I explained elliptical and parabolic by giving exampl......Sun, 06 Sep 2026 11:28:29 -0400https://pop.com.sg/probability-of-an-event-happening-over-period-of-time-with-100-certain.htmlhttps://pop.com.sg/probability-of-an-event-happening-over-period-of-time-with-100-certain.htmlIf I have a certain event that is $100 \%$ to happen exactly once in $10$ days, how can I calculate the probability each day? I cannot be $10 \%$ per day, since by the end of $10$ days, there still......Sun, 06 Sep 2026 11:27:15 -0400https://pop.com.sg/is-there-a-3-dimensional-matrix-by-matrix-product.htmlhttps://pop.com.sg/is-there-a-3-dimensional-matrix-by-matrix-product.htmlIs it possible to multiply A[m,n,k] by B[p,q,r]? Does the regular matrix product have generalized form? I would appreciate it if you could help me to find out some tutorials online or mathematical......Sun, 06 Sep 2026 11:21:29 -0400https://pop.com.sg/solving-the-roots-of-redlich-kwong-equation.htmlhttps://pop.com.sg/solving-the-roots-of-redlich-kwong-equation.htmlGood evening everyone. After studying some equations of state, I've read about the mathematical steps formulated to model some. In the particular case of Van der Waals, where $$P=\frac{RT}{v-b}-\f......Sun, 06 Sep 2026 11:17:58 -0400https://pop.com.sg/why-does-the-exactness-of-a-koszul-complex-require-commutativity.htmlhttps://pop.com.sg/why-does-the-exactness-of-a-koszul-complex-require-commutativity.htmlMost references on Koszul complexes seem to assume that the elements $x_1,\ldots, x_n$ live in a commutative ring or are central. It appears to me that the proof also works provided the weaker assu......Sun, 06 Sep 2026 11:17:42 -0400https://pop.com.sg/user-michael-zieve-mathematics-stack-exchange.htmlhttps://pop.com.sg/user-michael-zieve-mathematics-stack-exchange.htmlQ&A for people studying math at any level and professionals in related fields...Sun, 06 Sep 2026 11:17:33 -0400https://pop.com.sg/diagonalize-real-symmetric-matrix.htmlhttps://pop.com.sg/diagonalize-real-symmetric-matrix.htmlI believe that for any real symmetric matrix $A$, I am always be able to write in the eigen decomposition form $$A = P\Lambda P^T \tag{1}$$ whatever the rank of $A$. (see Real symmetric matrix is......Sun, 06 Sep 2026 11:13:19 -0400https://pop.com.sg/what-is-more-probable-one-50-percent-chance-or-ten-5-percent-chances.htmlhttps://pop.com.sg/what-is-more-probable-one-50-percent-chance-or-ten-5-percent-chances.htmlWhat is more probable; a single 50% chance at success, or ten 5% chances at success? If the former, how many additional 5% chances are required in order for it to be as likely to succeed using mul......Sun, 06 Sep 2026 11:09:18 -0400https://pop.com.sg/taking-derivatives-of-a-power-series.htmlhttps://pop.com.sg/taking-derivatives-of-a-power-series.htmlI've been working on understanding power series, and came across a problem asking for the derivative of a certain power series and for the derivative to be a summation with a lower limit equal to z......Sun, 06 Sep 2026 11:07:28 -0400https://pop.com.sg/a-simple-determinant-calculations.htmlhttps://pop.com.sg/a-simple-determinant-calculations.htmlI calculate the claim of the theorem below for a couple of times, always I get $\sum_{i=1}^{\infty} c_i z^{L+i}$ and not $\sum_{i=1}^{\infty} c_i z^{L+M+i}$ as the coefficient of RHS of last determ......Sun, 06 Sep 2026 11:04:38 -0400https://pop.com.sg/isometry-is-not-surjective.htmlhttps://pop.com.sg/isometry-is-not-surjective.htmlAccording to the definition I am using, an isometry is a mapping $f:X \rightarrow Y$ between two metric spaces $(X,d_{X})$ and $(Y,d_{Y})$: $$ d_{Y}(f(a),f(b)) = d_{X}(a,b) $$ for all $a,b \in X $ I ...Sun, 06 Sep 2026 11:01:23 -0400https://pop.com.sg/calculate-improvement-percentage-based-on-time-spent.htmlhttps://pop.com.sg/calculate-improvement-percentage-based-on-time-spent.htmlI am doing a task that it takes 2 hours. After some changes the same task takes 10 minutes. What formula I can use to calculate the improvement percentage between these 2 values?...Sun, 06 Sep 2026 11:01:10 -0400https://pop.com.sg/is-distance-between-two-graphs-defined-somehow.htmlhttps://pop.com.sg/is-distance-between-two-graphs-defined-somehow.htmlIf the two graphs are isomorphic, then their distance is zero. And this distance increases, if vertices or edges are added or removed to/from one of the graphs. Does this "distance" have a special......Sun, 06 Sep 2026 11:01:05 -0400https://pop.com.sg/limits-of-functions-and-left-hand-right-hand-limit.htmlhttps://pop.com.sg/limits-of-functions-and-left-hand-right-hand-limit.htmlUse the $\epsilon$,$\delta$-definition of limits to show that $\lim\limits_{x\to a} f(x)=L$ if and only if $\lim\limits_{x \to a^+} f(x)=\lim\limits_{x\to a^-}f(x)=L$ I started by defining the le......Sun, 06 Sep 2026 10:58:01 -0400https://pop.com.sg/teaching-myself-differential-topology-and-differential-geometry.htmlhttps://pop.com.sg/teaching-myself-differential-topology-and-differential-geometry.htmlI have a hazy notion of some stuff in differential geometry and a better, but still not quite rigorous understanding of basics of differential topology. I have decided to fix this lacuna once for ......Sun, 06 Sep 2026 10:51:04 -0400https://pop.com.sg/calculate-on-which-side-of-a-straight-line-is-a-given-point-located.htmlhttps://pop.com.sg/calculate-on-which-side-of-a-straight-line-is-a-given-point-located.htmlI am a programmer without really good knowledge in math. :/ So I have to write an algorithm that changes the color of pixel(dot) P to opposite if it's on left side of the straigt line in coordinate ...Sun, 06 Sep 2026 10:37:22 -0400https://pop.com.sg/what-is-the-size-of-a-set-of-sets-of-the-empty-set.htmlhttps://pop.com.sg/what-is-the-size-of-a-set-of-sets-of-the-empty-set.htmlWhat is the size of a set of sets of the empty set {{}, {{}}, {{{}}}}? I am not sure if the empty set, in this case, can be considered as a set and make the size 3 or if it is 2 or 0. thanks...Sun, 06 Sep 2026 10:25:32 -0400https://pop.com.sg/in-how-many-ways-can-we-arrange-4-letters-of-the-word-engine.htmlhttps://pop.com.sg/in-how-many-ways-can-we-arrange-4-letters-of-the-word-engine.htmlI need to know a combinatoric solution to this problem, with Generating functions in book, gives us 102. It might be a very simple problem, but Im very confused with this. Would be very nice of ......Sun, 06 Sep 2026 10:23:57 -0400https://pop.com.sg/solve-system-of-equations-using-calculus-linear-algebra.htmlhttps://pop.com.sg/solve-system-of-equations-using-calculus-linear-algebra.htmlI'm having trouble with this problem. I'm not sure how to solve it the "calculus way". My professor gave us the formula "$b = c + Dt$" to work with (y = mx + b), and then dealt with partial derivat......Sun, 06 Sep 2026 10:11:51 -0400https://pop.com.sg/why-is-only-a-square-matrix-invertible.htmlhttps://pop.com.sg/why-is-only-a-square-matrix-invertible.htmlCan anyone give a very simple proof (or explanation) as to why only square matrix can possibly be invertible?...Sun, 06 Sep 2026 10:02:55 -0400https://pop.com.sg/moment-generating-function-of-a-gamma-distribution.htmlhttps://pop.com.sg/moment-generating-function-of-a-gamma-distribution.htmlIf I have a variable $X$ that has a gamma distribution with parameters $s$ and $\lambda$, what is its momment generating function. I know that it is $\int_0^\infty e^{tx}\frac{1}{\Gamma(s)}\lambda......Sun, 06 Sep 2026 09:59:01 -0400https://pop.com.sg/proving-the-product-of-two-non-singular-matrices-is-also-non-singular.htmlhttps://pop.com.sg/proving-the-product-of-two-non-singular-matrices-is-also-non-singular.htmlI am having trouble with a proof for linear algebra. Could somebody explain to me how to prove that if $A$ and $B$ are both $n\times n$ non singular matrices, that their product $AB$ is also non si......Sun, 06 Sep 2026 09:55:32 -0400https://pop.com.sg/how-to-start-proving-a-times-b-times-c-ne-a-times-b-times-c.htmlhttps://pop.com.sg/how-to-start-proving-a-times-b-times-c-ne-a-times-b-times-c.htmlThis is a problem from Discrete Mathematics and its Applications: Explain why $A \times B \times C$ and $(A \times B) \times C$ are not the same. I understand the process behind the Cartesian pro......Sun, 06 Sep 2026 09:48:13 -0400https://pop.com.sg/factorization-of-an-invertible-symmetric-matrix.htmlhttps://pop.com.sg/factorization-of-an-invertible-symmetric-matrix.htmlGiven any invertible symmetric matrix: $A=\begin{bmatrix}a&b&c\\ b&d&e\\ c&e&f\end{bmatrix}$ over the complex number, Can be it factored as $A=T^\top T$? where $T^\top$ ......Sun, 06 Sep 2026 09:32:23 -0400https://pop.com.sg/what-is-the-measure-of-measuredangle-x-in-the-figure-below.htmlhttps://pop.com.sg/what-is-the-measure-of-measuredangle-x-in-the-figure-below.htmlFor reference: In the figure shown P and Q are points of tangency and $ \overset{\LARGE{\frown}}{BC}+\overset{\LARGE{\frown}}{FE}=130^o. $ Calculate "x". My progress: I think the starting ...Sun, 06 Sep 2026 09:32:09 -0400https://pop.com.sg/proving-that-a-point-is-a-global-minimum.htmlhttps://pop.com.sg/proving-that-a-point-is-a-global-minimum.htmlI have this cost function $$ f(x,y,z)=0.5 \left(x^2+y^2+0.1(z^2)\right)+0.55z $$ such that $$x,y,z>0$$ $$x+y+z=1$$ I was given the point $(0.5,0.5,0)$ as a local minimum. How to I go along to pr......Sun, 06 Sep 2026 09:26:03 -0400https://pop.com.sg/where-does-the-factor-y-2-1-come-from.htmlhttps://pop.com.sg/where-does-the-factor-y-2-1-come-from.htmlI keep looking and writing out the problem, but I don't know where $(y^2 + 1)$ comes from. I get $y^2(y-3)$ as the numerator instead, but I can't tell if I'm having a brain fart and am missing some......Sun, 06 Sep 2026 09:25:46 -0400https://pop.com.sg/how-is-the-pythagorean-theorem-related-to-the-equation-of-a-circle.htmlhttps://pop.com.sg/how-is-the-pythagorean-theorem-related-to-the-equation-of-a-circle.htmlMy teacher wants me to figure out how the Pythagorean Theorem and the Equation of a Circle are related. I can't figure this out because I view them as being two completely different things. I under......Sun, 06 Sep 2026 09:16:06 -0400https://pop.com.sg/how-do-i-write-3-6-11-18-27-38-in-set-builder-notation.htmlhttps://pop.com.sg/how-do-i-write-3-6-11-18-27-38-in-set-builder-notation.htmlFor the set $\{3,6,11,18,27,38,...\}$, the $(n+1)^\text{th}$ term, which I'll call $a_{n+1}$, is: $$a_{n+1} = a_n + 2n+1$$ How can I write this set in set-builder notation? My best guess doesn't seem ...Sun, 06 Sep 2026 09:07:39 -0400https://pop.com.sg/find-the-domain-of-the-function-it-039-s-a-square-root-polynomial.htmlhttps://pop.com.sg/find-the-domain-of-the-function-it-039-s-a-square-root-polynomial.htmlFind the domain of the function: $$f(x)= \sqrt{x^2 - 4x - 45}$$ I'm just guessing here; how about if I square everything and then put it in the graphing calculator? Thanks, Lauri...Sun, 06 Sep 2026 08:54:12 -0400https://pop.com.sg/what-is-the-difference-between-normal-and-perpendicular.htmlhttps://pop.com.sg/what-is-the-difference-between-normal-and-perpendicular.htmlWhat is the difference when a line is said to be normal to another and a line is said to be perpendicular to other?...Sun, 06 Sep 2026 08:52:07 -0400https://pop.com.sg/using-vector-identities-and-the-divergence-theorem-to-derive-simplifie.htmlhttps://pop.com.sg/using-vector-identities-and-the-divergence-theorem-to-derive-simplifie.htmlI am trying to follow along a review paper regarding the derivation of the Cahn-Hilliard paper. The question I have is related to the following section of the paper: I have been trying to follow a......Sun, 06 Sep 2026 08:42:20 -0400https://pop.com.sg/integral-power-rule-step-by-step.htmlhttps://pop.com.sg/integral-power-rule-step-by-step.htmlReal quick, two things: I'm sorry if my notation or terminology is incorrect, and I know what I'm asking isn't strictly necessary for my studies, but writing something out step by step helps me to ...Sun, 06 Sep 2026 08:41:16 -0400https://pop.com.sg/simple-application-of-prime-number-theorem.htmlhttps://pop.com.sg/simple-application-of-prime-number-theorem.htmlI was wondering if someone could possibly help me verify the following: For sufficiently large $n$ there is always a prime between $n- \sqrt{n}$ and $n$. I am not sure if this is true or not. If......Sun, 06 Sep 2026 08:26:51 -0400https://pop.com.sg/intuitive-explanation-for-why-the-15-and-290-theorems-should-be-true.htmlhttps://pop.com.sg/intuitive-explanation-for-why-the-15-and-290-theorems-should-be-true.htmlI recently read about the remarkable 15 and 290 theorems, the second of which states that any positive definite integral quadratic form represents all positive integers iff it represents all positive ...Sun, 06 Sep 2026 08:20:56 -0400https://pop.com.sg/what-039-s-the-geometrical-interpretation-of-the-magnitude-of-gradient.htmlhttps://pop.com.sg/what-039-s-the-geometrical-interpretation-of-the-magnitude-of-gradient.htmlIn the following picture, the author of the Field and Wave Electromagnetics shows the geometrical meaning of the direction of the gradient. That is, only by following the direction of the normal ve......Sun, 06 Sep 2026 08:19:43 -0400https://pop.com.sg/does-non-symmetric-positive-definite-matrix-have-positive-eigenvalues.htmlhttps://pop.com.sg/does-non-symmetric-positive-definite-matrix-have-positive-eigenvalues.htmlI found out that there exist positive definite matrices that are non-symmetric, and I know that symmetric positive definite matrices have positive eigenvalues. Does this hold for non-symmetric mat......Sun, 06 Sep 2026 08:18:29 -0400https://pop.com.sg/numerator-vs-denominator-vs-nominator.htmlhttps://pop.com.sg/numerator-vs-denominator-vs-nominator.htmlWhat is appropriate usage of "numerator", "denominator", and "nominator" to refer to parts of a fraction? I'm posting this question and answer here because I had little luck finding a clear answer ...Sun, 06 Sep 2026 08:18:02 -0400https://pop.com.sg/understanding-the-formula-for-curvature.htmlhttps://pop.com.sg/understanding-the-formula-for-curvature.htmlIn this khan academy article, they discuss how can define curvature as, $$ \bigg\| \frac{dT}{dS} \bigg\| = \kappa$$ In the post they write, "However, we don't want differences in the rate at ......Sun, 06 Sep 2026 08:16:03 -0400https://pop.com.sg/why-do-we-say-almost-surely-in-probability-theory.htmlhttps://pop.com.sg/why-do-we-say-almost-surely-in-probability-theory.htmlI recently asked a question and got a great answer that involved proving that "X is almost surely one of the roots of P". I know (now) that "almost surely" means "with probability 1", but I've never ...Sun, 06 Sep 2026 07:51:59 -0400https://pop.com.sg/define-strong-solution.htmlhttps://pop.com.sg/define-strong-solution.htmlLet $\Omega \subset \mathbb{R}^n$ be open bounded. We find a function $u:\overline\Omega \rightarrow R$ satisfying $-\Delta u + u = f$ in $\Omega$, $u=0$ on $\partial\Omega$ ---(*) Define strong ...Sun, 06 Sep 2026 07:49:59 -0400