https://pop.com.sgExplosive pop stories with broad entertainment appeal.en-usSun, 30 Aug 2026 09:37:23 -0400https://pop.com.sg/understanding-related-rates-ladder-problem.htmlhttps://pop.com.sg/understanding-related-rates-ladder-problem.htmlI'm having some trouble really understanding this related rates problem. I will first state the problem and then point out where I'm confused. The top of a 25 foot ladder leaning against a vertica......Sun, 30 Aug 2026 13:34:08 -0400https://pop.com.sg/how-to-compute-the-de-rham-cohomology-of-the-punctured-plane-just-by-c.htmlhttps://pop.com.sg/how-to-compute-the-de-rham-cohomology-of-the-punctured-plane-just-by-c.htmlI have a classmate learning algebra.He ask me how to compute the de Rham cohomology of the punctured plane $\mathbb{R}^2\setminus\{0\}$ by an elementary way,without homotopy type,without Mayer-Vie......Sun, 30 Aug 2026 13:32:27 -0400https://pop.com.sg/how-to-find-if-the-points-fall-in-a-straight-line-or-not.htmlhttps://pop.com.sg/how-to-find-if-the-points-fall-in-a-straight-line-or-not.htmlThree points $(x_1,y_1), (x_2,y_2)$ and $(x_3,y_3)$ whether fall in a straight line or not. How do I do that?...Sun, 30 Aug 2026 13:28:07 -0400https://pop.com.sg/induction-with-negative-step.htmlhttps://pop.com.sg/induction-with-negative-step.htmlWe've learned that we can use induction to show that a statement holds for all natural numbers (or for all natural numbers above n). The steps are: prove that the statement holds for a base number b ...Sun, 30 Aug 2026 13:11:59 -0400https://pop.com.sg/solving-trigonometric-word-problems-with-limited-information.htmlhttps://pop.com.sg/solving-trigonometric-word-problems-with-limited-information.htmlThis is a diagram of this problem The word Problem: Andrew wants to construct a new stairway to the front door of his house. The door is 3 feet above the ground and at an angle of $30^{\circ}$ to ......Sun, 30 Aug 2026 13:10:41 -0400https://pop.com.sg/surface-parameterization-of-a-cylinder.htmlhttps://pop.com.sg/surface-parameterization-of-a-cylinder.htmlSuppose I wanted to parameterize the cylinder $x^2 + y^2 = R^2$ (for the purpose of computing a surface integral). Say $z$ is in range $-z_0 \le z \le z_0$. The standard parameterization I see ...Sun, 30 Aug 2026 13:06:18 -0400https://pop.com.sg/evaluate-integral-of-sin-ln-x-dx.htmlhttps://pop.com.sg/evaluate-integral-of-sin-ln-x-dx.htmlCan somebody verify this solution for me? Thanks! Evaluate $\displaystyle \int \sin(\ln(x))dx$. First, lets simplify things a bit by making the substitution $y=ln(x)$. Then $dy = \frac{1}{x}dx$ and......Sun, 30 Aug 2026 13:05:56 -0400https://pop.com.sg/what-is-the-name-of-a-game-that-cannot-be-won-until-it-is-over.htmlhttps://pop.com.sg/what-is-the-name-of-a-game-that-cannot-be-won-until-it-is-over.htmlConsider the following game: The game is to keep a friend's secret. If you ever tell the secret, you lose. As long as you don't you are winning. Clearly, it's a game that takes a lifetime to win. ...Sun, 30 Aug 2026 13:03:57 -0400https://pop.com.sg/1e-0-8-what-what-does-e-mean-duplicate.htmlhttps://pop.com.sg/1e-0-8-what-what-does-e-mean-duplicate.htmlHello I came across a math equation and I was wondering what did the "e" stand for? the equation is : y=47931x-1E+0.8 Can someone please help me by showing what the E stands for and the answer for......Sun, 30 Aug 2026 13:01:15 -0400https://pop.com.sg/why-is-the-number-e-so-important-in-mathematics.htmlhttps://pop.com.sg/why-is-the-number-e-so-important-in-mathematics.htmlI've heard a lot about this number $e$. Why is it so important? How does it fit into the 'bigger picture' of mathematics? How is it calculated and used?...Sun, 30 Aug 2026 13:00:13 -0400https://pop.com.sg/direction-and-magnitude-of-a-dog-running.htmlhttps://pop.com.sg/direction-and-magnitude-of-a-dog-running.htmlProblem: A dog in an open field runs 12.0m east and then 30.0m in a direction 54 degrees west of north. Part A: In what direction must the dog then run to end up 12.0m south of her original start......Sun, 30 Aug 2026 12:59:31 -0400https://pop.com.sg/finding-slope-of-a-curve-by-finding-the-limits-of-secant-slopes.htmlhttps://pop.com.sg/finding-slope-of-a-curve-by-finding-the-limits-of-secant-slopes.htmlFind the slope of the curve $y=x^2-4x-5$ at the point $P(3,-8)$ by finding the limit of the secant slopes through point $P$. My try: I picked another point $Q$ to get the secant $PQ$. Since $P......Sun, 30 Aug 2026 12:58:38 -0400https://pop.com.sg/power-series-expansion.htmlhttps://pop.com.sg/power-series-expansion.htmlI recently had a problem. I know how to evaluate power series but I cannot seem to find an expansion for $\sqrt{x+1}$. I've tried differentiating it, in order to bring it in reciprocal form but t......Sun, 30 Aug 2026 12:48:15 -0400https://pop.com.sg/what-is-an-example-of-an-involutory-matrix-which-is-not-normal.htmlhttps://pop.com.sg/what-is-an-example-of-an-involutory-matrix-which-is-not-normal.htmlIn another question, a user kindly explained that not all Involutory matrices are Normal. I thought they were 'very Normal': both Unitary and Hermitian, but clearly they are neither always Unitary ......Sun, 30 Aug 2026 12:47:01 -0400https://pop.com.sg/finding-dy-dx-by-implicit-differentiation-with-the-quotient-rule.htmlhttps://pop.com.sg/finding-dy-dx-by-implicit-differentiation-with-the-quotient-rule.htmlI've been stuck on a certain implicit differentiation problem that I've tried several times now. $$ \frac{x^2}{x+y} = y^2+6 $$ I know to take the derivatives of both sides and got: $$ \frac{(x+y)2x-\...Sun, 30 Aug 2026 12:39:03 -0400https://pop.com.sg/every-conservative-vector-field-is-irrotational.htmlhttps://pop.com.sg/every-conservative-vector-field-is-irrotational.htmlI have done an example where I needed to show that every conservative $C^2$ vector field is irrotational. However, there is something unclear in the solutions: Namely, I am uncertain what does the ...Sun, 30 Aug 2026 12:38:55 -0400https://pop.com.sg/phi-the-greek-letter-and-it-039-s-use-in-mathematics.htmlhttps://pop.com.sg/phi-the-greek-letter-and-it-039-s-use-in-mathematics.htmlIn the following inequality $$\frac{1+c}{a+b}\leq\varphi$$ how do I say it in English? I think that I should say: "The ratio of the sum of $1$ and $c$ to the sum of $a$ and $b$ is less than or e......Sun, 30 Aug 2026 12:34:50 -0400https://pop.com.sg/what-are-real-life-examples-where-a-linear-equation-would-end-up-being.htmlhttps://pop.com.sg/what-are-real-life-examples-where-a-linear-equation-would-end-up-being.htmlIn Algebra, linear equations in one variable can have either one solution (i.e. $x=3$), infinitely many solutions (i.e. $2x+6=3(x+2)-x$ ), or no solution (i.e. $x+3=x+5$). What would be some real-l......Sun, 30 Aug 2026 12:34:23 -0400https://pop.com.sg/formula-for-the-n-th-derivative-of-e-1-x-2.htmlhttps://pop.com.sg/formula-for-the-n-th-derivative-of-e-1-x-2.html$f(x) = \begin{cases} e^{-1/x^2} & \text{ if } x \ne 0 \\ 0 & \text{ if } x = 0 \end{cases}$ so $\displaystyle f'(0) = \lim_{x \to 0} \frac{f(x) - f(0)}{x - 0} = \lim_{x \to 0} \frac {e^{-1......Sun, 30 Aug 2026 12:30:03 -0400https://pop.com.sg/subtraction-when-second-number-is-bigger-than-first-number.htmlhttps://pop.com.sg/subtraction-when-second-number-is-bigger-than-first-number.htmlI'm a bit new to this. I'm trying to figure out how subtraction works pen and paper wise. I have a bit of a program where I can't seem to find any answers online. What I want to do is use the borr......Sun, 30 Aug 2026 12:29:03 -0400https://pop.com.sg/how-to-use-polyfit-function-in-matlab.htmlhttps://pop.com.sg/how-to-use-polyfit-function-in-matlab.htmlFirst of all i am new to Matlab, and not sure whether to use polyfit or something else for my problem. I plotted a graph with the following matlab code: t=transpose(linspace(-1,1,50)) y=1./(1+......Sun, 30 Aug 2026 12:27:13 -0400https://pop.com.sg/how-to-calculate-the-nth-term-and-the-sum-in-this-series-if-the-common.htmlhttps://pop.com.sg/how-to-calculate-the-nth-term-and-the-sum-in-this-series-if-the-common.htmlI have this series comprised of $$1,2,5,10,17,26,...$$, and so on. So far i have found that they add up in intervals being odd numbers. But I don't know how to find, let's say the 16th term and t......Sun, 30 Aug 2026 12:22:04 -0400https://pop.com.sg/area-of-a-rectangle-using-a-double-integral-in-polar-coordinates.htmlhttps://pop.com.sg/area-of-a-rectangle-using-a-double-integral-in-polar-coordinates.htmlI'm trying to find the area (which we know is equal to $ab$) of the rectangle bounded by $0\leq x\leq a$ and $0\leq y\leq b$ where $0<a<b$ using a double-integral in polar coordinates. The do......Sun, 30 Aug 2026 12:17:18 -0400https://pop.com.sg/how-to-represent-the-edges-of-a-complete-graph.htmlhttps://pop.com.sg/how-to-represent-the-edges-of-a-complete-graph.htmlLet say I have a graph $\mathcal{G}$. Denote this graph by $\mathcal{G}=(\mathcal{V}, \mathcal{E})$ where $\mathcal{V}$ is the set of vertices and $\mathcal{E}$ is the set of edges. My question is ...Sun, 30 Aug 2026 12:17:06 -0400https://pop.com.sg/long-exact-sequence-of-a-fibration-center.htmlhttps://pop.com.sg/long-exact-sequence-of-a-fibration-center.htmlLet $p:E \rightarrow B$ be a fibration with fiber $F$ . Associated to this we have a long exact sequence $$\cdots \rightarrow \pi_n(F) \rightarrow \pi_n(E) \rightarrow \pi_n(B) \rightarrow \pi_{n-1......Sun, 30 Aug 2026 12:15:19 -0400https://pop.com.sg/what-is-a-two-dimensional-sphere-in-a-three-dimensional-space.htmlhttps://pop.com.sg/what-is-a-two-dimensional-sphere-in-a-three-dimensional-space.htmlI was reading about Hausdorff paradox and i met this statement "It involves the sphere S2 (a 2-dimensional sphere in R3)" and i don't know what it means or how it appears in a paper P.S I am a no......Sun, 30 Aug 2026 12:06:18 -0400https://pop.com.sg/is-the-proof-of-the-derivative-of-sin-x-circular.htmlhttps://pop.com.sg/is-the-proof-of-the-derivative-of-sin-x-circular.htmlThe proof of $\frac{d}{dx}\sin(x)$ goes something like this: $$\begin{aligned} \lim_{h\to0}\frac{\sin(x+h)-\sin(x)}{h}=\lim_{h\to0}\frac{\sin(x)\cos(h)+\cos(x)\sin(h)-\sin(x)}{h}\\ =\lim_{h\to0}\f......Sun, 30 Aug 2026 12:02:06 -0400https://pop.com.sg/convert-power-series-to-function.htmlhttps://pop.com.sg/convert-power-series-to-function.htmlI tried to solve the attached Power Series, however I can't get to the right answer. I wrote down the correct answer at the top-right of the page. Appreciate your help!...Sun, 30 Aug 2026 11:46:42 -0400https://pop.com.sg/height-of-the-hill.htmlhttps://pop.com.sg/height-of-the-hill.htmlThe angles of elevation of the top of a distant hill in the forest as seen from three consecutive km stones on a straight horizontal road are 30, 45 and 60 degrees. Find the height of the hill.\ My......Sun, 30 Aug 2026 11:39:35 -0400https://pop.com.sg/equation-of-a-plane-containing-a-point-and-a-line.htmlhttps://pop.com.sg/equation-of-a-plane-containing-a-point-and-a-line.htmlFind the equation of the plane containing the point (0, 7, -7) and the line $\frac{x+1}{-3} = \frac{y-3}{2} = \frac{z+2}{1}$ I'm not sure how to tackle this question, since the equation of the l......Sun, 30 Aug 2026 11:25:39 -0400https://pop.com.sg/2d-transformation-matrix-order.htmlhttps://pop.com.sg/2d-transformation-matrix-order.htmlRotate a triangle with vertices (2, 1). (4. 1). (3, 3) by 90 degrees (clock wise) and then Translate it by Tx 4 and Ty 3 and finally rotate it by 30 degrees (anti-clock wise). (cos90=0, sin90 1. co......Sun, 30 Aug 2026 11:25:06 -0400https://pop.com.sg/what-is-the-minimal-polynomial-of-a.htmlhttps://pop.com.sg/what-is-the-minimal-polynomial-of-a.htmlLet $f$ be an endomorphism of $\mathbb{R}^3$ with its matrix $A$ in the canonical basis $\mathcal{B}$ as $$ A = \begin{pmatrix} 3 & 1 & 1 \\ -1 & 1 & - 1 \\ 0 & 0 & 1 \end{...Sun, 30 Aug 2026 11:20:59 -0400https://pop.com.sg/derive-or-differentiate.htmlhttps://pop.com.sg/derive-or-differentiate.htmlWhen the action is: Taking the derivative what verb should be used? to differentiate to derive I feel that deriving is not the correct word here. In my mind it's more a synonym of deducing. Am I......Sun, 30 Aug 2026 11:16:54 -0400https://pop.com.sg/how-would-i-find-k-such-that-the-following-matrix-is-singular.htmlhttps://pop.com.sg/how-would-i-find-k-such-that-the-following-matrix-is-singular.html\begin{pmatrix}-4&1&4\\ 4&-2&-3\\ -34+k&7&18\end{pmatrix} I know that a singular matrix is not invertible and has a determinant of zero. But what I am confused about is get......Sun, 30 Aug 2026 11:15:56 -0400https://pop.com.sg/unitary-matrix-for-matrix-representation.htmlhttps://pop.com.sg/unitary-matrix-for-matrix-representation.htmlIn the book The Symmetric Group the author says: Let $\chi$ and $\psi$ be characters of the $G$-module $V$. By picking an orthonormal basis for $V$, we obtain a matrix representation $Y$ for $\ps......Sun, 30 Aug 2026 11:07:17 -0400https://pop.com.sg/optimization-word-problem-for-cost-effective-fence-enclosure.htmlhttps://pop.com.sg/optimization-word-problem-for-cost-effective-fence-enclosure.htmlHere's the question: A fence is to be built to enclose a rectangular area of 200 square feet. The fence along three sides is to be made of material that costs 5 dollars per foot, and the material f......Sun, 30 Aug 2026 11:01:50 -0400https://pop.com.sg/what-is-the-name-for-a-rectangle-that-is-not-a-square.htmlhttps://pop.com.sg/what-is-the-name-for-a-rectangle-that-is-not-a-square.htmlA square is a special case of a rectangle. What is a single-word term for a rectangle that is not a square? I am looking for a word that excludes squares. I am also looking for a word that is not "...Sun, 30 Aug 2026 10:40:05 -0400https://pop.com.sg/how-to-find-the-sum-of-x-bar-x-y-bar-y-closed.htmlhttps://pop.com.sg/how-to-find-the-sum-of-x-bar-x-y-bar-y-closed.htmlPlease I need help. I need the correct steps how to calculate: Sum $(x-\bar x)(y- \bar y)$? My numbers are: $x$: 2,4,6,8,10 $y$: 3,5,7,10,12 My results are: $\Sigma x=30$ $\Sigma y=37$ $\bar......Sun, 30 Aug 2026 10:31:59 -0400https://pop.com.sg/specification-of-2d-plane-in-4d-space.htmlhttps://pop.com.sg/specification-of-2d-plane-in-4d-space.htmlMy understanding is that for a given point of interception on a line in 3D space there is one plane such that all of its points are orthogonal to that line. Is there the same 1:1 correspondence in 4D ...Sun, 30 Aug 2026 10:24:45 -0400https://pop.com.sg/how-to-prove-the-sum-of-combination-is-equal-to-2-n-1.htmlhttps://pop.com.sg/how-to-prove-the-sum-of-combination-is-equal-to-2-n-1.htmlOne of the algorithm I learnt involve these steps: $1$. define a set $S$ of $n$ elements $2$. form a subset $S'$ of $k$ choice from $n$ elements of the set $S$ ($k$ starts with $1$), which is ...Sun, 30 Aug 2026 10:23:32 -0400https://pop.com.sg/how-many-9-digit-phone-numbers-possible-from-digits-1-2-3-4-5-5-5-5-5.htmlhttps://pop.com.sg/how-many-9-digit-phone-numbers-possible-from-digits-1-2-3-4-5-5-5-5-5.htmlMy daughter came home from school yesterday with a math quiz that we couldn't work out: How many 9-digit phone numbers can we compose from digits 1, 2, 3, 4, 5, 5, 5, 5, 5 ? I tried the basic ...Sun, 30 Aug 2026 10:20:11 -0400https://pop.com.sg/is-there-a-graph-that-has-7-vertices-and-each-vertex-has-a-degree-of-2.htmlhttps://pop.com.sg/is-there-a-graph-that-has-7-vertices-and-each-vertex-has-a-degree-of-2.htmlIs there a graph that has 7 vertices and each vertex has a degree of $2,2,3,5,5,5,6$? Any ideas on how to solve this one?...Sun, 30 Aug 2026 10:16:16 -0400https://pop.com.sg/how-to-tell-if-a-rubik-039-s-cube-is-solvable.htmlhttps://pop.com.sg/how-to-tell-if-a-rubik-039-s-cube-is-solvable.htmlHow can I determine if a certain Rubik's cube, that is in a certain state, is solveable? By "certain state" it could mean that the cube has been dismantled and put together again. And in my experie......Sun, 30 Aug 2026 10:12:08 -0400https://pop.com.sg/why-does-cross-cancellation-of-fractions-in-multiplication-work.htmlhttps://pop.com.sg/why-does-cross-cancellation-of-fractions-in-multiplication-work.htmlI'm reviewing my arithmetic and right now I'm at fractions, I'm just having a little bit of problem "visualizing" why cancellation works in multiplying fractions, I know how it works, it's just the......Sun, 30 Aug 2026 10:10:20 -0400https://pop.com.sg/how-can-we-describe-the-graph-of-x-x-for-negative-values.htmlhttps://pop.com.sg/how-can-we-describe-the-graph-of-x-x-for-negative-values.htmlWe usually only see the graph $y=x^x$ for $x>0$, because $x^x$ is a complex number for most negative values of $x$. Yet here is a full graph of $y=x^x$ on the real line: This graph may seem li......Sun, 30 Aug 2026 10:04:57 -0400https://pop.com.sg/integration-of-cosec-2x-method.htmlhttps://pop.com.sg/integration-of-cosec-2x-method.htmlHow to integrate $\csc (2x)$ by using substitution $t=\tan x$ ? I know a different way to solve it. But the book mentioned must be using substitution of tan x...Sun, 30 Aug 2026 10:00:45 -0400https://pop.com.sg/finding-the-equation-of-a-tangent-line-at-a-given-point.htmlhttps://pop.com.sg/finding-the-equation-of-a-tangent-line-at-a-given-point.htmlDoing my final review for the semester and just want to make sure I have all my bases covered. (I'll more than likely have another 4-5 question to come today, but I'll start here). Looking for som......Sun, 30 Aug 2026 09:57:54 -0400https://pop.com.sg/are-a-3-b-3-equivalent-to-1-3-a-1-3-b.htmlhttps://pop.com.sg/are-a-3-b-3-equivalent-to-1-3-a-1-3-b.htmlI have the following expression with two answers, I'm not sure if they're correct: $1/3(a+b) = 1/3 (a) + 1/3 (b)$ can the answer be: $a/3+ b/3$ too?...Sun, 30 Aug 2026 09:49:22 -0400https://pop.com.sg/can-a-function-be-increasing-at-a-point.htmlhttps://pop.com.sg/can-a-function-be-increasing-at-a-point.htmlFrom what I understand we say that a function is increasing on an interval $I$ if $$ x_1 < x_2 \quad\Rightarrow\quad f(x_1) < f(x_2). $$ for all $x_1,x_2\in I$. I understand that some might c......Sun, 30 Aug 2026 09:46:13 -0400https://pop.com.sg/what-is-exponentiation.htmlhttps://pop.com.sg/what-is-exponentiation.htmlIs there an intuitive definition of exponentiation? In elementary school, we learned that $$ a^b = a \cdot a \cdot a \cdot a \cdots (b\ \textrm{ times}) $$ where $b$ is an integer. Then later on......Sun, 30 Aug 2026 09:36:53 -0400https://pop.com.sg/calculating-bloch-wigner-dilogarithm.htmlhttps://pop.com.sg/calculating-bloch-wigner-dilogarithm.htmlIs there some tool/calculator (or some tables) for explicitly calculating values of the Bloch-Wigner dilogarithm?...Sun, 30 Aug 2026 09:36:07 -0400https://pop.com.sg/understanding-the-assumptions-in-the-reverse-fatou-039-s-lemma.htmlhttps://pop.com.sg/understanding-the-assumptions-in-the-reverse-fatou-039-s-lemma.htmlFatou's Lemma says the following: If $(f_n)$ is a sequence of extended real-valued, nonnegative, measurable functions defined on a measure space $\left(\mathbf{X},\mathcal{X},\mu\right)$, then ......Sun, 30 Aug 2026 09:33:45 -0400https://pop.com.sg/looking-for-induction-problems-that-are-not-formula-based.htmlhttps://pop.com.sg/looking-for-induction-problems-that-are-not-formula-based.htmlI am looking for problems that use induction in their proofs such as this one: Given a checker board with one square removed you can cover it with L-shaped pieces made out of three squares. This p......Sun, 30 Aug 2026 09:27:42 -0400https://pop.com.sg/point-set-topology.htmlhttps://pop.com.sg/point-set-topology.htmlSome time back, I tried reading Rudin's Principles of Mathematical Analysis and I found no trouble with the introductory chapter. In chapter II I encountered point set topology. The number of theor......Sun, 30 Aug 2026 09:27:26 -0400https://pop.com.sg/continuous-of-square-root-of-e-duplicate.htmlhttps://pop.com.sg/continuous-of-square-root-of-e-duplicate.htmlEvaluate $$\sqrt{e\sqrt{e\sqrt{e\sqrt{e\cdots}}}}$$ My attempt: Let $$x=\sqrt{e\sqrt{e\sqrt{e\sqrt{e\cdots}}}}$$ $$x^2=e\sqrt{e\sqrt{e\sqrt{e\cdots}}}$$ But I've no idea, how to proceed. Hope s......Sun, 30 Aug 2026 09:26:07 -0400https://pop.com.sg/del-partial-delta-nabla-correct-enunciation.htmlhttps://pop.com.sg/del-partial-delta-nabla-correct-enunciation.htmlI've come across various different symbols being pronounced as "del". What is the internationally accepted del? If not internationally, then what's the English/American(specify which one if they are ...Sun, 30 Aug 2026 09:18:35 -0400https://pop.com.sg/possible-range-of-a-function.htmlhttps://pop.com.sg/possible-range-of-a-function.htmlSuppose there is a function, $$f(x) = \frac{x}{1 + x^2}$$ And we have to find its domain and range. So first I calculated the domain:- Since the denominator of f(x) is always greater than 0 for a......Sun, 30 Aug 2026 09:10:38 -0400https://pop.com.sg/optimization-with-cylinder.htmlhttps://pop.com.sg/optimization-with-cylinder.htmlI have no idea how to do this problem at all. A cylindrical can without a top is made to contain V cm^3 of liquid. Find the dimensions that will minimize the cost of the metal to make the can. Si......Sun, 30 Aug 2026 09:04:52 -0400https://pop.com.sg/general-cartesian-rectangular-equation-for-polar-rose-r-sin-k-theta.htmlhttps://pop.com.sg/general-cartesian-rectangular-equation-for-polar-rose-r-sin-k-theta.htmlHow do I convert the Polar Equation $r=\sin(k \theta)$ to Cartesian Equation? I understand that $r^2=x^2+y^2$ and that $x=r\cos\theta$ and $y=r\sin\theta$, but no matter how I try to arrange them it ...Sun, 30 Aug 2026 08:56:38 -0400https://pop.com.sg/prove-the-formula-for-the-foci-of-an-ellipse.htmlhttps://pop.com.sg/prove-the-formula-for-the-foci-of-an-ellipse.htmlI have summarized the question below: If the vertices of an ellipse centered at the origin are $(a,0),(-a,0),(0,b),$ and $(0,-b)$, and $a>b$, prove that for foci at $(\pm c,0)$, $c^2=a^2-b^2$......Sun, 30 Aug 2026 08:52:48 -0400https://pop.com.sg/must-every-probability-distribution-over-a-countable-set-be-discrete.htmlhttps://pop.com.sg/must-every-probability-distribution-over-a-countable-set-be-discrete.htmlIntuitively I expect this to follow from countable additivity, but there are ideas I can't rule out such as: Select a real number r from the uniform distribution over [0, 1]. If r is exactly 0.5, ......Sun, 30 Aug 2026 08:47:21 -0400https://pop.com.sg/meaning-of-almost-everywhere-in-measure-theory.htmlhttps://pop.com.sg/meaning-of-almost-everywhere-in-measure-theory.htmlI'm slightly confused about the term almost everywhere as it is used in Folland's real analysis. Given a measure space $(X, \mathcal{M}, \mu)$ Suppose $f \equiv g$, $\mu$-almost everywhere where ......Sun, 30 Aug 2026 08:35:50 -0400https://pop.com.sg/taylor-series-expansion-of-sin-x-closed.htmlhttps://pop.com.sg/taylor-series-expansion-of-sin-x-closed.htmlI understand that Taylor series expansion for $\sin(x)$ is derived as follow: $$ \sin(x) = x - \frac{x^3}{3!}+\frac{x^5}{5!}-... $$ Now, what exactly is the first, second, and third term? Is the f......Sun, 30 Aug 2026 08:31:21 -0400https://pop.com.sg/what-is-the-inverse-of-2-x-duplicate.htmlhttps://pop.com.sg/what-is-the-inverse-of-2-x-duplicate.htmlNote: This may not be correct mathematical term, so in case of confusion, I mean what division is to multiplication. If not, just poke me in the comments. I was given this the other day: $2^x=8$ ......Sun, 30 Aug 2026 08:25:45 -0400https://pop.com.sg/derive-gaussian-quadrature-formula-for-three-points.htmlhttps://pop.com.sg/derive-gaussian-quadrature-formula-for-three-points.htmlAs Gaussian quadrature rule $$\int_{-1}^{1}f(x)dx \approx c_1f(x_1)+c_2f(x_2)+c_3f(x_3)\qquad(*)$$ for three nodes We have three weights, $c_1,c_2,c_3$ and three nodes ,$x_1,x_2,x_3$ to find. Now ...Sun, 30 Aug 2026 08:25:04 -0400https://pop.com.sg/definitional-question-difference-between-a-correspondence-and-a-functi.htmlhttps://pop.com.sg/definitional-question-difference-between-a-correspondence-and-a-functi.htmlIs there a difference between a correspondence and a function? For example, in game theory I am told that for a given strategy set, $\Sigma_i$, the best response given by $BR_i(\sigma_{-i})=\text{a......Sun, 30 Aug 2026 08:19:11 -0400https://pop.com.sg/in-statistics-why-do-you-reject-the-null-hypothesis-when-the-p-value-i.htmlhttps://pop.com.sg/in-statistics-why-do-you-reject-the-null-hypothesis-when-the-p-value-i.htmlThis is a question that I've always wondered in statistics, but never had the guts to ask the professor. The professor would say that if the p-value is less than or equal to the level of significan......Sun, 30 Aug 2026 08:13:00 -0400https://pop.com.sg/license-plate-permutations.htmlhttps://pop.com.sg/license-plate-permutations.htmlA state has changed its license plate numbering system for the three largest counties. Before the change, each plate had the number 1, 2, or 3, followed by either one or two letters, followed by 3 ......Sun, 30 Aug 2026 08:08:28 -0400https://pop.com.sg/linear-algebra-for-computer-scientists.htmlhttps://pop.com.sg/linear-algebra-for-computer-scientists.htmlI'm planning a linear algebra course for computer science freshmen. Do you have any good textbook to recommend?...Sun, 30 Aug 2026 08:08:20 -0400https://pop.com.sg/how-can-i-simplify-sqrt-50y-8-to-5y-4-sqrt-2.htmlhttps://pop.com.sg/how-can-i-simplify-sqrt-50y-8-to-5y-4-sqrt-2.htmlQuestion is in the title. My textbook has given me the question and solution but nothing in between. For this question I do not even know where to start. In this chapter I've been learning about ...Sun, 30 Aug 2026 08:06:12 -0400https://pop.com.sg/classification-of-finite-rank-abelian-groups.htmlhttps://pop.com.sg/classification-of-finite-rank-abelian-groups.htmlAn Abelian group $A$ is said to be finite rank if there is a natural number $n$ such that any finitely generated subgroup of $A$ can be generated by no more than $n$ elements. It is well-known that a ...Sun, 30 Aug 2026 08:03:15 -0400https://pop.com.sg/calculation-of-the-derivative-of-e-cos-x-from-first-principles.htmlhttps://pop.com.sg/calculation-of-the-derivative-of-e-cos-x-from-first-principles.htmlThe derivative of $e^{\cos(x)}$ is $-\sin(x)e^{\cos(x)}$. However I would like to prove it using first principles, i.e. by using $f'(x)=\lim_{h\to0}\frac{f(x+h)-f(x)}{h}$. I tried Taylor series but......Sun, 30 Aug 2026 08:01:16 -0400https://pop.com.sg/how-to-find-the-max-value-of-this-function-and-find-the-limit-to-infin.htmlhttps://pop.com.sg/how-to-find-the-max-value-of-this-function-and-find-the-limit-to-infin.htmlThe Question derivative of f(x) $$ 0 = (1-x)^n -n(1-x)^{n-1}x $$ $$ $$ finding the x in terms of n divide f '(x) by $$ (1-x)^{n-1}$$ $$ $$ $$ 0 = (1-x) - xn $$ $$ x = \frac{1} {1 + n} $$ i'm......Sun, 30 Aug 2026 07:49:53 -0400https://pop.com.sg/derivative-the-tangent-line-y-x-2-4x-5-2-7.htmlhttps://pop.com.sg/derivative-the-tangent-line-y-x-2-4x-5-2-7.htmlI was just introduced to derivatives. I have a problem, Find the slope of the tangent line to the graph at the indicated point. The question was, $$y=x^2 -4x -5 ; (-2,7)$$ So the indicated ......Sun, 30 Aug 2026 07:32:41 -0400https://pop.com.sg/solving-2-2-sin-x-2-sin-x-6-0.htmlhttps://pop.com.sg/solving-2-2-sin-x-2-sin-x-6-0.htmlCan someone help me with this question. The mark scheme says its C but i have no idea why or how you get to that....Sun, 30 Aug 2026 07:30:44 -0400https://pop.com.sg/how-is-graduate-abstract-algebra-different-from-undergraduate-abstract.htmlhttps://pop.com.sg/how-is-graduate-abstract-algebra-different-from-undergraduate-abstract.htmlI am currently an undergraduate math student. (In fact, freshmen.) I know that usually abstract algebra is taught somehow late in the undergraduate course, and curious how studies of abstract alge......Sun, 30 Aug 2026 07:21:30 -0400https://pop.com.sg/characteristic-map-of-a-n-cell-in-a-cw-complex.htmlhttps://pop.com.sg/characteristic-map-of-a-n-cell-in-a-cw-complex.htmlI have a problem in understanding the purpose of the definition of a CW complex. What really would help me is to understand the following: Let $\sigma$ be a n-cell and $\Phi_\sigma:\mathbb D^n \to......Sun, 30 Aug 2026 07:11:25 -0400https://pop.com.sg/how-to-find-the-equation-of-an-ellipsoid-given-the-center-and-two-trac.htmlhttps://pop.com.sg/how-to-find-the-equation-of-an-ellipsoid-given-the-center-and-two-trac.htmlIf a center of an ellipsoid $E$ is $(1,k,0)$ and the trace of $E$ when $z=0$ is $x^2+y^2-2x-8y+16=0$ and when $x=1$ it is $-9y^2-z^2+72y=135$. Determine the equation of E So, the ellipsoid has the ......Sun, 30 Aug 2026 06:53:17 -0400https://pop.com.sg/cubing-the-expression-of-a-complex-number.htmlhttps://pop.com.sg/cubing-the-expression-of-a-complex-number.htmlCalculate the solutions to $$\left(-8-8\sqrt{3}i\right)^3$$ I would really appreciate if you could help me with this. Thanks...Sun, 30 Aug 2026 06:47:43 -0400https://pop.com.sg/if-the-lateral-surface-area-of-a-square-based-pyramid-with-equal-edges.htmlhttps://pop.com.sg/if-the-lateral-surface-area-of-a-square-based-pyramid-with-equal-edges.htmlThe lateral surface area of a square based pyramid having equal edges is $144√3$ cm^2. Find the area of base. My Attempt: let, length of base =$a$ slant height=$l$ Then, $$L.S.A. of pyramid=144......Sun, 30 Aug 2026 06:44:20 -0400https://pop.com.sg/levi-civita-and-kronecker-delta-properties.htmlhttps://pop.com.sg/levi-civita-and-kronecker-delta-properties.htmlI'm trying to grasp Levi-civita and Kronecker del notation to use when evaluating geophysical tensors, but I came across a few problems in the book I'm reading that have me stumped. 1) $\delta_{i......Sun, 30 Aug 2026 06:29:34 -0400https://pop.com.sg/what-is-the-difference-between-mapping-and-morphism.htmlhttps://pop.com.sg/what-is-the-difference-between-mapping-and-morphism.htmlI wonder if there's differences between Mapping and Morphism. Although the terms are used in different context i.e. mapping for set theory and morphism for category theory, from my understanding th......Sun, 30 Aug 2026 06:22:21 -0400https://pop.com.sg/advantage-of-first-order-logic-over-second-order-logic.htmlhttps://pop.com.sg/advantage-of-first-order-logic-over-second-order-logic.htmlAs I look over the post that has the similar question, I began to wonder: The only reason I found is that first-order logic can prove validity of some second-order logic formula/sentences, as some......Sun, 30 Aug 2026 06:16:27 -0400https://pop.com.sg/almost-sure-convergence-of-a-sequence-of-random-variables.htmlhttps://pop.com.sg/almost-sure-convergence-of-a-sequence-of-random-variables.htmlOnce again I've encountered a problem, which might not be difficult: I'm given a sequence of random variables $ (X_n) $, each with density function $g_n(x) = nx^{n-1} \textbf{1}_{(0,1]} $. I am to ...Sun, 30 Aug 2026 06:08:52 -0400https://pop.com.sg/solving-differential-equation-y-039-cos-xy.htmlhttps://pop.com.sg/solving-differential-equation-y-039-cos-xy.htmlMy question is about solving differential equation $y'=\cos(xy)$. I tried to changing variable $u=xy$ \begin{align*} u &= xy \\ \frac{du}{dx} &= y+ x\frac{dy}{dx}\\ \frac{dy}{dx} &= \fr......Sun, 30 Aug 2026 05:55:38 -0400https://pop.com.sg/on-the-definition-of-locally-connected-spaces.htmlhttps://pop.com.sg/on-the-definition-of-locally-connected-spaces.htmlI have seen two a priori different definitions of locally connected space : 1) For all point $x$, and neighborhood $V$ of $x$, there is a connected neighborhood $C$ of $x$ such that $C\subseteq V......Sun, 30 Aug 2026 05:46:12 -0400https://pop.com.sg/find-the-domain-and-range-of-the-function-f-x-y-sqrt-x-y-and-sketch-th.htmlhttps://pop.com.sg/find-the-domain-and-range-of-the-function-f-x-y-sqrt-x-y-and-sketch-th.htmlI have found the domain to be $y \geq -x$. I have found the range to be $z \geq 0$, or $[ 0, \infty )$. I'm not sure how to sketch the domain in the xy-plane. I figured it would be a straight line ...Sun, 30 Aug 2026 05:36:54 -0400https://pop.com.sg/how-to-construct-a-genus-2-surface-from-8-gon.htmlhttps://pop.com.sg/how-to-construct-a-genus-2-surface-from-8-gon.htmlI am requesting some help or reference for visualization? I am having a hard time constructing a genus 2 surface from 8-gon. May I request for some reference? Here's the construction I used from H......Sun, 30 Aug 2026 05:36:32 -0400https://pop.com.sg/working-out-the-median-of-a-beta-function.htmlhttps://pop.com.sg/working-out-the-median-of-a-beta-function.htmlI am trying to work out the median of the beta function of $\mathrm{B}(1/2,1/6)$. I have been told the answer to this is $0.9510$ but i'm unsure to get there? Is there a simple formula in order to ......Sun, 30 Aug 2026 05:27:27 -0400https://pop.com.sg/is-x-x-a-proper-or-improper-fraction.htmlhttps://pop.com.sg/is-x-x-a-proper-or-improper-fraction.htmlIf the numerator is greater than the denominator, it's improper. The other way around, it's proper. What if the numerator is the same as the denominator?...Sun, 30 Aug 2026 05:21:11 -0400https://pop.com.sg/how-to-write-k-is-equal-to-any-integer-in-symbols.htmlhttps://pop.com.sg/how-to-write-k-is-equal-to-any-integer-in-symbols.htmlHow do you state that $k$ is equal to any integer in the following? The solutions to this equation $$2\sin(3x)-1=0$$ are $$ \left\{ \begin{array}{ll} x=\dfrac{\pi}{18}+\dfrac{2\pi}{3}k\\[4......Sun, 30 Aug 2026 05:18:47 -0400https://pop.com.sg/what-is-the-isomorphic-graph-of-k4-closed.htmlhttps://pop.com.sg/what-is-the-isomorphic-graph-of-k4-closed.htmlWhat is the isomorphic graph of a k4 graph? Is every complete graph isomorphic to itself? If there are any theorems related to it, it would be highly appreciated to be pointed out here. Thanks....Sun, 30 Aug 2026 05:09:06 -0400https://pop.com.sg/expectation-of-stopping-time-w-r-t-a-brownian-motion.htmlhttps://pop.com.sg/expectation-of-stopping-time-w-r-t-a-brownian-motion.htmlHow do you take the expectation of a stopping time with respect to a Brownian motion? The specific question is: $$ \tau = \inf\{ t \ge 0: B(t) \in \{-a, b\}\} $$ I understand the optional stopping ...Sun, 30 Aug 2026 04:55:54 -0400https://pop.com.sg/partial-differential-equation-change-of-variables.htmlhttps://pop.com.sg/partial-differential-equation-change-of-variables.htmlI am given the following; $u(x,y)$ satisfes, $$2\frac{\partial ^ 2u}{\partial x^2} +3\frac{\partial ^ 2u}{\partial y^2}-7\frac{\partial ^ 2u}{\partial x \partial y}=0 $$ Use a change of variables......Sun, 30 Aug 2026 04:54:29 -0400https://pop.com.sg/how-to-take-the-third-derivative-of-this-function-without-it-getting-e.htmlhttps://pop.com.sg/how-to-take-the-third-derivative-of-this-function-without-it-getting-e.htmlI have the equation $x^2 +xy + y^3 = 1$ and I'm being asked to find $y'''(x)$ at the point where $x=1$. I calculate $f'(x) = {{-y-2x}\over{x+3y^2}}$ When I try to take the second derivative of the ...Sun, 30 Aug 2026 04:43:04 -0400https://pop.com.sg/understanding-the-definition-of-the-center-z-g-of-a-group-g.htmlhttps://pop.com.sg/understanding-the-definition-of-the-center-z-g-of-a-group-g.htmlI'm having trouble understanding the definition of "center" in group theory. My textbook says: "The center, $Z(G)$, of a group is the subset of elements in $G$ that commute with every element of......Sun, 30 Aug 2026 04:29:48 -0400https://pop.com.sg/how-do-i-tell-wolfram-alpha-that-i-want-the-fourth-derivative-of-y-in-.htmlhttps://pop.com.sg/how-do-i-tell-wolfram-alpha-that-i-want-the-fourth-derivative-of-y-in-.htmlFor example, how would I enter y^(IV) - 16y = 0? typing out fourth derivative, and putting four ' marks does not seem to work....Sun, 30 Aug 2026 04:26:07 -0400https://pop.com.sg/induction-proof-dealing-with-geometric-series-duplicate.htmlhttps://pop.com.sg/induction-proof-dealing-with-geometric-series-duplicate.html$1+r+(r^2)+...+r^n= \frac{1-r^{n+1}} {1-r}$ Any help would be appreciated in solving the geometric series....Sun, 30 Aug 2026 04:24:12 -0400https://pop.com.sg/what-does-this-notation-mean-f-u.htmlhttps://pop.com.sg/what-does-this-notation-mean-f-u.htmlIf $F$ is a function and $U$ is a set, then what does $F|U$ mean? In http://ocw.mit.edu/courses/mathematics/18-101-analysis-ii-fall-2005/lecture-notes/lecture6.pdf, the Inverse Function Theorem, ......Sun, 30 Aug 2026 04:17:46 -0400https://pop.com.sg/why-are-modular-lattices-important.htmlhttps://pop.com.sg/why-are-modular-lattices-important.htmlA lattice $(L,\leq)$ is said to be modular when $$ (\forall x,a,b\in L)\quad x \leq b \implies x \vee (a \wedge b) = (x \vee a) \wedge b, $$ where $\vee$ is the join operation, and $\wedge$ is the ......Sun, 30 Aug 2026 04:03:24 -0400