https://pop.com.sgExplosive pop stories with broad entertainment appeal.en-usSun, 30 Aug 2026 14:52:41 -0400https://pop.com.sg/continuously-changing-dimensions-of-a-rectangle.htmlhttps://pop.com.sg/continuously-changing-dimensions-of-a-rectangle.htmlThe dimension of a rectangle are continousl changing. The width increases at a rate of 3 in/s while the length decreases at the rate of 2 in/s. At one instant the rectangle is a 20-in square. How ......Sun, 30 Aug 2026 18:40:12 -0400https://pop.com.sg/show-that-b-x-n-1-p-1-b-n-x-1-n-p.htmlhttps://pop.com.sg/show-that-b-x-n-1-p-1-b-n-x-1-n-p.htmlIn my statistics book $B$ is defined as: $$B(x; \ n, \ p) = \sum_{y = 0}^{x}b(y; \ n, \ p) = \sum_{y = 0}^{x}{n \choose y}p^{y}(1 - p)^{n - y}$$ I want to show that: $$B(x; \ n, \ 1 - p) = 1 - B(n......Sun, 30 Aug 2026 18:33:28 -0400https://pop.com.sg/how-to-find-explicit-formula-for-a-sequence-seems-to-be-geometric-but-.htmlhttps://pop.com.sg/how-to-find-explicit-formula-for-a-sequence-seems-to-be-geometric-but-.htmlI recently came across this interesting problem, where the progression was neither arithmetic nor geometric. The problem asks for an explicit formula for the following recursive formula: $a_{1} = ......Sun, 30 Aug 2026 18:28:23 -0400https://pop.com.sg/how-does-full-row-rank-imply-column-space-is-r-m-for-a-m-times-n-matri.htmlhttps://pop.com.sg/how-does-full-row-rank-imply-column-space-is-r-m-for-a-m-times-n-matri.htmlFrom Gilbert Strang's textbook Introduction to Linear Algebra (p.159) Every matrix with full row rank has these properties $Ax=b$ has a solution for every right side $b$. The column space is the......Sun, 30 Aug 2026 18:25:42 -0400https://pop.com.sg/expected-number-of-steps-between-states-in-a-markov-chain.htmlhttps://pop.com.sg/expected-number-of-steps-between-states-in-a-markov-chain.htmlSuppose I am given a state space $S=\{0,1,2,3\}$ with transition probability matrix $\mathbf{P}= \begin{bmatrix} \frac{2}{3} & \frac{1}{3} & 0 & 0 \\[0.3em] \frac{2}{......Sun, 30 Aug 2026 18:22:19 -0400https://pop.com.sg/confusion-regarding-fermat-039-s-theorem-and-closed-interval-method.htmlhttps://pop.com.sg/confusion-regarding-fermat-039-s-theorem-and-closed-interval-method.htmlIn James Stewart's Calculus (8th edition) Fermat's Theorem is stated as If $f$ has a local maximum or minimum at $c$, and if $f'(c)$ exists, then $f'(c) = 0$. The author cautions that the convers......Sun, 30 Aug 2026 18:10:59 -0400https://pop.com.sg/why-divide-diameter-by-square-root-of-2-to-get-a-diameter-of-a-circle-.htmlhttps://pop.com.sg/why-divide-diameter-by-square-root-of-2-to-get-a-diameter-of-a-circle-.htmlI would be very grateful if you can help me with this problem. I am trying to explain in the simplest terms possible the sequence of f-stops in photography. The common f-stop rounded sequence is: f......Sun, 30 Aug 2026 18:06:05 -0400https://pop.com.sg/definition-for-covariant-derivative.htmlhttps://pop.com.sg/definition-for-covariant-derivative.htmlWhat is simple definition of the covariant derivative that looks like the definition of the derivative of a function in calculus? definition of the derivative of a function in calculus is: $$\fra......Sun, 30 Aug 2026 18:01:24 -0400https://pop.com.sg/has-there-ever-been-an-application-of-dividing-by-0.htmlhttps://pop.com.sg/has-there-ever-been-an-application-of-dividing-by-0.htmlRegarding the expression $a/0$, according to Wikipedia: In ordinary arithmetic, the expression has no meaning, as there is no number which, multiplied by $0$, gives $a$ (assuming $a\not= 0$), a......Sun, 30 Aug 2026 18:01:13 -0400https://pop.com.sg/proving-this-limit-lim-limits-n-to-infty-sqrt-n-n-1.htmlhttps://pop.com.sg/proving-this-limit-lim-limits-n-to-infty-sqrt-n-n-1.htmlThis is supposed to be proven: $$\lim\limits_{n\to\infty}\sqrt [n]n=1$$ The sequence is monotone decreasing and has a lower bound of $1$. So $\epsilon=0$ With the Archimedean Property we get: $......Sun, 30 Aug 2026 17:45:50 -0400https://pop.com.sg/finding-maclaurin-series-of-a-function.htmlhttps://pop.com.sg/finding-maclaurin-series-of-a-function.htmlThe prompt is to find the Maclaurin series of the function $$f(x) = xe^{x^2-1}$$ and evaluate the 100th derivative. At what value of x does this series converge? We know that $$e^x = \sum \frac{x......Sun, 30 Aug 2026 17:28:40 -0400https://pop.com.sg/how-do-derivatives-work-in-the-xyz-3d-plane.htmlhttps://pop.com.sg/how-do-derivatives-work-in-the-xyz-3d-plane.htmlIn just the XY plane, a derivative is the slope of the tangent line. How does it work when you have a 3D shape? I recall 3D math, and the Z-axis. Was this Calc 3? Let's say you have a "mou......Sun, 30 Aug 2026 17:22:04 -0400https://pop.com.sg/ode-theory-are-centers-and-linear-centers-the-same-for-reversible-syst.htmlhttps://pop.com.sg/ode-theory-are-centers-and-linear-centers-the-same-for-reversible-syst.htmleveryone! I'm trying to prove that a linear center of a planar system IS a center when the system is reversible (invariant under the change of variables $t\mapsto -t$ and $y \mapsto -y$). I fooled ...Sun, 30 Aug 2026 17:16:12 -0400https://pop.com.sg/solve-the-equation-x-2-3-6x-1-3-8-0.htmlhttps://pop.com.sg/solve-the-equation-x-2-3-6x-1-3-8-0.htmlI'm going around in circles on this one... Solve the equation $$x^{2/3}-6x^{1/3} + 8 = 0$$ According to the book, the answer is 8, 64, but that makes no sense to me either... Thanks Gary...Sun, 30 Aug 2026 17:10:20 -0400https://pop.com.sg/meaning-of-commutative-diagram.htmlhttps://pop.com.sg/meaning-of-commutative-diagram.htmlWhat is the meaning of a commutative diagram in mathematics? For example, if a map translate an object, then rotate it around the origin and then translate it again, is this a commutative diagram?...Sun, 30 Aug 2026 17:04:02 -0400https://pop.com.sg/law-of-total-expectation-with-conditional-expectations.htmlhttps://pop.com.sg/law-of-total-expectation-with-conditional-expectations.htmlI am confused between these two equations which use the law of total expectation: $$E(X|Y)=E(X|Y,Z)P(Z)+E(X|Y,Z')P(Z')$$ $$E(X|Y)=E(X|Y,Z)P(Z|Y)+E(X|Y,Z')P(Z'|Y)$$ where Z' is the complement of Z. ......Sun, 30 Aug 2026 16:44:41 -0400https://pop.com.sg/subset-of-a-finite-set-is-finite.htmlhttps://pop.com.sg/subset-of-a-finite-set-is-finite.htmlWe define $A$ to be a finite set if there is a bijection between $A$ and a set of the form $\{0,\ldots,n-1\}$ for some $n\in\mathbb N$. How can we prove that a subset of a finite set is finite? It......Sun, 30 Aug 2026 16:43:18 -0400https://pop.com.sg/confused-about-the-definition-and-properties-of-a-sigma-algebra.htmlhttps://pop.com.sg/confused-about-the-definition-and-properties-of-a-sigma-algebra.htmlSo I am taking an intro to measure theoretic probability course and am a bit confused by how we defined sigma-algebras and measurable sets. In particular if you have a sigma-algebra is that suffic......Sun, 30 Aug 2026 16:42:53 -0400https://pop.com.sg/composition-of-convex-and-power-function.htmlhttps://pop.com.sg/composition-of-convex-and-power-function.htmlLet $g$ be a convex nonegative function, and $p\ge1$. To show: $f(x)=g(x)^p$ is convex. Let $h(x)=x^p$. Then clearly $f=h \circ g$. Denote $\tilde{h}$ as the extended-value extension of $h$, which ...Sun, 30 Aug 2026 16:37:31 -0400https://pop.com.sg/finding-area-of-a-triangle-with-integrals.htmlhttps://pop.com.sg/finding-area-of-a-triangle-with-integrals.htmlI am suppose to find the area of a triangle using integrals with vertices 0,0 1,2 and 3,1 This gives me $y= 2x$ $y=\frac{1}{3}x$ $y= \frac{-1}{2}x+\frac{5}{2}$ for my slopes I know that I can ...Sun, 30 Aug 2026 16:37:22 -0400https://pop.com.sg/injectivity-surjectivity-and-cardinality.htmlhttps://pop.com.sg/injectivity-surjectivity-and-cardinality.htmlIt is known that if $f:A\to B$ is injective, then the $\mathrm{Card}(A)\leq\mathrm{Card}(B)$. Similarly if $f$ is surjective, the opposite inequality holds. Can one please show me how to write down a ...Sun, 30 Aug 2026 16:31:22 -0400https://pop.com.sg/how-to-simplify-a-function-with-fractional-exponents.htmlhttps://pop.com.sg/how-to-simplify-a-function-with-fractional-exponents.htmlI want to understand how to simplify fractional exponents in the formula $$g(x) = (4/3)x^{-1/3}-(5/3)x^{2/3}$$ My text book says the answer is $$\frac{4 -5x}{3x^{1/3}}$$ but I can only simplify ......Sun, 30 Aug 2026 16:29:06 -0400https://pop.com.sg/length-of-a-spiral.htmlhttps://pop.com.sg/length-of-a-spiral.htmlI need to find the length of a spiral. The spiral start at a certain radius R1 (25mm) and ends at a larger radius R2(unknown). As the spiral spins outwards, the distance between each arm of the spi......Sun, 30 Aug 2026 16:19:01 -0400https://pop.com.sg/solving-the-trig-inequality-sin-x-cos-x-1.htmlhttps://pop.com.sg/solving-the-trig-inequality-sin-x-cos-x-1.html$|\sin{x} + \cos{x} |> 1$ How to solve this kind of question? Is there any websites to learn trigonometry inequalities? My teacher only taught us the simple question but not the complicated one.......Sun, 30 Aug 2026 16:15:34 -0400https://pop.com.sg/optimality-conditions-necessary-vs-sufficient.htmlhttps://pop.com.sg/optimality-conditions-necessary-vs-sufficient.htmlFor a convex optimization problem, the first-order necessary condition says that at an optimum, the gradient is equal to zero. $\triangledown L(w*) = 0 $ The second-order sufficient condition ens......Sun, 30 Aug 2026 16:07:58 -0400https://pop.com.sg/what-does-9-with-a-line-above-the-9-mean.htmlhttps://pop.com.sg/what-does-9-with-a-line-above-the-9-mean.htmlWhat does this mean? $$\Large.\overline9 $$ I've never seen this notation before....Sun, 30 Aug 2026 15:55:10 -0400https://pop.com.sg/an-amazing-approximation-of-e.htmlhttps://pop.com.sg/an-amazing-approximation-of-e.htmlAs we can read in Wolfram Mathworld's article on approximations of $e$, the base of natural logarithm, An amazing pandigital approximation to e that is correct to $18457734525360901453873570$ de......Sun, 30 Aug 2026 15:48:53 -0400https://pop.com.sg/union-of-two-3x3-matrices.htmlhttps://pop.com.sg/union-of-two-3x3-matrices.htmlI'm trying to find the Union of two sets of matrices. The first set is all diagonal matrices, for example $$A = \begin{bmatrix} a & 0 & 0 \\ 0 & b & 0 \\ ......Sun, 30 Aug 2026 15:38:39 -0400https://pop.com.sg/what-is-the-difference-between-factoring-and-dividing-an-equation.htmlhttps://pop.com.sg/what-is-the-difference-between-factoring-and-dividing-an-equation.htmlNewbie question: Is pulling out factors and dividing the same? Please explain the difference in the examples below. Example: $$2x-4 \to 2(x-2)$$ Is it the same as $$4x^2 - 8x + 1 = 0$$ $$\fr......Sun, 30 Aug 2026 15:15:58 -0400https://pop.com.sg/solving-2-y-615-x-2-for-y.htmlhttps://pop.com.sg/solving-2-y-615-x-2-for-y.htmlHow can I transform the following equation: $$2^y = 615+x^2$$ to something of the form $$y = \cdots$$ SideQuest: What would the values of $x$ and $y$ that make the equality True, and how would you ......Sun, 30 Aug 2026 15:06:42 -0400https://pop.com.sg/proof-of-concavity-of-log-function.htmlhttps://pop.com.sg/proof-of-concavity-of-log-function.htmlDoes anybody have a proof of the concavity of the $\log{x}$ that does not use calculus?...Sun, 30 Aug 2026 15:06:27 -0400https://pop.com.sg/abc-is-a-triangle-such-that-bc-74-cm-and-point-d-is-on-bc-so-bd-14cm-i.htmlhttps://pop.com.sg/abc-is-a-triangle-such-that-bc-74-cm-and-point-d-is-on-bc-so-bd-14cm-i.htmlIn the figure below, $ABC$ is a triangle such that $\angle BAC = 150^\circ$ , $BC = 74 cm$ and point $D$ is on $BC$ such that $BD=14cm$.If $\angle ADB=60^\circ$,then what is the area, in $cm^2$, of ...Sun, 30 Aug 2026 15:04:05 -0400https://pop.com.sg/examples-with-not-periodic-functions.htmlhttps://pop.com.sg/examples-with-not-periodic-functions.htmlI'm looking for examples of non-periodic functions $f\in C^{\infty}$ satifying $|f^{(n)}(x)|\leq 1 \forall n\in \mathbb{N},\forall x\in\mathbb{R}$ I know only examples with periodic functions as......Sun, 30 Aug 2026 15:02:21 -0400https://pop.com.sg/is-zero-even-or-odd-duplicate.htmlhttps://pop.com.sg/is-zero-even-or-odd-duplicate.htmlThis question is already answered on here. head over to that link. I have been sitting here wondering if zero is even or odd and if it is even why is it even. I have no justification for why it is......Sun, 30 Aug 2026 15:00:27 -0400https://pop.com.sg/gram-schmidt-process-on-a-polynomial.htmlhttps://pop.com.sg/gram-schmidt-process-on-a-polynomial.htmlI'm currently learning Linear Algebra and I was asked to calculate the orthogonal projection of the vector $x^3$ on a subspace of $R_5[x]$ - $U = Sp\{1, x, x^2\}$ with the integral dot product in $......Sun, 30 Aug 2026 14:53:18 -0400https://pop.com.sg/calculus-related-rates-rectangle-area-problem-closed.htmlhttps://pop.com.sg/calculus-related-rates-rectangle-area-problem-closed.htmlQuestion: The length of a rectangle is increasing at a rate of $4$ inches per second while its width is decreasing at a rate of $3$ inches per second. At what rate, in square inches per second, is ......Sun, 30 Aug 2026 14:36:56 -0400https://pop.com.sg/natural-deduction-help-solving.htmlhttps://pop.com.sg/natural-deduction-help-solving.htmlI have this issue in natural deduction: B -> (A & ~A) ├ ~B I am having a really hard time understanding the rules in natural deduction - but this is what I got so far - can someone help me t......Sun, 30 Aug 2026 14:35:33 -0400https://pop.com.sg/evaluating-the-product-of-three-kronecker-delta.htmlhttps://pop.com.sg/evaluating-the-product-of-three-kronecker-delta.htmlI am trying to evaluate the following product of these 3 Kronecker delta: $\delta_{ij}\delta_{jk}\delta_{ki}$ I am not sure how to proceed. I understand that the Kronecker delta acts as a ...Sun, 30 Aug 2026 14:31:43 -0400https://pop.com.sg/the-least-upper-bound-and-the-greatest-lower-bound.htmlhttps://pop.com.sg/the-least-upper-bound-and-the-greatest-lower-bound.htmlI am taking math class, and I am not sure about LUB and GLB. I need someone to give this dummy a short explanation about them.... On interval (0,10), 0 is a lower bound and 10 is upper bound, but ......Sun, 30 Aug 2026 14:21:23 -0400https://pop.com.sg/trapezoidal-rule-for-system-of-ode.htmlhttps://pop.com.sg/trapezoidal-rule-for-system-of-ode.htmlI studied the trapezoidal rule for first-order differential equations. If the problem is $$y' = f(t, y)$$ the rule is given by the recursive formula $$y_{n + 1} = y_n + \frac12h(f(t_n, y_n) + f(t_{......Sun, 30 Aug 2026 14:20:59 -0400https://pop.com.sg/subgroups-of-s-n.htmlhttps://pop.com.sg/subgroups-of-s-n.htmlI am pretty confused right now with this. There should be a subgroup of $S_n$ with order x. So for example, for which n does $S_n$ contain a subgroup of order 60? What about if we are looking for a ...Sun, 30 Aug 2026 14:03:57 -0400https://pop.com.sg/why-are-natural-numbers-countable.htmlhttps://pop.com.sg/why-are-natural-numbers-countable.htmlI get how Cantor's diagonalization argument works for real numbers, but I don't see why you can't apply the same logic to natural numbers. I was reading this thread, but the explanations weren't ...Sun, 30 Aug 2026 14:02:46 -0400https://pop.com.sg/how-different-is-beta-computation-using-covariance-and-linear-regressi.htmlhttps://pop.com.sg/how-different-is-beta-computation-using-covariance-and-linear-regressi.htmlI wanted to compute Beta for a Stock against an Index (Say Stock X against S&P 500). I computed the daily returns for over one year applied the following logic : Beta = COVAR(X, S&P 500......Sun, 30 Aug 2026 13:47:41 -0400https://pop.com.sg/how-to-convert-radicals-to-decimals-without-a-calculator.htmlhttps://pop.com.sg/how-to-convert-radicals-to-decimals-without-a-calculator.htmlHow can one convert radicals to decimals(approximate value) when the number is not perfect such as $\sqrt2$, $\sqrt3$, $\sqrt5$, etc. Without the use of calculators....Sun, 30 Aug 2026 13:47:26 -0400https://pop.com.sg/proving-sequence-convergence.htmlhttps://pop.com.sg/proving-sequence-convergence.htmlI'm pretty confused...I understand bits and pieces but not how it all comes together...I would appreciate some help, either a written out example (you can make up one) and/or comments on how to fix......Sun, 30 Aug 2026 13:38:54 -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 -0400