Divergence Theorem to evaluate integral where S is a sphere
Use the Divergence Theorem to evaluate $$\iint_S x^2y^2+y^2z^2+z^2x^2 dS$$ where $S$ is the surface of the sphere $x^2+y^2+z^2=1$. So the divergence of F ...
20 Apr 2026 1 READS
Archive: General
Rules for Product and Summation Notation
When we deal with summation notation, there are some useful computational shortcuts, e.g.: $$\sum\limits_{i=1}^{n} (2 + 3i) = \sum\limits_{i=1}^{n} 2 + \s...
20 Apr 2026 1 READS
Proof that aleph null is the smallest transfinite number?
The wikipedia page on the cardinal numbers says that $\aleph_0$, the cardinality of the set of natural numbers, is the smallest transfinite number. It doe...
20 Apr 2026 1 READS
Why is the outer unit normal unique for $C^k$ boundaries
Let $U$ be a bounded open set with $C^k$-boundary, i.e. there is a $C^k$-function $f:\mathbb{R}^n\rightarrow\mathbb{R}$ s.t. $$U=\{x\in\mathbb{R}^n:f(x)&g...
20 Apr 2026 1 READS
The exact value of $\cot\frac{7\pi}{6}$?
I am working on a trigonometry question at the moment and am very stuck. I have looked through all the tips to solving it and I cant seem to come up with ...
20 Apr 2026 1 READS
Unit step function differential equation
Suppose we model a physical phenomenon with a 2nd order linear differential equation: $a_2$(t)$y''$+$a_1$(t)$y'$+$a_0$(t)$y$=$f(t)$, where ...
20 Apr 2026 1 READS
quotient of complex numbers?
so I was wondering if you have two different equations having denominators $2+i$ and $2-i$ respectively how came the denominator of the quotient in standa...
20 Apr 2026 1 READS
How do I correctly measure the circumference of a circle
I found How exactly do you measure circumference or diameter? but it was more related to how people measured circumference and diameter in old days. BUT n...
20 Apr 2026 1 READS
Parametrization of a unit 2 sphere
Here is the parametrization for a unit 2 sphere locating at the center of a Euclidean 3 dimensional space: $$x=x(u,v)= \cos u\sin v,\ \ y=y(u,v)=\sin u\si...
20 Apr 2026 1 READS
Prove one standard deviation lies on inflection points
Is my conjecture correct that one standard deviation lies on the inflection points of the normal distribution curve (of the probability density function)?...
20 Apr 2026 1 READS
