Ocean of Math

In multivariable calculus courses, one usually first encounters the divergence as: $${\displaystyle \mathrm{\nabla }\cdot \overrightarrow{u}=\frac{\partial u_x}{\partial x}+\frac{\partial u_y}{\partial y}+\frac{\partial u_z}{\partial z}}$$ where $\overrightarrow{u}=[u_x,u_y,u_z]$ in cartesian coordinates. Then one learns the divergence theorem: $${\displaystyle {\int }_{\mathrm{\Omega }}\mathrm{\nabla }\cdot \overrightarrow{u}dx={\int }_{\partial \mathrm{\Omega }}\overrightarrow{u}\cdot \hat{n}dS}$$ and then, since cylindrical coordinates have usually already been introduced, one … Read More “Divergence in Cylindrical Coordinates – The Right Way” »

The Arithmetic and Geometric Means You’re probably familiar with the arithmetic mean, which is most people mean (heh) when they say “average”: $${\displaystyle A(x_i)=\frac{1}{n}\sum _i{x}_i}$$ where $x_i$ is a set of $n$ real numbers. The arithmetic mean answers the question “If all these numbers were equal, what would they have to be … Read More “Means and Concavity” »

A cyclic quadrilateral is a four-sided polygon whose vertices all lie on a circle, like this one: The center of this circle (the “circumscribed circle” or “circumcircle”) is called the “circumcenter”, marked by a dot in the above figure. Quadrilaterals have many special points that get called centers, such as the circumcenter, but the most … Read More “When is the centroid of a cyclic quadrilateral also its circumcenter?” »

Suppose you have a set $S$ of $N$ numbers: $${\displaystyle S=\{x_1,x_2,x_3,\cdots ,x_N\}}$$ How many different products can be formed by multiplying at most $F$ members of $S$, allowing repetitions? For example, take $S=\{a,b,c,d\}$. Then the distinct products are: Number of factors Products 1 $${\displaystyle a,b,c,d}$$ … Read More “Counting Products When Factors Count” »