Archive of posts in Mar, 2020

• Mar 27, 2020

  The frequency assignment of musical notes

  This article explores the concept which I call the frequency assignment, which is a mapping from $N$ (the set of notes) to $\mathbb R^+$ (the set of frequencies). Concepts such as octaves, intervals, and equal temperaments are introduced.

  • Categories: music
  • Tags: music theory, number sequence, long paper, abstract algebra
• Mar 3, 2020

  Normal vectors of a scalar field

  This article gives the formula for the normal vectors of a surface defined by a scalar field on $\mathbb R^n$. The normal vector of the graph of the function $y=f\!\left(\mathbf x\right)$ at $\left(\mathbf x_0,f\!\left(\mathbf x_0\right)\right)$ is $\left(\nabla f\!\left(\mathbf x_0\right),-1\right)$. This also provides us a way to recover a scalar field from the normal vectors of its graph: normalizing the vectors so that the last component is $-1$, and then integrate the rest components.

  • Categories: math
  • Tags: calculus, vector analysis