Archive of posts with tag “vector analysis”

• Dec 22, 2023

  The duality between two plane trajectories related by a conformal map

  The conformal map $\fc wz$ transforms the trajectory with energy $-B$ in potential $\fc Uz\ceq A\v{\d w/\d z}^2$ into the trajectory with energy $-A$ in potential $\fc Vw\ceq B\v{\d z/\d w}^2$. I will prove this beautiful result and show some implications of it.

  • Categories: physics
  • Tags: complex, classical mechanics, canonical transformation, kepler problem, mathematical physics, vector analysis, long paper
• 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