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.
/physics #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.
/math #calculus #vector analysis