PolarDerivative {KEPTED}

Derivative of the polar coordinate transformation

Description

This function compute the Jacobian matrix of the polar transformation theta=g(v), i.e., the transformation from the the rectangular coordinate representation of the directional vector to its angular representation.

Usage

PolarDerivative(v)

Arguments

v

A d-dimensional directional vector of length 1.

Details

See Lemma 3 of Tang and Li (2024).

Value

The Jacobian matrix of the polar transformation theta=g(v), with d-1 rows and d columns.

References

Tang, Y. and Li, B. (2024), “A nonparametric test for elliptical distribution based on kernel embedding of probabilities,” https://arxiv.org/abs/2306.10594

Examples

X=c(3,1,3)
V=X/sqrt(sum(X^2))
PolarDerivative(V)