Skew gradient

In mathematics, a skew gradient of a harmonic function over a simply connected domain with two real dimensions is a vector field that is everywhere orthogonal to the gradient of the function and that has the same magnitude as the gradient.

Definition[]

The skew gradient can be defined using complex analysis and the Cauchy–Riemann equations.

Let ${\displaystyle f(z(x,y))=u(x,y)+iv(x,y)}$ be a complex-valued analytic function, where u,v are real-valued scalar functions of the real variables x, y.

A skew gradient is defined as:

${\displaystyle \nabla ^{\perp }u(x,y)=\nabla v(x,y)}$

and from the Cauchy–Riemann equations, it is derived that

${\displaystyle \nabla ^{\perp }u(x,y)=(-{\frac {\partial u}{\partial y}},{\frac {\partial u}{\partial x}})}$

Properties[]

The skew gradient has two interesting properties. It is everywhere orthogonal to the gradient of u, and of the same length:

${\displaystyle \nabla u(x,y)\cdot \nabla ^{\perp }u(x,y)=0,\rVert \nabla u\rVert =\rVert \nabla ^{\perp }u\rVert }$

References[]