Skew gradient

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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[edit]

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

Let  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 xy.

A skew gradient is defined as:

\nabla^\perp u(x,y)=\nabla v(x,y)

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

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

Properties[edit]

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

\nabla u(x,y) \cdot \nabla^\perp u(x,y)=0 ,  \rVert \nabla u\rVert =\rVert \nabla^\perp u\rVert

References[edit]