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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

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

Let 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:

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

Properties

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

References

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