In mathematics, a skew gradient of a

harmonic function In mathematics, mathematical physics and the theory of stochastic processes, a harmonic function is a twice continuously differentiable function f: U \to \mathbb R, where is an open subset of that satisfies Laplace's equation, that is, ...

over a

simply connected domain In topology, a topological space is called simply connected (or 1-connected, or 1-simply connected) if it is path-connected and every path between two points can be continuously transformed (intuitively for embedded spaces, staying within the sp ...

with two real dimensions is a vector field that is everywhere

orthogonal In mathematics, orthogonality is the generalization of the geometric notion of '' perpendicularity''. By extension, orthogonality is also used to refer to the separation of specific features of a system. The term also has specialized meanings in ...

to the

gradient In vector calculus, the gradient of a scalar-valued differentiable function of several variables is the vector field (or vector-valued function) \nabla f whose value at a point p is the "direction and rate of fastest increase". If the gr ...

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 In the field of complex analysis in mathematics, the Cauchy–Riemann equations, named after Augustin Cauchy and Bernhard Riemann, consist of a system of two partial differential equations which, together with certain continuity and differen ...

. 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 ''x'', ''y''. A skew gradient is defined as: :

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

and from the

, it is derived that :

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

Properties

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

* Peter Olver
Introduction to Partial Differential Equations, ch. 7, p. 232
{{Refend Differential calculus Generalizations of the derivative Linear operators in calculus Vector calculus