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
f
(
z
(
x
,
y
)
)
=
u
(
x
,
y
)
+
i
v
(
x
,
y
)
{\displaystyle f(z(x,y))=u(x,y)+iv(x,y)}
u ,v are real-valued scalar functions of the real variables x , y .
A skew gradient is defined as:
∇
⊥
u
(
x
,
y
)
=
∇
v
(
x
,
y
)
{\displaystyle \nabla ^{\perp }u(x,y)=\nabla v(x,y)}
and from the Cauchy–Riemann equations , it is derived that
∇
⊥
u
(
x
,
y
)
=
(
−
∂
u
∂
y
,
∂
u
∂
x
)
{\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:
∇
u
(
x
,
y
)
⋅
∇
⊥
u
(
x
,
y
)
=
0
,
‖
∇
u
‖
=
‖
∇
⊥
u
‖
{\displaystyle \nabla u(x,y)\cdot \nabla ^{\perp }u(x,y)=0,\rVert \nabla u\rVert =\rVert \nabla ^{\perp }u\rVert }
References
Categories :
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
**DISCLAIMER** We are not affiliated with Wikipedia, and Cloudflare.
The information presented on this site is for general informational purposes only and does not constitute medical advice.
You should always have a personal consultation with a healthcare professional before making changes to your diet, medication, or exercise routine.
AI helps with the correspondence in our chat.
We participate in an affiliate program. If you buy something through a link, we may earn a commission 💕
↑
be a complex-valued analytic function, where u,v are real-valued scalar functions of the real variables x, y.
