Does-anyone-know-how-this-works-I-have-d-x-2-cy-2-dy-And-my-physics-teacher-says-it-is-or-can-be-a-harmonic-function-0-Can-anyone-explain-

Question Number 206273 by EmGent last updated on 10/Apr/24

Does anyone know how this works ?

I have d ψ = (x^{2} - {c y}^{2}) d y

And my physics teacher says it is (or can

be) a harmonic function (Δ ψ = 0)

Can anyone explain ?

Answered by aleks041103 last updated on 11/Apr/24

S i n c e \exists d ψ, t h e n

d ψ = \frac{\partial ψ}{\partial x} d x + \frac{\partial ψ}{\partial y} d y = (x^{2} - {c y}^{2}) d y

\Rightarrow {\begin{cases} \partial_{x} ψ = 0 \\ \partial_{y} ψ = x^{2} - {c y}^{2} \end{cases}

i t i s o b v i o u s t h a t \partial_{x} ψ, \partial_{y} ψ a r e c o n t i n u o u s .

\Rightarrow ψ \in C^{1}

A l s o i t i s e a s i l y s e e n t h a t \exists \partial_{x}^{2} ψ, \partial_{x} \partial_{y} ψ, \partial_{y} \partial_{x} ψ, \partial_{y}^{2} ψ .

S o, ψ i s t w i c e d i f f e r e n t i a b l e a n d i s o n c e

c o n t i n o u s l y d i f f e r e n t i a b l e .

T h e r e f o r e S c h w a r t z r u l e a p p l i e s :

\frac{\partial^{2} ψ}{\partial x \partial y} = \frac{\partial^{2} ψ}{\partial y \partial x}

B U T

\frac{\partial^{2} ψ}{\partial x \partial y} = \frac{\partial}{\partial x} (\frac{\partial ψ}{\partial y}) = \frac{\partial}{\partial x} (x^{2} - {c y}^{2}) = 2 x

\frac{\partial^{2} ψ}{\partial y \partial x} = \frac{\partial}{\partial y} (\frac{\partial ψ}{\partial x}) = \frac{\partial}{\partial y} (0) = 0

B U T 2 x \neq 0

\Rightarrow ∄ ψ : d ψ = (x^{2} - {c y}^{2}) d y

T h e r e f o r e, s u c h ψ c a n n o t b e h a r m o n i c,

s i n c e s u c h ψ {d o e s n}^{'} t e x i s t .