lambdaway :: maxwell

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{uncover https://cdn.pixabay.com/photo/2020/11/02/15/20/fractal-5707021_1280.jpg 100 1000 Spirales de Cornu}

_h1 [[formulas]] | maxwell | [[maxwell_equations]] | [[chatgpt_maxwell]]

_h2 vector operators
_p Defining the vector operators {code divergence, laplacian and curl} :

{{CODE}
∇.V = {frac ∂Vx ∂x 10} + {frac ∂Vy ∂y 10} + {frac ∂Vz ∂z 10} 
∇{sup 2}.f = {frac ∂{sup 2}F ∂x{sup 2} 10} + {frac ∂{sup 2}F ∂y{sup 2} 10} + {frac ∂{sup 2}F ∂z{sup 2} 10}
∇^V = [ {frac ∂Vz ∂y 10} - {frac ∂Vy ∂z 10} , {frac ∂Vx ∂z 10} - {frac ∂Vz ∂x 10} , {frac ∂Vy ∂x 10} - {frac ∂Vx ∂y 10} ]
}

_h2 maxwell's equations (in free space)
_p The [[Maxwell's equations|https://fr.wikipedia.org/wiki/James_Clerk_Maxwell]] linking the electrical {b E} and magnetic {b H} fields can be written this way :
{{CODE}
∇.E = 0
∇.H = 0
∇^E + {frac 1 c 2} {frac ∂H ∂t 10} = 0
∇^H - {frac 1 c 2} {frac ∂E ∂t 10} = 0
}

_ul Equations 1) and 2) say : E and H are invariant. 
_ul Equation 3) says that a rotating electrical field creates a magnetic field orthogonal to the rotation. 
_ul Equation 4) says that a rotating magnetic field creates an electrical field orthogonal to the rotation, in opposite direction.

_h2 the wave equations

_p With these equations, Maxwell discovered the electromagnetic waves. Taking the curl (∇^) of the curl equations, and using the curl of the curl identity

{{CODE}
∇^(∇^X) = ∇(∇·X) − ∇{sup 2}X 
}

_p he discovered the wave equations

{{CODE}
{frac 1 c{sup 2} 4} {frac ∂{sup 2}E ∂t{sup 2} 11} - ∇{sup 2}E = 0
{frac 1 c{sup 2} 4} {frac ∂{sup 2}H ∂t{sup 2} 11} - ∇{sup 2}H = 0

or, with ψ = [E,H]

{b {frac ∂{sup 2}ψ ∂t{sup 2} 11} - c{sup 2}∇{sup 2}ψ = 0 }

or, with □ = ({frac ∂{sup 2} ∂t{sup 2} 11} - c{sup 2}∇{sup 2})

□ψ = 0

}

_p Funny isnt'it?

_h2 displaying maths

_p The [[mathML|../alphawiki_2/?view=stock_maths_mathML]] tags don't work in Chrome and so lambdatalk forgets them. This page is built on the basic lambdatalk operators, {code '{sub ...}} and {code '{sup ...}}, and this single customized user function:

{pre 
'{def frac
 {lambda {:a :b :w}
  {table 
   {@ style="display:inline-block; 
      vertical-align:middle; 
      width::wpx; 
      text-align:center;
      margin-right:{* 3 :w}px;"} 
   {tr {td {@ style="border:0;"}:a}}
   {tr {td {@ style="border:0; border-top:1px solid; line-height:0"}}}
   {tr {td {@ style="border:0;"}:b}}
}}}
-> {def frac
 {lambda {:a :b :w}
  {table 
   {@ style="display:inline-block; 
      vertical-align:middle; 
      width::wpx; 
      text-align:center;
      margin-right:{* 3 :w}px;
  "}
   {tr {td {@ style="border:0;"}:a}}
   {tr {td {@ style="border:0; border-top:1px solid; line-height:0"}}}
   {tr {td {@ style="border:0;"}:b}}
}}}
}

_h2 the dirac equation

_p Another one beautiful equation, the [[dirac equation|https://en.wikipedia.org/wiki/Dirac_equation]], for the fun:

{div {@ style="font:italic 1.2em georgia; text-align:center;"}
i{del h}{quotient 30 ∂ψ ∂t}(x,t) = 
{paren 3 (}mc{sup 2}α{sub 0} - i{del h}c {sigma 30 j=1 3} α{sub j}{quotient 30 ∂ ∂x{sub j}}{paren 3 )} ψ(x,t)
}

_h2 the Wheeler-DeWitt equation

_p The [[Wheeler-DeWitt equation|https://en.wikipedia.org/wiki/Wheeler%E2%80%93DeWitt_equation]] for theoretical physics and applied mathematics, is a field equation attributed to John Archibald Wheeler and Bryce DeWitt. The equation attempts to mathematically combine the ideas of quantum mechanics and general relativity, a step towards a theory of quantum gravity. 
{div {@ style="font:normal 2.2em georgia; text-align:center;"}
{i H}{sup {span {@ style="margin-left:-0.6em"} ˆ}}({i x})|{i ψ}⟩ = 0 
}

;; _img https://wikimedia.org/api/rest_v1/media/math/render/svg/089983ba10cf7abf03e9f91d1b75b70ca873c0ed

_p No, it's not pictures! See also [[coques minces|http://marty.alain.free.fr/recherche/articles/coques.pdf]].

{iframe {@ width="580" height="315" src="https://www.youtube.com/embed/MNPwfQDrK5Q?si=qTPs1qTf3ItLllKC" title="YouTube video player" frameborder="0" allow="accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share" referrerpolicy="strict-origin-when-cross-origin" allowfullscreen}}

{{hide}

{def CODE pre {@ style="text-align:center; font:normal 1.5em optima;"}}

{def quotient {lambda {:s :num :denom}
{table {@ style="width::spx; display:inline-block; vertical-align:middle; text-align:center;"}
 {tr {td {@ style="border:0 solid; border-bottom:1px solid;"}:num}}
 {tr {td {@ style="border:0 solid;"}:denom}}
}}}

{def sigma {lambda {:s :one :two}
{table {@ style="width::spx; display:inline-block; vertical-align:middle; text-align:center;"}
 {tr {td {@ style="border:0 solid;"}:two}}
 {tr {td {@ style="border:0 solid; font-size:2em; line-height:0.7em;"}Σ}}
 {tr {td {@ style="border:0 solid;"}:one}}
}}}

{def paren {lambda {:s :p} 
 {span {@ style="font:normal :sem arial; vertical-align:-0.15em;"}:p}}}

}