Hilbert Book Model Project/Quaternionic Field Equations/Nabla Operators

From Wikiversity
Jump to navigation Jump to search

HansVanLeunen respectfully asks that people use the discussion page, their talk page or email them, rather than contribute to this page at this time. This page might be under construction, controversial, used currently as part of a brick and mortar class by a teacher, or document an ongoing research project. Please RESPECT their wishes by not editing this page.

Nabla playground[edit]

The spatial nabla and the quaternionic nabla are particular operators that play an important role in the partial differential equations that control the behavior of fields in the Hilbert Book Model.

Here we treat three kinds of nabla operators.

  1. spatial nabla
  2. quaternionic nabla
  3. Dirac nabla

The Dirac plays a role in the interpretation of the Dirac equation.

Properties of the spatial nabla operator[edit]

The nabla product is not necessarily associative. ThusString Module Error: function rep expects a number as second parameter, received "

"

 

 

 

 

(1)

String Module Error: function rep expects a number as second parameter, received "

"

Nabla in different coordinate systems[edit]

The spatial nabla exists in several coordinate systems. This section shows the representation of the quaternionic nabla for Cartesian coordinate systems and for polar coordinate systems.String Module Error: function rep expects a number as second parameter, received "

"

 

 

 

 

(2)

String Module Error: function rep expects a number as second parameter, received "

"String Module Error: function rep expects a number as second parameter, received "

"

 

 

 

 

(3)

String Module Error: function rep expects a number as second parameter, received "

" Here are the coordinates with as coordinate axes.String Module Error: function rep expects a number as second parameter, received "

"

 

 

 

 

(4)

String Module Error: function rep expects a number as second parameter, received "

"String Module Error: function rep expects a number as second parameter, received "

"

 

 

 

 

(5)

String Module Error: function rep expects a number as second parameter, received "

"String Module Error: function rep expects a number as second parameter, received "

"

 

 

 

 

(6)

String Module Error: function rep expects a number as second parameter, received "

"String Module Error: function rep expects a number as second parameter, received "

"

 

 

 

 

(7)

String Module Error: function rep expects a number as second parameter, received "

"String Module Error: function rep expects a number as second parameter, received "

"

 

 

 

 

(8)

String Module Error: function rep expects a number as second parameter, received "

"

Special formulas[edit]

The spatial nabla operator shows behavior that is valid for all quaternionic functions for which the first order partial differential equation exists.

Here the quaternionic field obeys the requirement that the first order partial differential exists.String Module Error: function rep expects a number as second parameter, received "

"

 

 

 

 

(9)

String Module Error: function rep expects a number as second parameter, received "

"String Module Error: function rep expects a number as second parameter, received "

"

 

 

 

 

(10)

String Module Error: function rep expects a number as second parameter, received "

"String Module Error: function rep expects a number as second parameter, received "

"

 

 

 

 

(11)

String Module Error: function rep expects a number as second parameter, received "

"String Module Error: function rep expects a number as second parameter, received "

"

 

 

 

 

(12)

String Module Error: function rep expects a number as second parameter, received "

"String Module Error: function rep expects a number as second parameter, received "

"

 

 

 

 

(13)

String Module Error: function rep expects a number as second parameter, received "

"String Module Error: function rep expects a number as second parameter, received "

"

 

 

 

 

(14)

String Module Error: function rep expects a number as second parameter, received "

"String Module Error: function rep expects a number as second parameter, received "

"

 

 

 

 

(15)

String Module Error: function rep expects a number as second parameter, received "

"String Module Error: function rep expects a number as second parameter, received "

"

 

 

 

 

(16)

String Module Error: function rep expects a number as second parameter, received "

" For constant and parameter holdsString Module Error: function rep expects a number as second parameter, received "

"

 

 

 

 

(17)

String Module Error: function rep expects a number as second parameter, received "

"String Module Error: function rep expects a number as second parameter, received "

"

 

 

 

 

(18)

String Module Error: function rep expects a number as second parameter, received "

"String Module Error: function rep expects a number as second parameter, received "

"

 

 

 

 

(19)

String Module Error: function rep expects a number as second parameter, received "

"String Module Error: function rep expects a number as second parameter, received "

"

 

 

 

 

(20)

String Module Error: function rep expects a number as second parameter, received "

"String Module Error: function rep expects a number as second parameter, received "

"

 

 

 

 

(21)

String Module Error: function rep expects a number as second parameter, received "

"String Module Error: function rep expects a number as second parameter, received "

"

 

 

 

 

(22)

String Module Error: function rep expects a number as second parameter, received "

"This indicates the relation between the Poisson equation and the Green's function.String Module Error: function rep expects a number as second parameter, received "

"

 

 

 

 

(23)

String Module Error: function rep expects a number as second parameter, received "

"

The term indicates the curvature of field .

The term indicates the stress of field

First order partial differential equation[edit]

String Module Error: function rep expects a number as second parameter, received "

"

 

 

 

 

(24)

String Module Error: function rep expects a number as second parameter, received "

"

The equation is a quaternionic first order partial differential equation.

The five terms on the right side show the components that constitute the full first-order change.

They represent subfields of field , and often they get special names and symbols.

Subfields[edit]

is the gradient of

is the divergence of .

is the curl of String Module Error: function rep expects a number as second parameter, received "

"

 

 

 

 

(25)

String Module Error: function rep expects a number as second parameter, received "

"

(Equation (25

) has no equivalent in Maxwell's equations!)String Module Error: function rep expects a number as second parameter, received "

"

 

 

 

 

(26)

String Module Error: function rep expects a number as second parameter, received "

"String Module Error: function rep expects a number as second parameter, received "

"

 

 

 

 

(27)

String Module Error: function rep expects a number as second parameter, received "

"String Module Error: function rep expects a number as second parameter, received "

"

 

 

 

 

(28)

String Module Error: function rep expects a number as second parameter, received "

"String Module Error: function rep expects a number as second parameter, received "

"

 

 

 

 

(29)

String Module Error: function rep expects a number as second parameter, received "

"

Derivation of higher order equations[edit]

With the help of the properties of the spatial nabla operator follows an interesting second-order partial differential equation.String Module Error: function rep expects a number as second parameter, received "

"

 

 

 

 

(30)

String Module Error: function rep expects a number as second parameter, received "

"String Module Error: function rep expects a number as second parameter, received "

"

 

 

 

 

(31)

String Module Error: function rep expects a number as second parameter, received "

"String Module Error: function rep expects a number as second parameter, received "

"

 

 

 

 

(32)

String Module Error: function rep expects a number as second parameter, received "

"String Module Error: function rep expects a number as second parameter, received "

"

 

 

 

 

(33)

String Module Error: function rep expects a number as second parameter, received "

"String Module Error: function rep expects a number as second parameter, received "

"

 

 

 

 

(34)

String Module Error: function rep expects a number as second parameter, received "

"String Module Error: function rep expects a number as second parameter, received "

"

 

 

 

 

(35)

String Module Error: function rep expects a number as second parameter, received "

"String Module Error: function rep expects a number as second parameter, received "

"

 

 

 

 

(36)

String Module Error: function rep expects a number as second parameter, received "

"String Module Error: function rep expects a number as second parameter, received "

"

 

 

 

 

(37)

String Module Error: function rep expects a number as second parameter, received "

"String Module Error: function rep expects a number as second parameter, received "

"

 

 

 

 

(38)

String Module Error: function rep expects a number as second parameter, received "

"String Module Error: function rep expects a number as second parameter, received "

"

 

 

 

 

(39)

String Module Error: function rep expects a number as second parameter, received "

" FurtherString Module Error: function rep expects a number as second parameter, received "

"

 

 

 

 

(40)

String Module Error: function rep expects a number as second parameter, received "

"

From the above formulas follows that the Maxwell equations do not form a complete set. Physicists use gauge equations to make Maxwell equations more complete.

Derivation of second order partial differential equation 1[edit]

String Module Error: function rep expects a number as second parameter, received "

"

 

 

 

 

(41)

String Module Error: function rep expects a number as second parameter, received "

"String Module Error: function rep expects a number as second parameter, received "

"

 

 

 

 

(42)

String Module Error: function rep expects a number as second parameter, received "

"String Module Error: function rep expects a number as second parameter, received "

"

 

 

 

 

(43)

String Module Error: function rep expects a number as second parameter, received "

"String Module Error: function rep expects a number as second parameter, received "

"

 

 

 

 

(44)

String Module Error: function rep expects a number as second parameter, received "

"String Module Error: function rep expects a number as second parameter, received "

"

 

 

 

 

(45)

String Module Error: function rep expects a number as second parameter, received "

"String Module Error: function rep expects a number as second parameter, received "

"

 

 

 

 

(46)

String Module Error: function rep expects a number as second parameter, received "

"String Module Error: function rep expects a number as second parameter, received "

"

 

 

 

 

(47)

String Module Error: function rep expects a number as second parameter, received "

"String Module Error: function rep expects a number as second parameter, received "

"

 

 

 

 

(48)

String Module Error: function rep expects a number as second parameter, received "

"String Module Error: function rep expects a number as second parameter, received "

"

 

 

 

 

(49)

String Module Error: function rep expects a number as second parameter, received "

"String Module Error: function rep expects a number as second parameter, received "

"

 

 

 

 

(50)

String Module Error: function rep expects a number as second parameter, received "

"String Module Error: function rep expects a number as second parameter, received "

"

 

 

 

 

(51)

String Module Error: function rep expects a number as second parameter, received "

"String Module Error: function rep expects a number as second parameter, received "

"

 

 

 

 

(52)

String Module Error: function rep expects a number as second parameter, received "

"String Module Error: function rep expects a number as second parameter, received "

"

 

 

 

 

(53)

String Module Error: function rep expects a number as second parameter, received "

" Most of the terms vanish.String Module Error: function rep expects a number as second parameter, received "

"

 

 

 

 

(54)

String Module Error: function rep expects a number as second parameter, received "

"

Derivation of second order partial differential equation 2[edit]

We add the complex imaginary base number to the spatial nabla operator .String Module Error: function rep expects a number as second parameter, received "

"

 

 

 

 

(55)

String Module Error: function rep expects a number as second parameter, received "

"String Module Error: function rep expects a number as second parameter, received "

"

 

 

 

 

(56)

String Module Error: function rep expects a number as second parameter, received "

"String Module Error: function rep expects a number as second parameter, received "

"

 

 

 

 

(57)

String Module Error: function rep expects a number as second parameter, received "

"String Module Error: function rep expects a number as second parameter, received "

"

 

 

 

 

(58)

String Module Error: function rep expects a number as second parameter, received "

"String Module Error: function rep expects a number as second parameter, received "

"

 

 

 

 

(59)

String Module Error: function rep expects a number as second parameter, received "

"String Module Error: function rep expects a number as second parameter, received "

"

 

 

 

 

(60)

String Module Error: function rep expects a number as second parameter, received "

"String Module Error: function rep expects a number as second parameter, received "

"

 

 

 

 

(61)

String Module Error: function rep expects a number as second parameter, received "

"String Module Error: function rep expects a number as second parameter, received "

"

 

 

 

 

(62)

String Module Error: function rep expects a number as second parameter, received "

"String Module Error: function rep expects a number as second parameter, received "

"

 

 

 

 

(63)

String Module Error: function rep expects a number as second parameter, received "

"String Module Error: function rep expects a number as second parameter, received "

"

 

 

 

 

(64)

String Module Error: function rep expects a number as second parameter, received "

"String Module Error: function rep expects a number as second parameter, received "

"

 

 

 

 

(65)

String Module Error: function rep expects a number as second parameter, received "

"String Module Error: function rep expects a number as second parameter, received "

"

 

 

 

 

(66)

String Module Error: function rep expects a number as second parameter, received "

"String Module Error: function rep expects a number as second parameter, received "

"

 

 

 

 

(67)

String Module Error: function rep expects a number as second parameter, received "

"String Module Error: function rep expects a number as second parameter, received "

"

 

 

 

 

(68)

String Module Error: function rep expects a number as second parameter, received "

"String Module Error: function rep expects a number as second parameter, received "

"

 

 

 

 

(69)

String Module Error: function rep expects a number as second parameter, received "

" Thus also this quaternionic second order partial differential equation splits in two first order partial differential equations. But these are no quaternionic partial differential equations!String Module Error: function rep expects a number as second parameter, received "

"

 

 

 

 

(70)

String Module Error: function rep expects a number as second parameter, received "

"