Difference between revisions of "Function Conjunction"

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(→‎The Anatomy of a Physical Expression)
(→‎The Anatomy of a Physical Expression)
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==The Anatomy of a Physical Expression==
 
==The Anatomy of a Physical Expression==
  
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<div style="overflow-x: auto">
 
{| class="wikitable"
 
{| class="wikitable"
 
|-
 
|-
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! Direction
 
! Direction
 
|-
 
|-
−
|valign=top align=center| '''Examples:'''<br><math>\mu_0, \epsilon_0</math><br><math>k_B, \alpha, c</math>
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|valign=top align=center| '''Examples:'''<br><math>\mu_0, \epsilon_0</math><br><math>k_B, \alpha, c</math><br>or<br><math>1</math>
−
|valign=top align=center| '''Examples:'''<br><math>\mu_r, \epsilon_r</math><br><math>N</math>
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|valign=top align=center| '''Examples:'''<br><math>\mu_r, \epsilon_r</math><br><math>N</math><br>or<br><math>1</math>
−
|valign=top align=center| '''Examples:'''<br><math>q,\lambda_q,\sigma_q,\rho_q</math><br><math>m,\rho</math>
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|valign=top align=center| '''Examples:'''<br><math>q,\lambda_q,\sigma_q,\rho_q</math><br><math>m,\rho</math><br>or<br><math>1</math>
−
|valign=top align=center| '''Examples:'''<br><math>\frac{1}{|\mathbf{r}|}, \frac{1}{|\mathbf{r}|^2}</math>
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|valign=top align=center| '''Examples:'''<br><math>\frac{1}{|\mathbf{r}|}, \frac{1}{|\mathbf{r}|^2}</math><br>or<br><math>1</math>
−
|valign=top align=center| '''Examples:'''<br><math>\mathbf{r}, \frac{d\mathbf{r}}{dt}, \frac{d^2\mathbf{r}}{dt^2}</math><br><math>\mathbf{r'}, \frac{d\mathbf{r'}}{dt}, \frac{d^2\mathbf{r'}}{dt^2}</math><br><math>\mathbf{x}, \mathbf{v}, \mathbf{a}, \beta</math>
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|valign=top align=center| '''Examples:'''<br><math>\mathbf{r}, \frac{d\mathbf{r}}{dt}, \frac{d^2\mathbf{r}}{dt^2}</math><br><math>\mathbf{r'}, \frac{d\mathbf{r'}}{dt}, \frac{d^2\mathbf{r'}}{dt^2}</math><br><math>\mathbf{x}, \mathbf{v}, \mathbf{a}, \beta</math><br>or<br><math>1</math>
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|valign=top align=center| '''Examples:'''<br><math>\mathbf{\hat{r}},\mathbf{\hat{\dot{r}}},\mathbf{\hat{\ddot{r}}}</math><br><math>\mathbf{\hat{r'}},\mathbf{\hat{\dot{r'}}},\mathbf{\hat{\ddot{r'}}}</math><br><math>\mathbf{\hat{x}}, \mathbf{\hat{v}}, \mathbf{\hat{a}}</math>
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|valign=top align=center| '''Examples:'''<br><math>\mathbf{\hat{r}},\mathbf{\hat{\dot{r}}},\mathbf{\hat{\ddot{r}}}</math><br><math>\mathbf{\hat{r'}},\mathbf{\hat{\dot{r'}}},\mathbf{\hat{\ddot{r'}}}</math><br><math>\mathbf{\hat{x}}, \mathbf{\hat{v}}, \mathbf{\hat{a}}</math><br>or<br><math>1</math>
 
|}
 
|}
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</div>
  
 
===Constants===
 
===Constants===
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* <math>\alpha</math> = Fine Structure Constant
 
* <math>\alpha</math> = Fine Structure Constant
 
* <math>c</math> = Speed of Light
 
* <math>c</math> = Speed of Light
 +
 +
===Coefficients===
  
 
===Quantities===
 
===Quantities===
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* <math>m</math> = mass
 
* <math>m</math> = mass
 
* <math>\rho</math> = volume mass density
 
* <math>\rho</math> = volume mass density
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 +
===Proximities===
  
 
===Dislocations===
 
===Dislocations===
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* <math>\frac{d\mathbf{r'}}{dt}</math> = velocity a charge <math>q'</math> was at the retarded time <math>t' = t - |\mathbf{r}-\mathbf{r'}|/c</math>, when it emitted a light signal which has now reached <math>q</math> at position <math>\mathbf{r}</math> and time <math>t</math>
 
* <math>\frac{d\mathbf{r'}}{dt}</math> = velocity a charge <math>q'</math> was at the retarded time <math>t' = t - |\mathbf{r}-\mathbf{r'}|/c</math>, when it emitted a light signal which has now reached <math>q</math> at position <math>\mathbf{r}</math> and time <math>t</math>
 
* <math>\frac{d^2\mathbf{r'}}{dt^2}</math> = acceleration a charge <math>q'</math> was at the retarded time <math>t' = t - |\mathbf{r}-\mathbf{r'}|/c</math>, when it emitted a light signal which has now reached <math>q</math> at position <math>\mathbf{r}</math> and time <math>t</math>
 
* <math>\frac{d^2\mathbf{r'}}{dt^2}</math> = acceleration a charge <math>q'</math> was at the retarded time <math>t' = t - |\mathbf{r}-\mathbf{r'}|/c</math>, when it emitted a light signal which has now reached <math>q</math> at position <math>\mathbf{r}</math> and time <math>t</math>
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===Directions===
  
 
==Functions Composed of Physical Expressions==
 
==Functions Composed of Physical Expressions==

Revision as of 17:21, 23 April 2016

The Anatomy of a Physical Expression

Constant × Coefficient × Quantity × Proximity × Dislocation × Direction
Examples:
μ0,ϵ0
kB,α,c
or
1 		Examples:
μr,ϵr
N
or
1 		Examples:
q,λq,σq,ρq
m,ρ
or
1 		Examples:
1|r|,1|r|2
or
1 		Examples:
r,drdt,d2rdt2
r′,dr′dt,d2r′dt2
x,v,a,β
or
1 		Examples:
ˆr,ˆ˙r,ˆ¨r
^r′,^˙r′,^¨r′
ˆx,ˆv,ˆa
or
1

Constants

  • μ0 = Magnetic Permeability of Free Space
  • ϵ0 = Electric Permittivity of Free Space
  • kB = Boltzmann's constant
  • α = Fine Structure Constant
  • c = Speed of Light

Coefficients

Quantities

  • q = point charge
  • λq = linear charge density (for continuous charge)
  • σq = surface charge density (for continuous charge)
  • ρq = volume charge density (for continuous charge)
  • m = mass
  • ρ = volume mass density

Proximities

Dislocations

  • r = position of a charge q at time t, when it receives a light signal from q′ that was emitted earlier at time t′=t−|r−r′|/c
  • drdt = velocity of a charge q at time t, when it receives a light signal from q′ that was emitted earlier at time t′=t−|r−r′|/c
  • d2rdt2 = acceleration of a charge q at time t, when it receives a light signal from q′ that was emitted earlier at time t′=t−|r−r′|/c
  • r′ = position a charge q′ was at the retarded time t′=t−|r−r′|/c, when it emitted a light signal which has now reached q at position r and time t
  • dr′dt = velocity a charge q′ was at the retarded time t′=t−|r−r′|/c, when it emitted a light signal which has now reached q at position r and time t
  • d2r′dt2 = acceleration a charge q′ was at the retarded time t′=t−|r−r′|/c, when it emitted a light signal which has now reached q at position r and time t

Directions

Functions Composed of Physical Expressions

Functions for a point charge q′

The electric scalar potential φ at (r,t) due to a point charge q′ at (r′,t′) is:

φ(r,r′)=q′4π ϵ0constant×1|r−r′|proximity

The magnetic vector potential A at (r,t) due to a point charge q′ which had a velocity v′ at (r′,t′) is:

A(r,r′,v)=q′4π ϵ0constant×1|r−r′|proximity×v′/c2dislocation

A(r,r′,v)=μ0 q′4πconstant×1|r−r′|proximity×v′dislocation

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