Difference between revisions of "Function Conjunction"

From S.H.O.
Jump to: navigation, search
(→‎The Anatomy of a Physical Expression)
(→‎The Anatomy of a Physical Expression)
Line 1: Line 1:
 
==The Anatomy of a Physical Expression==
 
==The Anatomy of a Physical Expression==
  
 +
<div style="overflow-x: auto">
 
{| class="wikitable"
 
{| class="wikitable"
 
|-
 
|-
Line 15: Line 16:
 
! Direction
 
! Direction
 
|-
 
|-
−
|valign=top align=center| '''Examples:'''<br><math>\mu_0, \epsilon_0</math><br><math>k_B, \alpha, c</math>
+
|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>
+
|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>
+
|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>
+
|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>
+
|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>
−
|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>
+
|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>
 
|}
 
|}
 +
</div>
  
 
===Constants===
 
===Constants===
Line 29: Line 31:
 
* <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===
Line 37: Line 41:
 
* <math>m</math> = mass
 
* <math>m</math> = mass
 
* <math>\rho</math> = volume mass density
 
* <math>\rho</math> = volume mass density
 +
 +
===Proximities===
  
 
===Dislocations===
 
===Dislocations===
Line 45: Line 51:
 
* <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>
 +
 +
===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 [math]\times[/math] Coefficient [math]\times[/math] Quantity [math]\times[/math] Proximity [math]\times[/math] Dislocation [math]\times[/math] Direction
Examples:
[math]\mu_0, \epsilon_0[/math]
[math]k_B, \alpha, c[/math]
or
[math]1[/math] 		Examples:
[math]\mu_r, \epsilon_r[/math]
[math]N[/math]
or
[math]1[/math] 		Examples:
[math]q,\lambda_q,\sigma_q,\rho_q[/math]
[math]m,\rho[/math]
or
[math]1[/math] 		Examples:
[math]\frac{1}{|\mathbf{r}|}, \frac{1}{|\mathbf{r}|^2}[/math]
or
[math]1[/math] 		Examples:
[math]\mathbf{r}, \frac{d\mathbf{r}}{dt}, \frac{d^2\mathbf{r}}{dt^2}[/math]
[math]\mathbf{r'}, \frac{d\mathbf{r'}}{dt}, \frac{d^2\mathbf{r'}}{dt^2}[/math]
[math]\mathbf{x}, \mathbf{v}, \mathbf{a}, \beta[/math]
or
[math]1[/math] 		Examples:
[math]\mathbf{\hat{r}},\mathbf{\hat{\dot{r}}},\mathbf{\hat{\ddot{r}}}[/math]
[math]\mathbf{\hat{r'}},\mathbf{\hat{\dot{r'}}},\mathbf{\hat{\ddot{r'}}}[/math]
[math]\mathbf{\hat{x}}, \mathbf{\hat{v}}, \mathbf{\hat{a}}[/math]
or
[math]1[/math]

Constants

  • [math]\mu_0[/math] = Magnetic Permeability of Free Space
  • [math]\epsilon_0[/math] = Electric Permittivity of Free Space
  • [math]k_B[/math] = Boltzmann's constant
  • [math]\alpha[/math] = Fine Structure Constant
  • [math]c[/math] = Speed of Light

Coefficients

Quantities

  • [math]q[/math] = point charge
  • [math]\lambda_q[/math] = linear charge density (for continuous charge)
  • [math]\sigma_q[/math] = surface charge density (for continuous charge)
  • [math]\rho_q[/math] = volume charge density (for continuous charge)
  • [math]m[/math] = mass
  • [math]\rho[/math] = volume mass density

Proximities

Dislocations

  • [math]\mathbf{r}[/math] = position of a charge [math]q[/math] at time [math]t[/math], when it receives a light signal from [math]q'[/math] that was emitted earlier at time [math]t' = t - |\mathbf{r}-\mathbf{r'}|/c[/math]
  • [math]\frac{d\mathbf{r}}{dt}[/math] = velocity of a charge [math]q[/math] at time [math]t[/math], when it receives a light signal from [math]q'[/math] that was emitted earlier at time [math]t' = t - |\mathbf{r}-\mathbf{r'}|/c[/math]
  • [math]\frac{d^2\mathbf{r}}{dt^2}[/math] = acceleration of a charge [math]q[/math] at time [math]t[/math], when it receives a light signal from [math]q'[/math] that was emitted earlier at time [math]t' = t - |\mathbf{r}-\mathbf{r'}|/c[/math]
  • [math]\mathbf{r'}[/math] = position 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]

Directions

Functions Composed of Physical Expressions

Functions for a point charge [math]q'[/math]

The electric scalar potential [math]\mathbf{\varphi}[/math] at [math]\left(\mathbf{r},t\right)[/math] due to a point charge [math]q'[/math] at [math]\left(\mathbf{r'},t'\right)[/math] is:

[math]\mathbf{\varphi}\left(\mathbf{r},\mathbf{r'}\right) = \underset{constant}{\frac{q'}{4\pi\ \epsilon_0}} \times \underset{proximity}{\frac{1}{|\mathbf{r}-\mathbf{r'}|}}[/math]

The magnetic vector potential [math]A[/math] at [math]\left(\mathbf{r},t\right)[/math] due to a point charge [math]q'[/math] which had a velocity [math]\mathbf{v'}[/math] at [math]\left(\mathbf{r'},t'\right)[/math] is:

[math]\mathbf{A}\left(\mathbf{r},\mathbf{r'},\mathbf{v}\right) = \underset{constant}{\frac{q'}{4\pi\ \epsilon_0}} \times \underset{proximity}{\frac{1}{|\mathbf{r}-\mathbf{r'}|}} \times \underset{dislocation}{\mathbf{v'}/c^2}[/math]

[math]\mathbf{A}\left(\mathbf{r},\mathbf{r'},\mathbf{v}\right) = \underset{constant}{\frac{\mu_0\ q'}{4\pi}} \times \underset{proximity}{\frac{1}{|\mathbf{r}-\mathbf{r'}|}} \times \underset{dislocation}{\mathbf{v'}}[/math]

Retrieved from "http://www.sho.wiki/index.php?title=Function_Conjunction&oldid=465"