Difference between revisions of "Function Conjunction"

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|valign=top| '''Examples:'''<br><math>q,\lambda_q,\sigma_q,\rho_q</math><br><math>m,\rho</math>
 
|valign=top| '''Examples:'''<br><math>q,\lambda_q,\sigma_q,\rho_q</math><br><math>m,\rho</math>
 
|valign=top| '''Examples:'''<br><math>\frac{1}{|\mathbf{r}|}, \frac{1}{|\mathbf{r}|^2}</math>
 
|valign=top| '''Examples:'''<br><math>\frac{1}{|\mathbf{r}|}, \frac{1}{|\mathbf{r}|^2}</math>
−
|valign=top| '''Examples:'''<br><math>\mathbf{r}, \frac{d^2\mathbf{r}}{dt^2}, \frac{d^3\mathbf{r}}{dt^3}</math><br><math>\mathbf{r'}, \frac{d^2\mathbf{r'}}{dt^2}, \frac{d^3\mathbf{r'}}{dt^3}</math><br><math>\mathbf{x}, \mathbf{v}, \mathbf{a}, \beta</math>
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|valign=top| '''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| '''Examples:'''<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| '''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>
 
|}
 
|}
 +
 +
===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
 +
 +
===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
 +
 +
===Dislocations===
 +
* <math>\mathbf{r'}</math> = position at which a light signal is emitted at the retarded time <math>t' = t - |\mathbf{r}-\mathbf{r'}|/c</math>
 +
* <math>\frac{d\mathbf{r'}}{dt}</math> = velocity of the source of the light signal at the retarded time <math>t' = t - |\mathbf{r}-\mathbf{r'}|/c</math>
 +
* <math>\frac{d^2\mathbf{r'}}{dt^2}</math> = acceleration of the source of the light signal at the retarded time <math>t' = t - |\mathbf{r}-\mathbf{r'}|/c</math>
  
 
==Functions Composed of Physical Expressions==
 
==Functions Composed of Physical Expressions==

Revision as of 12:06, 21 April 2016

The Anatomy of a Physical Expression

Constant Coefficient Quantity Proximity Dislocation Direction
Examples:
[math]\mu_0, \epsilon_0[/math]
[math]k_B, \alpha, c[/math] 		Examples:
[math]\mu_r, \epsilon_r[/math]
[math]N[/math] 		Examples:
[math]q,\lambda_q,\sigma_q,\rho_q[/math]
[math]m,\rho[/math] 		Examples:
[math]\frac{1}{|\mathbf{r}|}, \frac{1}{|\mathbf{r}|^2}[/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] 		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]

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

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

Dislocations

  • [math]\mathbf{r'}[/math] = position at which a light signal is emitted at the retarded time [math]t' = t - |\mathbf{r}-\mathbf{r'}|/c[/math]
  • [math]\frac{d\mathbf{r'}}{dt}[/math] = velocity of the source of the light signal at the retarded time [math]t' = t - |\mathbf{r}-\mathbf{r'}|/c[/math]
  • [math]\frac{d^2\mathbf{r'}}{dt^2}[/math] = acceleration of the source of the light signal at the retarded time [math]t' = t - |\mathbf{r}-\mathbf{r'}|/c[/math]

Functions Composed of Physical Expressions

Electric scalar potential [math]\mathbf{\varphi}[/math]

[math]\mathbf{\varphi}[/math] of a point charge [math]q[/math]:

[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]

Magnetic vector potential [math]A[/math]

[math]\mathbf{A}[/math] of a moving point charge [math]q[/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]

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