View source for Function Conjunction

==The Anatomy of a Physical Expression==

<div style="overflow-x: auto">
{| class="wikitable"
|-
! Constant
!rowspan=2|<math>\times</math>
! Coefficient
!rowspan=2|<math>\times</math>
! Quantity
!rowspan=2|<math>\times</math>
! Proximity
!rowspan=2|<math>\times</math>
! Dislocation
!rowspan=2|<math>\times</math>
! Direction
|-
|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>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><br>or<br><math>1</math>
|valign=top align=center| '''Examples:'''<br><math>\frac{1}{|\mathbf{r}-\mathbf{r'}|}, \frac{1}{|\mathbf{r}-\mathbf{r'}|^2}</math><br>or<br><math>1</math>
|valign=top align=center| '''Examples:'''<br><math>\mathbf{x}, \mathbf{v}, \mathbf{a}</math><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>or<br><math>1</math>
|valign=top align=center| '''Examples:'''<br><math>\mathbf{\hat{x}}, \mathbf{\hat{v}}, \mathbf{\hat{a}}</math><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>or<br><math>1</math>
|}
</div>

===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===
* <math>\mu_r</math> = Relative Magnetic Permeability of Free Space
* <math>\epsilon_r</math> = Relative Electric Permittivity of Free Space

===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===
* <math>\frac{1}{|\mathbf{r}-\mathbf{r'}|}</math> = inverse of the magnitude of the separation between positions <math>\mathbf{r}</math> and <math>\mathbf{r'}</math>
* <math>\frac{1}{|\mathbf{r}-\mathbf{r'}|^2}</math> = inverse square of the magnitude of the separation between positions <math>\mathbf{r}</math> and <math>\mathbf{r'}</math>

===Dislocations===
* <math>\mathbf{\hat{x}}</math> = position
* <math>\mathbf{\hat{v}}</math> = velocity
* <math>\mathbf{\hat{a}}</math> = acceleration
* <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===
* <math>\mathbf{\hat{x}}</math> = position unit vector
* <math>\mathbf{\hat{v}}</math> = velocity unit vector
* <math>\mathbf{\hat{a}}</math> = acceleration unit vector
* <math>\mathbf{\hat{r}}</math> = position unit vector of <math>q</math>
* <math>\mathbf{\hat{\dot{r}}}</math> = velocity unit vector of <math>q</math>
* <math>\mathbf{\hat{\ddot{r}}}</math> = acceleration unit vector of <math>q</math>
* <math>\mathbf{\hat{r'}}</math> = position unit vector of <math>q'</math>
* <math>\mathbf{\hat{\dot{r'}}}</math> = velocity unit vector of <math>q'</math>
* <math>\mathbf{\hat{\ddot{r'}}}</math> = acceleration unit vector of <math>q'</math>

==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'}\right) = \mathbf{\varphi}\left(\mathbf{r},\mathbf{r'}\right) \times \underset{constant}{\frac{1}{c^2}} \times \underset{dislocation}{\frac{d\mathbf{r'}}{dt}}</math>

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

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