Difference between revisions of "Function Conjunction"

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(→‎The Anatomy of a Physical Expression)
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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_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><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>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><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{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{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><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><br>or<br><math>1</math>
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===Coefficients===
 
===Coefficients===
 +
* <math>\mu_r</math> = Relative Magnetic Permeability of Free Space
 +
* <math>\epsilon_r</math> = Relative Electric Permittivity of Free Space
  
 
===Quantities===
 
===Quantities===
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===Proximities===
 
===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===
 
===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>\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\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>
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* <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>\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\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===
 
===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 Composed of Physical Expressions==
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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:
 
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'}\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'},\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>
+
<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>

Revision as of 20:14, 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]
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}-\mathbf{r'}|}, \frac{1}{|\mathbf{r}-\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

  • [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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