Difference between revisions of "Function Conjunction"

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==The Anatomy of a Physical Expression==
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{{#seo:
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|title=Function Conjunction @ Sho Drives Wiki
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|keywords=electricity,magnetism,motor,generator
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|description=To facilitate understanding of the S.H.O. Drive, this place catalogs the ideas, creations, and techniques underlying its invention.
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<div style="overflow-x: auto">
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To facilitate understanding of the S.H.O. Drive, this '''Function Conjunction''' will catalog the ideas, creations, and techniques underlying its invention.
{| class="wikitable"
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|-
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! Constant
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!rowspan=2|<math>\times</math>
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! Coefficient
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!rowspan=2|<math>\times</math>
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! Quantity
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!rowspan=2|<math>\times</math>
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! Proximity
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!rowspan=2|<math>\times</math>
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! Dislocation
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!rowspan=2|<math>\times</math>
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! Direction
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|-
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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>
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|valign=top align=center| '''Examples:'''<br><math>\mu_r, \epsilon_r</math><br>or<br><math>1</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>
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|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>
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|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>
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|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>
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|}
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</div>
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===Constants===
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'''[[Magnetic Energy]]'''
* <math>\mu_0</math> = Magnetic Permeability of Free Space
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* <math>\epsilon_0</math> = Electric Permittivity of Free Space
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* <math>k_B</math> = Boltzmann's constant
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* <math>\alpha</math> = Fine Structure Constant
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* <math>c</math> = Speed of Light
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===Coefficients===
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: The Magnetic Energy is the energy existing in magnetic fields.
* <math>\mu_r</math> = Relative Magnetic Permeability of Free Space
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* <math>\epsilon_r</math> = Relative Electric Permittivity of Free Space
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===Quantities===
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'''[[The Anatomy of a Physical Expression]]'''
* <math>q</math> = point charge
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* <math>\lambda_q</math> = linear charge density (for continuous charge)
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* <math>\sigma_q</math> = surface charge density (for continuous charge)
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* <math>\rho_q</math> = volume charge density (for continuous charge)
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* <math>m</math> = mass
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* <math>\rho</math> = volume mass density
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===Proximities===
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: A physical expression is a product of factors, each with their own distinct role in defining a property of a physical system. Types include constants, coefficients, quantities, proximities, dislocations, and directions.
* <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>
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* <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>
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===Dislocations===
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'''[[Functions composed of Physical Expressions]]'''
* <math>\mathbf{\hat{x}}</math> = position
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* <math>\mathbf{\hat{v}}</math> = velocity
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* <math>\mathbf{\hat{a}}</math> = acceleration
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* <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>
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* <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>\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>
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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>
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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>
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* <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===
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: A function composed of physical expressions is simply the result of the summations, differences, exponentiations, logarithms, or distributed multiplications or divisions of these physical expressions, or in the simplest case, a function is simply equal to an expression, such as <math>E(m) = mc^2</math>, where <math>E</math> is a function of <math>m</math>.
* <math>\mathbf{\hat{x}}</math> = position unit vector
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* <math>\mathbf{\hat{v}}</math> = velocity unit vector
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* <math>\mathbf{\hat{a}}</math> = acceleration unit vector
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* <math>\mathbf{\hat{r}}</math> = position unit vector of <math>q</math>
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* <math>\mathbf{\hat{\dot{r}}}</math> = velocity unit vector of <math>q</math>
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* <math>\mathbf{\hat{\ddot{r}}}</math> = acceleration unit vector of <math>q</math>
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* <math>\mathbf{\hat{r'}}</math> = position unit vector of <math>q'</math>
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* <math>\mathbf{\hat{\dot{r'}}}</math> = velocity unit vector of <math>q'</math>
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* <math>\mathbf{\hat{\ddot{r'}}}</math> = acceleration unit vector of <math>q'</math>
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==Functions Composed of Physical Expressions==
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'''[[Electromagnetic Potentials]]'''
  
===Functions for a point charge <math>q'</math>===
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: The basic idea here is that the electromagnetic potentials <math>\phi</math> and <math>A</math> and their derivatives can be used to derive all electromagnetism.
  
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:
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{{Site map}}
  
<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>
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[[Category:Function Conjunction| ]]
 
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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:
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<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>
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<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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{{Site map}}
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Latest revision as of 19:22, 14 July 2017

To facilitate understanding of the S.H.O. Drive, this Function Conjunction will catalog the ideas, creations, and techniques underlying its invention.

Magnetic Energy

The Magnetic Energy is the energy existing in magnetic fields.

The Anatomy of a Physical Expression

A physical expression is a product of factors, each with their own distinct role in defining a property of a physical system. Types include constants, coefficients, quantities, proximities, dislocations, and directions.

Functions composed of Physical Expressions

A function composed of physical expressions is simply the result of the summations, differences, exponentiations, logarithms, or distributed multiplications or divisions of these physical expressions, or in the simplest case, a function is simply equal to an expression, such as [math]E(m) = mc^2[/math], where [math]E[/math] is a function of [math]m[/math].

Electromagnetic Potentials

The basic idea here is that the electromagnetic potentials [math]\phi[/math] and [math]A[/math] and their derivatives can be used to derive all electromagnetism.

Site map

HQGlossaryApril 2016 Presentation