User:Thermochap/Sandbox

Did you know?

• ...that Kikuchi lines, formed in diffraction patterns by diffusely scattered electrons, are useful tools in electron microscopy of crystalline and nanocrystalline materials? (from Wikipedia's main page on 4/28/2008)

Editing tips

Note: The markup to create the following items will be visible in the editing window, if you click on [edit] in the upper right corner of this section.

Here's a sample reference that can be cited more than once. Multiple formats for automatically numbered footnotes are available, although only one of these formats is used below.

And then there is the question of inter-wikilinks. For example, w:Apple is the article on the (English) Wikipedia for Apple, while wikt:Apple would be the analogous one on Wiktionary. There is probably a list of prefixes, e.g. for wikiversity, wikibooks, wikicommons, etc.

Don't forget that you can insert fractions into text like this: 7⁄8. You can also put TeX math into text like this: ${\frac {7}{8}}$.

An example of TeX math in its own indented line is this equation from Miller index for the normal to the (hkl) plane of a hexagonal crystal in terms of both covariant reciprocal-lattice basis one-forms or co-vectors a*, b*, c*, and contravariant direct-lattice basis vectors a, b, c:

$(hkl)=h{\vec {a^{*}}}+k{\vec {b^{*}}}+l{\vec {c^{*}}}={\frac {2}{3a^{2}}}(2h+k){\vec {a}}+{\frac {2}{3a^{2}}}(h+2k){\vec {b}}+{\frac {1}{c^{2}}}(l){\vec {c}}$.

Two fun equations from scattering theory are the kinematic equation from reciprocal lattice for single-scattering of a coherent beam by a cluster of atoms,

$I[{\vec {g}}]=\sum _{j=1}^{N}\sum _{k=1}^{N}f_{j}[{\vec {g}}]f_{k}[{\vec {g}}]e^{2\pi i{\vec {g}}\cdot {\vec {r}}_{jk}}$,

and its angle-averaged sister from powder diffraction, the Deybe equation for scattering from a random collection of such clusters:

$I_{powder}[g]=\sum _{j=1}^{N}\sum _{k=1}^{N}f_{j}[g]f_{k}[g]{\frac {\sin[2\pi gr_{jk}]}{2\pi gr_{jk}}}$.

A pretty example from proper acceleration is this equation for relativistic coordinate acceleration in terms of proper (physical) acceleration A as well as the various geometric accelerations that arise from the coordinate system's affine connection:

${\frac {dU^{\lambda }}{d\tau }}=A^{\lambda }-\Gamma ^{\lambda }{}_{\mu \nu }U^{\mu }U^{\nu }$.

Experimenting with equations

columnar growth on a cylinder

How would you solve this equation for φ_local? Here φ_local is the local azimuth of a lattice fringe's tilt axis with respect to a nanotube on which it resides, ζ is the angle between that lattice plane's normal and a reference or growth plane, θ is the angle between the growth plane and the viewing direction, and φ_view is the azimuth of the projected fringe and the tube axis. If you have a closed-form solution, by all means post it in this section!

$\phi _{view}=\arctan \left[{\frac {\cos[\zeta ]\sin[\theta ]+\cos[\theta ]\sin[\zeta ]\sin[\phi _{local}]}{\cos[\phi _{local}]\sin[\zeta ]}}\right]$

It would also be nice to solve this equation for beam angle from face-on, set equal to π/2, for θ:

${\frac {\pi }{2}}=\arctan \left[{\frac {\sqrt {(\cos[\phi ]\sin[\zeta ])^{2}+(\cos[\zeta ]\sin[\theta ]+\cos[\theta ]\sin[\phi ]\sin[\zeta ])^{2}}}{\cos[\theta ]\cos[\zeta ]-\sin[\theta ]\sin[\phi ]\sin[\zeta ]}}\right]$.

Suggestions invited.

fringe visibility / bend contour rocking curves

Bend contour and lattice fringe visibility as a function of specimen thickness and beam tilt.

The expression for fringe visibility half-angle looks like:

$\alpha _{max}=\sin ^{-1}\left(d{\frac {\Gamma }{t}}+{\frac {\lambda }{2d}}\left\{1-\left(d{\frac {\Gamma }{t}}\right)^{2}\right\}\right)$.

In the expression above, d is the spacing of the lattice planes, t is the crystal thickness, λ is the wavelength of the electrons, and Γ is a “visibility factor” on the order of 1 that empirically accounts for the signal-to-noise ratio in the method used to detect fringes.

Replacing Γ/t with deviation parameter or excitation error s in the above equation, one can solve for deviation parameter in terms of off-edge angle α to get:

$s[\alpha ]=\left|{\frac {{\sqrt {1-2(\lambda /d)\sin {\alpha }+(\lambda /d)^{2}}}-1}{\lambda }}\right|$.

The Fourier shape transform of a spherical crystal in three dimensions, in terms of spatial frequency g and sphere diameter t, may be written:

$\Xi [g]={\frac {\sin {\pi gt}-\pi gt\cos {\pi gt}}{2\pi ^{2}g^{3}}}$.

Consider a lattice plane canted by angle α radians from the edge-on position along the electron-beam direction. Since all reciprocal lattice spots will be convolved with the shape transform, the intensity of Bragg scattering from the brightfield (unscattered beam) image can be estimated by adding amplitudes (in the coherent kinematic scattering case) from both sides of the lattice plane by adding Ξ[s] values for deviation s[α] evaluated at ±α.

some KL divergence stuff

measures of useful information

Surprisal: $s[p]=k\ln \left[{\frac {1}{p}}\right]\geq 0$ where $0\leq p\leq 1$.

When k=1/ln[2], surprisal units are bits and we can say that probability p = 2^sBits. If k is 1 then the units are nats, while if k is Boltzmann's constant 1.38×10^-23 then the units are Joules/Kelvin.

Evidence for a True-False Assertion: $e[p]=s[1-p]-s[p]$.

When k=1/ln[2], evidence units are bits and we can say that odds ratio = 1/2^eBits.

Entropy, uncertainty, or average surprisal: $S=\sum _{i=1}^{N}p_{i}s[p_{i}]\geq 0$ where $\sum _{i=1}^{N}p_{i}=1$.

Kullback-Leibler divergence or net surprisal of {p_o} from {p}: $I=\sum _{i=1}^{N}p_{i}\left(s[p_{oi}]-s[p_{i}]\right)\geq 0$.

• In ecology and related fields the KL divergence of "model from reality" is useful in ranking models against experimental data, with help from Akaike Information Criterion according to the residuals that the models do not account for.

• In communications theory, in clade analysis, and in quantum computing the KL divergence of "uncorrelated from correlated" measures the mutual information associated with fidelity, inheritance, and entanglement.

• In thermodynamics, the KL divergence of "ambient from actual" measures distance to equilibrium and (multiplied by ambient temperature) available work.

In this project, we show that the foregoing are special cases of KL divergence as a measure of "useful information". Each is, however, typically applied on only one level of organization at a time. In addition to offering some new and surprising applications, we also show how the most interesting future applications may be more explicit about their relationship to correlations on multiple levels.

A fun fact for physics students: All measurable values of "useful information" may have to be less than the mass of the observable universe times lightspeed squared over 2.715 Kelvin, or about 10⁹² bits.

ideal gas applications

For a monatomic ideal gas at fixed temperature can write the available work or free energy with respect to ambient as:

$W=\Delta G=NkT_{o}\Theta \left[{\frac {V}{V_{o}}}\right]$.

where Θ[x]≡x-1-lnx≥0 and

$V_{o}={\frac {NkT_{o}}{P_{o}}}$.

More generally, we can write the thermodynamic availability in information units as

$\Delta I={\frac {W}{T_{o}}}=Nk\left(\Theta \left[{\frac {V}{V_{o}}}\right]+{\frac {3}{2}}\Theta \left[{\frac {T}{T_{o}}}\right]\right)$.

layered niche-network applications

The niche-layer multiplicity for an individual metazaon might be defined in terms of the fractional resource allocation f_i to each of six layers (directed inward/outward from skin, gene-pool and meme-pool) via:

$k\ln M_{i}=\sum _{j=1}^{6}f_{ij}s[f_{ij}]$ where $\sum _{j=1}^{6}f_{ij}=1$ and $0\leq f_{ij}\leq 1$.

If we put the intermediate quantity into bits, we might also write this as:

$M_{i}=2^{C_{i}}$ where $C_{i}=\sum _{j=1}^{6}f_{ij}\ln _{2}[{\frac {1}{f_{ij}}}]$ and $\sum _{j=1}^{6}f_{ij}=1$.

The associated community niche-layer multiplicity might then be defined as a harmonic average of the M_i values. The harmonic average M is written

$M=\left(\prod _{i=1}^{N}M_{i}\right)^{\frac {1}{N}}=2^{\left({\frac {1}{N}}\sum _{i=1}^{N}\sum _{j=1}^{6}f_{ij}\ln _{2}[{\frac {1}{f_{ij}}}]\right)}$.

In terms of C_i in bits, for a community of N individuals this could also be written as:

$M=2^{\langle {C\rangle }}$ where $\langle {C\rangle }={\frac {1}{N}}\sum _{i=1}^{N}C_{i}$.

More recent work suggests that a linear average (rather than harmonic) might be more appropriate, since for simple systems the mutual information in a correlated multi-layer system is proportional to multiplicity M rather than to ln₂M. This linear average is often written as:

$\langle {M\rangle }={\frac {1}{N}}\sum _{i=1}^{N}M_{i}={\frac {1}{N}}\sum _{i=1}^{N}2^{C_{i}}={\frac {1}{N}}\sum _{i=1}^{N}2^{\left(\sum _{j=1}^{6}f_{ij}\ln _{2}[{\frac {1}{f_{ij}}}]\right)}$.

In either case the multiplicity values (M_i, M, and <M>) fall between 1 and 6, while the various C values above fall between 0 and ln₂[6] ~ 2.58 bits.

quantifying multilayer correlation

For example, consider an L×n system having L layers each with n possible states per layer. Let M be the multiplicity of correlated layers, where 0 ≤ M ≤ L-1. The simplest matrix of all n^L probabilities, for integer values of correlated-layer multiplicity M, has n^L-M probabilities equal to p_max=1/n^L-M with the remaining n^L-n^L-M probabilities equal to zero. Total state uncertainty is S_tot = (L-M) ln₂n, the marginal probability of any given layer j={0,L-1} is p_j = 1/n with associated uncertainty of S_j = ln₂n. Finally, the mutual information is I_net = ΣS_j - S_tot = M ln₂n.

The simplest example of this might be for L=3 and n=2. For instance, imagine level 0 substrate states {land, sea}, first level locomotion states {legs, fins}, and second level color states {brown,blue}. In this case the uncorrelated M=0 probability set (all mixtures of states from each level) has these eight probability assignments:

$\left({\begin{array}{ll}\left\{{\frac {1}{8}},{\frac {1}{8}}\right\}&\left\{{\frac {1}{8}},{\frac {1}{8}}\right\}\\\left\{{\frac {1}{8}},{\frac {1}{8}}\right\}&\left\{{\frac {1}{8}},{\frac {1}{8}}\right\}\end{array}}\right)$.

In this uncorrelated case, all eight different types of organism are found with equal probability e.g. blue land dwellers with fins are as likely as brown sea dwellers with legs. Hence the total uncertainty ln₂8 = 3 bits equals the sum of the three 1 bit marginal uncertainties, and the mutual information is zero.

For the M=1 set, legs always go with land and fins always with sea while colors remain random. This has a probability array that looks like:

$\left({\begin{array}{ll}\left\{{\frac {1}{4}},{\frac {1}{4}}\right\}&\{0,0\}\\\{0,0\}&\left\{{\frac {1}{4}},{\frac {1}{4}}\right\}\end{array}}\right)$.

In this partially correlated case, total state uncertainty is only 2 bits (four types of organism) even though the marginal uncertainties still sum to 3 bits i.e. you still have one bit of uncertainty about the state of any randomly picked organism on a given niche level. Hence the mutual information is now 3-2=1 bit.

In the fully-correlated M=2 set, land, legs, and brown color always go together as do sea, fins and blue. The probability array now looks like:

$\left({\begin{array}{ll}\left\{{\frac {1}{2}},0\right\}&\{0,0\}\\\{0,0\}&\left\{0,{\frac {1}{2}}\right\}\end{array}}\right)$.

In this fully correlated case, total state uncertainty is only 1 bit (i.e. there are only two types of organism), the marginal uncertainties still sum to 3 bits, and therefore the mutual information is now 3-1=2 bits. Thus here as elsewhere the niche multiplicity and mutual information go hand in hand.

Note that both non-substrate levels (i.e. locomotion and color) relate to individual fitness. In that sense they do NOT represent different niche layers, since separate layers must point inward or outward with respect to distinct physical boundary types (like molecule surfaces, cell membranes, metazoan skins, or gene-pool and meme-pool edges). Nonetheless the example does illustrate correlated-state mathematics reasonably well.

For comparison, a table of properties for a true 7×n multilayer model is provided below. The first two rows refer to the various layers, while the remaining rows refer to various values of inter-layer correlation. The self and pair columns look inward and outward (respectively) with respect to metazoan skins. The family and hierarchy columns similarly orient with respect to molecule code-pool boundaries, while culture and profession columns predicate themselves on idea-pool boundaries instead.

Simply-correlated 7-layer model with n states per layer:

layer names	substrate	self	pair	family	hierarchy	culture	profession
relevant properties	land, air, water	nutrition, fitness, learning	friend, partner, mentor	ancestors, offspring, inlaws	community, employer, government	tradition, arts/sports, religion	specialty, archives, field/study
niche-layer multiplicity	0	1	2	3	4	5	6
pmax	1/n7	1/n6	1/n5	1/n4	1/n3	1/n2	1/n
Stotal	7 ln2[n]	6 ln2[n]	5 ln2[n]	4 ln2[n]	3 ln2[n]	2 ln2[n]	ln2[n]
pmarginal	1/n	1/n	1/n	1/n	1/n	1/n	1/n
ΣSmarginal	7 ln2[n]	7 ln2[n]	7 ln2[n]	7 ln2[n]	7 ln2[n]	7 ln2[n]	7 ln2[n]
Inet=ΣSm-St	0	ln2[n]	2 ln2[n]	3 ln2[n]	4 ln2[n]	5 ln2[n]	6 ln2[n]

A variety of bioscience issues are put into an integrative context with this layered-niche network approach. Included in these, for example, are the quantitative relevance of gene and meme pool diversity, the fidelity of molecule and idea code expression, and even the importance of industrial QA!

some four-vector rearrangements

The four-vector equation in the previous section can be broken into timelike and spacelike parts. If w is the proper velocity dx/dτ, γ is as usual dt/dτ, we might the write the following:

$c^{2}\gamma {\frac {d\gamma }{dt}}=\gamma {\vec {v}}\cdot {\frac {d{\vec {w}}}{dt}}=\gamma {\vec {v}}\cdot ({\vec {a_{o}}}+{\vec {a_{g}}})$.

$\gamma {\frac {d{\vec {w}}}{dt}}=\gamma ({\vec {a_{o}}}+{\vec {a_{g}}})$.

In the above equations, a_o is an acceleration due to proper forces and a_g is presumably due to geometric forces. At low speeds they track the familiar coordinate acceleration vector. For unidirectional motion at any speed, a_o's magnitude tracks proper acceleration's magnitude. How do these quantities relate to the 4-vector terms above more generally?

If we multiply the above equations by mass m and divide by γ=dt/dτ, one obtains:

${\frac {dE}{dt}}={\vec {v}}\cdot {\frac {d{\vec {p}}}{dt}}=m{\vec {v}}\cdot {\frac {d{\vec {w}}}{dt}}$ (timelike) and ${\frac {d{\vec {p}}}{dt}}=m{\frac {d{\vec {w}}}{dt}}$ (spacelike).

The map frame rate of change of proper velocity dw/dt can, in turn, be broken down into proper and geometric force components based on the original 4-vector equation as follows:

${\frac {d{\vec {w}}}{dt}}={\vec {a_{o}}}+{\vec {a_{g}}}={\frac {1}{m}}(\Sigma {\vec {f_{o}}}+\Sigma {\vec {f_{g}}})$

Are f_o and f_g here similarly proper and geometric force components seen from the map-frame coordinate system, which respectively sum to cause the observed motion? If so, how do these components relate to the 4-vector components above, and the frame invariant proper force F_o=mα seen by the object?

In short, we might therefore write...

${\frac {dE}{dt}}={\vec {v}}\cdot {\frac {d{\vec {p}}}{dt}}$ (timelike) and ${\frac {d{\vec {p}}}{dt}}=\Sigma {\vec {f_{o}}}+\Sigma {\vec {f_{g}}}=m({\vec {a_{o}}}+{\vec {a_{g}}})$ (spacelike).

some geometric force rearrangements

Coordinate acceleration a_rot associated with an object from the perspective of a rotating frame adds to the object's physical or proper acceleration a_o a series of geometric terms:

${\vec {a}}_{rot}={\vec {a}}_{o}-{\vec {\omega }}\times ({\vec {\omega }}\times {\vec {r}})-2{\vec {\omega }}\times {\vec {v}}_{rot}-{\frac {d{\vec {\omega }}}{dt}}\times {\vec {r}}$.

The first "centrifugal acceleration" term depends only on the radial position r and not velocity of our object, the second "Coriolis acceleration" term depends only on the object's velocity in the rotating frame v_rot but not its position, and the third "Euler acceleration" term depends only on position and the rate of change of the frame's angular velocity ω.

For an object observed at low speed from the vantage point of an accelerating frame, the coordinate acceleration observed depends on the acceleration of the frame. If the object is being accelerated in the same way as the frame, it appears to have no acceleration at all.

${\vec {a}}_{acc}={\vec {a}}_{o}-{\vec {a}}_{frame}$

In the Schwarzschild shell-frame case, we might similarly write:

${\vec {a}}_{shell}={\vec {a}}_{o}-{\sqrt {\frac {r}{r-r_{s}}}}{\frac {GM}{r^{2}}}{\hat {r}}$

where Schwarzschild radius r_s=2GM/c². Thus for r>>r_s, an upward proper force of magnitude GMm/r² is needed to prevent one from accelerating downward. At the earth's surface this becomes:

${\vec {a}}_{shell}={\vec {a}}_{o}-g{\hat {r}}$

where g is the downward 9.8 m/s² acceleration due to gravity, and r-hat is a unit vector in the radially outward direction from the center of the gravitating body. Thus here an outward proper force of mg is needed to keep one from accelerating downward.

Note: The foregoing results follow if one first calculates the Christoffel symbols:

$\left({\begin{array}{llll}\left\{\Gamma _{t,t}^{t},\Gamma _{t,r}^{t},\Gamma _{t,\theta }^{t},\Gamma _{t,\phi }^{t}\right\}&\left\{\Gamma _{r,t}^{t},\Gamma _{r,r}^{t},\Gamma _{r,\theta }^{t},\Gamma _{r,\phi }^{t}\right\}&\left\{\Gamma _{\theta ,t}^{t},\Gamma _{\theta ,r}^{t},\Gamma _{\theta ,\theta }^{t},\Gamma _{\theta ,\phi }^{t}\right\}&\left\{\Gamma _{\phi ,t}^{t},\Gamma _{\phi ,r}^{t},\Gamma _{\phi ,\theta }^{t},\Gamma _{\phi ,\phi }^{t}\right\}\\\left\{\Gamma _{t,t}^{r},\Gamma _{t,r}^{r},\Gamma _{t,\theta }^{r},\Gamma _{t,\phi }^{r}\right\}&\left\{\Gamma _{r,t}^{r},\Gamma _{r,r}^{r},\Gamma _{r,\theta }^{r},\Gamma _{r,\phi }^{r}\right\}&\left\{\Gamma _{\theta ,t}^{r},\Gamma _{\theta ,r}^{r},\Gamma _{\theta ,\theta }^{r},\Gamma _{\theta ,\phi }^{r}\right\}&\left\{\Gamma _{\phi ,t}^{r},\Gamma _{\phi ,r}^{r},\Gamma _{\phi ,\theta }^{r},\Gamma _{\phi ,\phi }^{r}\right\}\\\left\{\Gamma _{t,t}^{\theta },\Gamma _{t,r}^{\theta },\Gamma _{t,\theta }^{\theta },\Gamma _{t,\phi }^{\theta }\right\}&\left\{\Gamma _{r,t}^{\theta },\Gamma _{r,r}^{\theta },\Gamma _{r,\theta }^{\theta },\Gamma _{r,\phi }^{\theta }\right\}&\left\{\Gamma _{\theta ,t}^{\theta },\Gamma _{\theta ,r}^{\theta },\Gamma _{\theta ,\theta }^{\theta },\Gamma _{\theta ,\phi }^{\theta }\right\}&\left\{\Gamma _{\phi ,t}^{\theta },\Gamma _{\phi ,r}^{\theta },\Gamma _{\phi ,\theta }^{\theta },\Gamma _{\phi ,\phi }^{\theta }\right\}\\\left\{\Gamma _{t,t}^{\phi },\Gamma _{t,r}^{\phi },\Gamma _{t,\theta }^{\phi },\Gamma _{t,\phi }^{\phi }\right\}&\left\{\Gamma _{r,t}^{\phi },\Gamma _{r,r}^{\phi },\Gamma _{r,\theta }^{\phi },\Gamma _{r,\phi }^{\phi }\right\}&\left\{\Gamma _{\theta ,t}^{\phi },\Gamma _{\theta ,r}^{\phi },\Gamma _{\theta ,\theta }^{\phi },\Gamma _{\theta ,\phi }^{\phi }\right\}&\left\{\Gamma _{\phi ,t}^{\phi },\Gamma _{\phi ,r}^{\phi },\Gamma _{\phi ,\theta }^{\phi },\Gamma _{\phi ,\phi }^{\phi }\right\}\end{array}}\right)$

for the far-coordinate Schwarzschild metric (c dτ)² = (1-a/r)(c dt)² - (1/(1-a/r))dr² - r² dθ² - (r sin[θ])² dφ², where a is the Schwarzschild radius 2GM/c², to get...

$\left({\begin{array}{llll}\left\{0,-{\frac {a}{2ar-2r^{2}}},0,0\right\}&\left\{-{\frac {a}{2ar-2r^{2}}},0,0,0\right\}&\{0,0,0,0\}&\{0,0,0,0\}\\\left\{{\frac {ac^{2}(r-a)}{2r^{3}}},0,0,0\right\}&\left\{0,{\frac {a}{2ar-2r^{2}}},0,0\right\}&\{0,0,a-r,0\}&\left\{0,0,0,(a-r)\sin ^{2}(\theta )\right\}\\\{0,0,0,0\}&\left\{0,0,{\frac {1}{r}},0\right\}&\left\{0,{\frac {1}{r}},0,0\right\}&\{0,0,0,-\cos(\theta )\sin(\theta )\}\\\{0,0,0,0\}&\left\{0,0,0,{\frac {1}{r}}\right\}&\{0,0,0,\cot(\theta )\}&\left\{0,{\frac {1}{r}},\cot(\theta ),0\right\}\end{array}}\right)$.

You can obtain the shell-frame proper acceleration, then, by setting the proper acceleration to cancel the geometric acceleration of a stationary object i.e. $A^{\lambda }=\Gamma ^{\lambda }{}_{\mu \nu }U^{\mu }U^{\nu }$ = {0,GM/r²,0,0}, and then showing that its frame-invariant (object-frame) magnitude is α=Sqrt[1/(1-a/r)]GM/r².

Inserting collapsible animations

This is a useful trick, that I think I first encountered on the kinematics page, which allows one to make animations (or static figures for that matter as well) available only on demand. It also has potential for illustrating how the concepts on a given page can be used to address sample challenges. The simpler the solution, the better the correlation between page concepts and the problem at hand.

The perspective of a linearly-accelerated frame might be illustrated with specific calculated examples. For instance:

Worked example: Driving from block to block

Map and car frame perspectives of physical (red) and fictitious (blue) forces for a car driving from one stop sign to the next. In this illustration the car accelerates after a stop sign until midway up the block, at which point the driver is immediately off the accelerator and onto the brake so as to make the next stop. Leaving the stop sign In the following analyses, only forces forward or backward in the direction of travel will be considered. Both analyses predict that the forward force of the car on the passenger will equal the passenger mass m times the acceleration of the car. Thus if the acceleration of an 80 kg passenger is 1 gee (9.8 m/s2), then the forward force needed to do it is nearly 800 Newtons. Map frame during speed up $F_{road}=\Sigma F_{map}=ma_{map}=ma_{seat}.$ Here from the map frame perspective the passenger is seen to be continually accelerated forward, with an acceleration equal to the unbalanced forward force Froad of the car on the passenger, divided by the passenger's mass m. The fact that the road force on the passenger is unbalanced means that in this frame there are no other forces in the direction of travel on the passenger. Car frame during speed up $F_{road}-ma_{seat}=\Sigma F_{car}=ma_{car}=0.$ From the car frame perspective the passenger may be said to experience a forward road force canceled by a backward g-force, which results in no acceleration at all from the perspective of that frame. Unlike the forward road force exerted by the car on the passenger, the backward g-force acts on every bit of the passenger. Moreover the g-force magnitude is proportional to the passenger's mass so that, if allowed to cause acceleration, the acceleration would be mass-independent. Approaching the stop sign Map frame while braking $-F_{brake}=\Sigma F_{map}=ma_{map}=-ma_{belt}.$ Car frame while braking $-F_{brake}+ma_{belt}=\Sigma F_{car}=ma_{car}=0.$

The rotating frame perspective might be illustrated with specific calculated examples. For instance:

Worked example: Constant speed slingshot

Map and spin frame perspectives of forces associated with a stone released after being spun around on a massless rope. Forces on the stone include the inward centripetal (red) force seen in both frames, as well as the geometric (blue) force seen in the spin frame. Before the stone is released, the blue geometric force is purely centrifugal (pointing radially outward), while after release the geometric force is a sum of centrifugal and Coriolis components. Note that the centrifugal component (light blue) is always radial, while the Coriolis component (green) is always perpendicular to spin frame velocity. Also seen in both frames is the force on the rope's anchor point (magenta) caused by Newton's 3rd Law action-reaction to the centripetal force on the stone. Before projectile launch The following alternate analyses of motion before the stone is released consider only forces acting in the radial direction. Both analyses predict that string tension T=mv2/r. For example if the radius of the sling is r=1 metre, the velocity of the stone in the map frame is v=25 metres per second, and the stone's mass m=0.2 kilogram, then the tension in the string will be 125 Newtons. Map frame story before launch $-T_{centripetal}=\Sigma F_{radial}=ma_{radial}=-m{\frac {v^{2}}{r}}.$ Here the stone is seen to be continually accelerated inward so as to follow a circular path of radius r. The inward radial acceleration of aradial=v2/r is caused by a single unbalanced centripetal force T. The fact that the tension force is unbalanced means that, in this frame, the centrifugal (radially-outward) force on the stone is zero. Spin frame story before launch $m{\frac {v^{2}}{r}}-T_{centripetal}=\Sigma F_{rot}=ma_{rot}=0.$ From the spin frame perspective the stone may be said to experience balanced inward centripetal (T) and outward centrifugal (mv2/r) forces, which result in no acceleration at all from the perspective of that frame. Unlike the centripetal force, the frame-dependent centrifugal force acts on every bit of the circling stone much as gravity acts on every ounce of you. Moreover the centrifugal force magnitude is proportional to the stone's mass so that, if allowed to cause acceleration, the acceleration would be mass-independent. After projectile launch After the stone is released, in the spin frame both centripetal and Coriolis forces act in a delocalized way on all parts of the stone with accelerations that are independent of the stone's mass. By comparison in the map frame, after release no forces are acting on the projectile at all. The foregoing three properties: (i) ability to be eliminated by a suitable choice of frame, (ii) accelerations that are independent of mass, and (iii) delocalized action on every bit of mass are common to forces that arise from the connection term in a coordinate system's covariant derivative. Such geometric or improper forces include g-forces, Euler forces and gravity as well as Coriolis and centrifugal. They are in this way distinct from physical or proper forces that result e.g. from direct contact or electrostatic interaction.

Blockquotes, and code repair

Aside: The following paragraphs were used to temporarily replace a consensus introduction after it was unceremoniously deleted. How does this relate to mechanisms for regulation and repair of molecular codes similarly offered up for community access in eukaryotic cell interiors? This also illustrates use of Wikipedia's blockquote qualifier:

When an object is constrained to move in circular motion, the outward radial force seen to be acting on that object from its rotating vantage point is known as the centrifugal force (from Latin centrum "center" and fugere "to flee").

Because this force arises from the connection term in the accelerated coordinate system's covariant derivative, it may be referred to as a geometric or fictitious force (as distinct from a proper or physical force) even though its consequences from the perspective of that frame are very real. Such geometric forces allow one to apply Newton's laws locally in accelerated frames, and they act on every ounce of an object's being rather than e.g. via direct contact or electrostatic repulsion.

Centrifugal force should not be confused with the inward-acting centripetal force that causes a moving object to follow a circular path. The proper reaction to this centripetal force, exerted by such revolving objects on their surroundings, was in earlier times also called centrifugal although this use is less common today.

Who has these interests?

Complex-system informatics, nanoscience, materials astronomy, and the thermo-chapters of introductory physics?

Other useful wikicode

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• <ref>{{cite journal |url= |title= | last = | first = | journal = | volume = | issue = | pages = |accessdate=2024-11-18 |quote= |publisher= }}</ref>
• <ref>{{cite book |last= |first= |authorlink= |coauthors= |title= |year= |publisher= |quote= | url= |isbn= }}</ref>
• <ref>{{cite news |first= |last= |authorlink= |coauthors= |title= |url= |quote= |publisher=[[New York Times]] |date= |accessdate=2024-11-18 }}</ref>
• <ref>{{cite news |first= |last= |authorlink= |coauthors= |title= |url= |quote= |publisher=[[Time (magazine)]] |date= |accessdate=2024-11-18 }}</ref>
• <ref>{{cite news |first= |last= |authorlink= |coauthors= |title= |url= |quote= |publisher=[[National Public Radio]] |date= |accessdate=2024-11-18 }}</ref>
• <ref>{{cite news |first= |last= |authorlink= |coauthors= |title= |url= |quote= |publisher=[[Rolling Stone]] |date= |accessdate=2024-11-18 }}</ref>
• <ref>{{cite journal | last = | first = | authorlink = | coauthors = | date = | year = | month = | title = | journal = | volume = | issue = | pages = | publisher = | location = | issn = | pmid = | doi = | bibcode = | oclc = | id = | url = | language = | format = | accessdate = | laysummary = | laysource = | laydate = | quote = }}</ref>
• {{Lifetime|1900|2000|Last, First}}
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• [[Category:National Aviation Hall of Fame]]
• [[Category:Missing middle or first names]]
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