Infinite Energy Field

Size: px
Start display at page:

Download "Infinite Energy Field"

Transcription

1 Infinite Energy Field Jerry E. Bayles January 1, 008 Tis paper presents te possibility tat te so-called static electric field is not static. Furter, tat te field is evolving as well as self sustaining. Also, if it is considered as an open and tus not terminated field, it approaces an infinite energy over a nearly infinite volume and time of expansion. Tis requires an input energy from wat I call "Energy Space". Tis is te same Energy Space tat created our universe during te Big Bang primordial creation event. In te concept presented in tis paper, all electrons and protons are still connected to tat energy source troug teir centers of origin. Te mass of te electron and proton is embodied in teir standing wave geometry wile te dynamic field moves outwards in discrete steps. Eac step distance is a fundamental lengt equal to te classic Compton radius derived from te particle's rest mass energy. Sells of energy will be formed at a rate of expansion equal to te speed of ligt outwards from te carge-particle, if te field is unterminated. In tis tougt scenario, a particle is suddenly placed into an oterwise empty and nearly infinite volume of space vacuum. Initially, a sell of field energy forms around te particle equal to its rest mass energy. Tis requires a time suc tat te Plank frequency times te Compton wavelengt is equal to te speed of ligt. Wen completed, instantly, te next sell begins to form werein te same frequency times wavelengt is equal to te speed of ligt constant. Eac sell will ave a total energy equal to te particle rest mass energy wile te energy density will be decreasing due to te increasing volume of eac sell. Tus te field is quantized into layers of energy, muc like an onion is built outwards in layers tat form individual sells. It will be developed matematically in te main body of tis paper ow suc a construction may exist and yet be measured by conventional metods to arrive at wat is incorrectly considered by contemporary science as a static field. Tere is no suc ting as a static field if force can be associated wit te field's existence. Tat is, a carge-field is not built as a one-sot event. It continues to flow or it does not exist at all. If it is terminated by a conjugate field, it still flows from source to receptor and te result is force. It is a steady state flow if te carges are not moved relative to eac oter, but te field is flowing at all times. Te return for te field is back into energy space. I envision te actual electron and proton as being in te sape of a standing wave field tat forms a torus. Ten te field must also build into te same sape. Te major axis of eac torus is expanding as te sells form outwards wile te minor radius is fixed at te Compton radius wic is also te connection to energy space. In fact, it will be developed tat te necessary volume will utilize te volume of a torus instead of te volume associated wit te classical geometric parameters of te electrostatic field. Te connection to te Golden Ratio will be te result. It may be argued tat since energy cannot be created or destroyed, wat I am advocating is scientific eresy. My counter argument is tat my concept above does not involve a closed system. Te creation of te universe was not te result of a closed system energy event and neiter is my concept of field as presented above and below. In fact, I furter propose tat once establised as a quantum sell of energy, te sell engenders te next sell and so on. Tis will allow for no voids, if for example, te electron or proton is accelerated from its state of constant velocity. Te old field will radiate away as a poton wile a new field is formed relative to te new position of te electron or proton. Te energy relative to te old field is radiated away but measurement indicates tis does not destroy te electron's or proton's ability to engender a new field. Te action causing te carge displacement cannot directly create a poton. Te poton is erein viewed a loop of field energy tat transfers its energy in alternate alf-cycles between negative and positive energy space, never gaining or losing energy overall. Furter, it completes eac cycle at a time related to its Plank energy and ten jumps in quantum fasion instantly to its next quantum wavelengt position. Tus a poton is a field of quanta and is not really a wave. However, it can exist ortogonally to its direction instantly along a line 90 degrees to its direction of travel so tat in a two slit experiment, it will appear to go troug two slits at once, tus being interpreted as a wave action. Tis applies to particles in general and not just potons by te DeBroglie wave mecanics principle.

2 InfiniteEnergyField_Add1.MCD 1 Te required parameters of calculation in S.I. units are: l q := m c m := sec E p := 16 ε o l q π := coul Vol( n) := π m e := kg Te geneses of te first 10 field layers is establised as: 3 ( n) l q 3 π ( n 1) l q ε o farad := m α := n := 1,.. 10 E d ( n) := E p Vol( n) 1) Te product of te sell volume times te sell energy density is a constant equal to E p above. VOLUME X ENERGY DENSITY = SHELL ENERGY CONSTANT, E p Vol( n) = m 3 E d ( n) = joule m 3 Vol( n) E d ( n) = joule ) Wat will be developed is tat te energy measured at increasing distances from te source falls off as 1/r wic agrees wit classic pysics measurements. However, te quantized metod will arrive at te same conclusion by taking into account a variable area torus interface tat will yield te same results as te classic metod utilizing a constant area probe at increasing distances. Tus, te field is quantized into sells of constant energy, eac being a widt equal to twice te classic radius of te electron. Tis relates to te Heisenberg expression E x T = so tat te field frequency related to te rest mass energy of te electron does not degenerate wit distance. For a so called static field, a degenerate frequency or energy field by E = f would amount to destruction of energy wic is not allowed since energy in a closed system (or establised sell in tis case) cannot be created or destroyed. In te classic sense, since a continuous field of decreasing energy per unit volume wit increasing distance is assumed, tis would amount to a decreasing amount of energy in a linear fasion and tus radiation of te field in te form of E = f. It is terefore likely tat in te quantized teory presented erein, if we consider te field tat reaces outwards from a single electron into an infinitely large unterminated space vacuum, were also te electron is suddenly created in tat same isolated space, te total field created over time approaces an infinite energy. Finally, it is also my proposal tat te energy tat creates te electron and proton field via te geometry of te related torus comes from te same infinite energy space tat created te original Big Bang.

3 In tis work, torus volume divided into energy potential yields energy density as sown below: 16 ε o l q π 3 l q π simplifies to 1 8 π 3 ε o l q equals energy density A variable major radius torus ring is formed as sown in te below equation. Te area of a torus is given as times π times te variable major radius n(l q ) times te minor radius l q. Te variable (n) increases te classic electron radius by wole integer steps to form te increasing major radius. Tis increases te major (circular) radius of te energy torus accordingly. Te constant /π is sown as a part of te total particle/field geometry and also yields a fundamental angle related to te rise of te apotem of te Great Pyramid at Giza. 8 π 3 ε o l q 3) Ionized Hydrogen would build an energy field, via te proton torus geometry connection to energy space, greater tan te atomic form would allow in te closely terminated atomic form. Wen te proton recaptures te electron, a great deal of energy stored in te expanded field would suddenly be released. Below is sown a comparison of te conventional expectation field energy measurement E p (n) compared to te area gated form Ψ e (n) involving te added terms related to te square root of te golden ration (/π) and te increasing major torus radius (n. l q ) multiplied by te electron classic minor radius (l q ). E p ( n) := 16 ε o ( ) nl q π Ψ e ( n) := Vol( n) E d ( n) π π l q π π ( nl q ) l q ( ) ) E p ( n) = joule Ψ e ( n) = joule Te above equation does not cancel like terms to illustrate te possible geometry of te field related to te golden ratio. Te terms inside of te above equation's bracket derive a ratio of te increasing torus area in te denominator divided into a fixed probe area corresponding to te fixed area of a measurement of te field. Te first 1000 classic (l q ) electron quantum radii field layers will yield te total summation of energy: n := 1, Ψpot e ( n) := Vol( n) E d ( n) Ψpot e ( n) = joule n 5) Note tat te electron rest mass energy is given as: E e := m e c Obviously, te field cannot be static, as te above analysis sows. E e = joule 6)

4 3 Now let: := joule sec and Ed torus := 7) 8 π 3 ε o l q Energy density times te velocity of ligt yields te Poynting vector power of: S:= Ed torus c α or, S watt = m Te utilization of t e signifies tat te electron pulses its energy into expanding and tus sequential field layers at te rate of te inverse of te time of t e. Tis approximates an onion skin appearance to te field if an onion were saped like a torus. Te torus sape of te field and of te source electron and proton is arrived at as sown below. Te contemporary or classic form of te electron field arrives at an energy density equation of: (Alpa (α) reduces c to te velocity of te n1 orbital of Hydrogen.) Let te Compton electron 'time' be stated as: t e := t e = sec 9) m e c Finally, allowing for a time x area gate to be applied to te tremendous Poynting vector power above: E:= ( S) t e π l q E = joule = electron rest mass energy. 10) 8) 1 E d ε = o E v were E v is te field in volts/meter: E v = π ε o r 11) 1 Ten: E d ε o = simplifies to π ε o r E d = 1) 3 π ε o r (= Classic volume.) 3 π ε o l q 8 π 3 ε o l q simplifies to: (= torus volume) π Tis is very close to te square root of te golden ratio wic is ubiquitous in te design and growt of natural processes. 13) Te /π form for te volume ratios of te classic field to te quantum torus field reveals a field tat is te basis of unlimited energy by revealing an evolving quantum torus structured field instead of a static field. Terefore, te following is postulated: Te universe is not static, since it is expanding. It is growing, or evolving. As a result, it is not a closed system. It as been said, "As above, so below!" Ten it sould be no surprise to find tat te basic particles suc as te electron and proton, are not separate from te same energy field tat not only created all matter but drives te universe to expand as well. Tis takes us all te way back to my first publication involving my electrogravitational field teory. Tis work is titled "Electrogravitation As A Unified Field Teory," Capter 1, specifically, pp 8 troug 9. Terein I developed te torus model of te electron and te mat tat supported tat conclusion. Tis work is available for free downloading in Adobe Acrobat PDF format at: ttp://

5 For te case of te magnetic field, te electron or proton is first considered as moving along a circular (elix) pat in tat infinite space vacuum mentioned above. Its rate of inline progress related to its distance of rotation is connected to te /π ratio as outlined above. It is continually creating a magnetic field relative to its carge and velocity. No energy is applied to te electron to aid its motion since it began its existence and yet, te magnetic field is building outwards along te pat of te moving carge. It is obvious tat tis cannot be a static energy scenario even for te simple case of linear non-accelerated motion. As te electron or proton continues to build te field, te electron or proton does not lose any rest mass or carge in te process, but simply continues to generate more magnetic and electric field layers as it moves along.. Now if we accelerate te isolated carge-particle, te field forms a poton and radiates away. Yet, te carge-particle's rest mass energy does not crease to exist or even in te least diminis. Furter, te carge associated wit tat particle does still exist and it also continues to create more field energy witout itself being in te least diminised. Te 'pus' tat accelerated te carge created a rift in te field tat will be selfealing. Finally, it is a pysical force apart from te rest mass or carge of te particle itself. Tus, we must conclude tat considering a carge field as being 'static' is incorrect. Terefore, a field associated wit carge is most likely a dynamic energy, capable of evolving over time. As a result, te field associated wit particle carge is supported from te same energy space tat created te universe. Even energy space may be viewed as a torus structure. In fact, it as been suggested by some tat our universe may also be a torus structure. If an observer is standing inside of a torus so tat te observer is looking along te axis of te torus, we can imagine tat te rotation of te torus wall is moving clockwise as te observer sees it. Now, if te observer is turned 180 degrees in te opposite direction, te wall of te torus will appear to be moving counterclockwise. Ten we can ascribe tese two different viewpoints as not only positive and negative time but also positive and negative energy, respectively. As a result, te same energy space in te form of a torus can ave te economy of exibiting eiter positive and negative energy (and time) depending on te orientation of te observer wen entering tat space. Ten, field energy may be considered as a connection to torus energy space via te center of te electron or proton, werein a single alf cycle is allowed, eiter positive or negative according to te direction of te connection inside of te torus. A poton owever, alternates in successive alf cycles between 'positive' and 'negative' energy space by continually flipping its orientation in te torus geometry of energy space. Te net poton energy yields no rest mass energy. However, at any instant, energy related to quantum frequency by E = f must exist. Carge, on te oter and, is associated wit rest mass energy since te field is monopolar in energy sign and time wit respect to direction of orientation along te axis in torus energy space. Te direction of tis torus axis must be considered as dependent on te spin of te carge-particle and as a result, te possible orientations in a quantum sense must be finite. A alf-cycle is equal to π radians = 180 degrees and te angle of rise of te great Pyramid at Giza is equal to te atan(/π). It as been suggested by some investigators tat te Great Pyramid olds a secret tat is key to te construct of te universe and also tat te Great Pyramid is peraps as old as 1,000 years or more. Te Spinx may also fall into tis category of being far older tan contemporary researc is able to admit. Te above conceptual view of so-called static fields actually being dynamic and capable of growing to infinite energies in an unterminated scenario is stunningly different tan contemporary science will allow us to believe. More on te Great Pyramid later. If we are to believe tat te entire universe was suddenly created from an incompreensible amount of energy, it is natural to conceive of matter still being connected to tat same beginning source, albeit in a muc more subtle way. Tat concept demands growt of structure small as well as large and tus te field energy cannot 'rest' in a static form, and if allowed to grow unterminated, will grow to infinite proportions according to te summation of increasing layers of energy over all time. Contemporary science's measurement of te field amounts to a blind man placing a stick into a steadily moving current of water. He will say, "ere is a force! I can feel it against my measuring stick!" He will not see te steady movement of te river. He may rationally conclude tat te force must be a 'static' force if it does not cange wit time.

6 Te Quantum Magnetic Field: It is developed below tat te quantum magnetic field related to uncanging velocity of a carge in open space is far weaker tan te corresponding electric field as developed above. Te energy density is correspondingly muc less for te magnetic field tan for te electric field per unit of volume. For a poton, it is establised tat te energy density for bot te electric and magnetic fields are equal. However, tis is for te case of a poton or electromagnetic wave werein te fields were born of accelerating carges and tus give rise to sympatetic and equal energy density field structures. Te case for te isolated and non-accelerated carge magnetic field genesis is given below. Te magnetic expression is based on te least quantum velocity related to wat I call te least quantum electrogravitational energy equal to Hz Tis frequency and energy is fundamental to my teory of electrogravitational action Firstly, te necessary parameters for calculation are stated: 5 f LM := Hz µ o π enry := µ o enry = 1) m m f LM V LM := or, V LM = m sec -1 15) m e Te potential magnetic energy related to V LM at te classic radius of te electron is: µ o E LM := VLM E LM = joule 16) π l q Next we determine te magnetic flux energy density B at te classic radius as follows: First, te E LM energy above is set equal to te equation tat includes B LM at te same classic radius. Ten Matcad's symbolic equation solver is utilized to find te expression tat solves for te magnetic flux density B. We will ten utilize te usual basic equation E d = B /µ o to find te magnetic field energy density. µ o q o µ o V LM 17) VLM = ( ) ( V LM ) ( B LM ) ( l q ) as solution(s): π l q π l q [Equal to force x distance = expended energy] Finally: were, 1 tesla = 1 weber m B LM := µ o V LM B LM = tesla 18) π l q volt sec Also: 1 tesla = 1 m Comparing te above magnetic energy density to te electric field energy density: Magnetic Energy Density Electric Energy Density B LM Em d := Em d joule = µ o m 3 Ed torus joule = 19) m 3 It is immediately apparent tat te magnetic field tat arises from te least quantum electrogravitational magnetic field is muc weaker tan te quantum electric field wen non-accelerated carge motion is concerned. Tis least quantum magnetic field is also associated wit te observable gravitational reaction as well. Note tat V LM is a least quantum constant angular 'velocity' and is terefore related to constant angular momentum in te quantum sense as in te Heisenberg expression mvr =in/π.

7 Te volume is solved for below related to te magnetic field potential energy divided by te magnetic field energy density as sown above. 6 E LM Volume = Ten: Em d µ o π l q µ o π l q µ o VLM V LM = E LM simplifies to = B LM µ o 3 8 π l q 0) Te ratio of te above volume to te torus volume volume expression is: 3 8 π l q π 3 l q simplifies to π 1) If we return to te contemporary classical form of expressing electric field energy potential divided by energy field density we ave: π ε o l q 3 π ε o l q Electric Field Energy Potential simplifies to Electric Field Energy Density (Non-torus field) 3 8 π l q wic is te same volume we arrived at for te magnetic field volume above related to it magnetic energy potential divided by its energy density. ) Te reason te beginning analysis utilized te volume of a torus is so tat equation (10) above would arrive at te correct energy for te rest mass energy of te electron. If te strict classical non-torus form of volume is used, te electric field energy will be lower tan required to arrive at te correct rest mass energy for te electron by 1/(/π). Tis was te fundamental reason wy te electron was ascribed te geometry of a torus in my beginning as well as present field analysis. Utilizing te torus energy density, te torus volume is sown for te electric field as sown below: π ε o l q 8 π 3 ε o l q simplifies to π 3 l q Tis is te area of a circle times a circumference troug te axis of rotation of tat same circle. 3) Ten we may find te energy density expression tat relates to te quantum magnetic field torus as we did for te electric field above.

8 7 µ o π l q π 3 l q VLM simplifies to µ o 8 π 3 l q wic is similar to te electric field energy density of equation (3) above of: Ten let te torus form of energy density be stated below as: V LM joule = ) m 3 8 π 3 ε o l q 5) µ o Emt d := 8 π 3 V LM or, Emt d joule = 6) l q m 3 Te ratio of te torus electric field energy density to te torus magnetic field energy density is: 8 π 3 ε o l q µ o 8 π 3 l q V LM Electric Field Torus Energy Density simplifies to 1 ε o µ o V LM Magnetic Field Torus Energy Density = ) wic is a 19 magnitudes number as te ratio of electric to magnetic field torus energy density. Multiplying te inverse of te above ratio ratio by te square of te speed of ligt and taking te square root of te result, we ave: 1 ε o µ o V LM 1 c = m sec -1 were: V LM m sec -1 = wic is te least quantum magnetic and electrogravitational velocity as stated in te beginning. 8) It turns out tat te least quantum magnetic field energy related to te electrogravitational (gravitational) action is at a temperature in degrees Kelvin in wat is called te Bose-Einstein condensate range were very weird quantum tings appen suc as Helium forming geysers and climbing up te sides of its container. Also, te particles at tis temperature act as one large particle saring te same quantum wavelengt. Solving for te temperature T in degrees Kelvin related to te least quantum magnetic energy: 3 E LM = k T as solution(s) E LM were, k joule := 9) 3 k K

9 8 E LM Ten: T LM := or, T LM = K = 31 pico Kelvins. 30) 3 k A Helsinki group reportedly as recently reaced te vicinity of 100 pico Kelvins. Electrogravitational effects may quite likely appear at or near te above temperature since it represents te least quantum electrogravitational energy posited not only by tis paper but oters as presented by myself previously at te above named website. Tis includes geysering of Helium as well as crawling up and out of te container. Also, te repulsion of magnetic fields may be expected if te magnetic field is of te open form and not part of a torus geometry wic is te closed standing wave field in te superfluid condensate temperature range. It is even possible tat a least quantum ponon energy less tan E LM = m x V LM may sed V LM and convert to mass only since V LM is te absolute lower limit of angular momentum. As in te manner of equations (8) and (10) above, we will solve for te Poynting vector related to te magnetic energy density x c. Area x time yields an energy just below te electrogravitational energy, E LM. S LM := Emt d c α S LM watt = 31) m ( ) t e Area x time gate of S LM : E SLM := S LM π l q E SLM = joule 3) Finally: E LM = E SLM 33) Tis may also be te secret as to wy te Great Pyramid as a /π ratio of te eigt to 1/ te lengt of te base. It is suggested tat it may be of interest to measure te energy available from a good conductor stretced from te base of te pyramid to its apex. Especially a superconductor. Te electrogravitational force between two electrons separated by te first energy sell (n1) radius of te Bor H1 atom is given as: E SLM F EG α l q µ E SLM := o α F EG = newton enry newton 3) l q m Te above equation sows tat E SLM is equal to E LM. Tis implies tat te gravitational frequency tat actually sows up to an observer is extremely close to wat is called te Scuman frequency between 7.83 and 8.00 Hz. Tis is peraps incorrectly assumed to be electrical/magnetic resonance between te ionospere and te Eart but is actually te electrogravitational frequency f LM divided by /π.. E SLM f ESLM := or, f ESLM π 1 = Hz f LM π 1 = Hz 35) Te product of /π times /π yields a number extremely close to wat is called te Golden Ratio. Ten it is suggested by above result tat te action of gravity embodies te structure of te geometry connected to te ubiquitous Golden Ratio found in most all processes involving natural growt. Eq. 19 above is te open field in deference to a standing wave torus structure connected wit te quantum om wic does not radiate electromagnetically and is considered to be a closed field. Te open field is connected to te free space resistance wic is connected to electromagnetic radiation. S rad := Em d c α S rad watt = 36) m

10 9 E Srad := ( S rad ) t e π l q E Srad = joule 37) f Srad := E Srad f Srad = Hz Wic is very close to te well known Scuman frequency tat is considered to occur via an Eart-ionispere resonance. 38) Note tat: f LM = f Srad were f LM is te closed volume torus (mass) and tus quantum om related frequency wile f Srad is te open free space resistance related frequency aving no rest mass. 39) were, π = ) Terefore, te 7.83 Hz Scuman frequency may be expected to exist on oter planets aving atmosperes tat can be jostled about by te electrogravitational action as outlined above. Finally, we would expect a different Scuman frequency cavity resonance according to te size of te cavity formed by te volume between te upper atmospere and te ground mass. Also, due to te rise and fall of te distance between te upper atmospere and te ground due to nigt and day solar effects, te frequency sould sift accordingly. I ave not eard of tis as being te case ere on Eart wic suggests tat if te Scuman frequency is fairly constant, terefore, it is not likely due to Eart to sky cavity resonance wic sould be affected by day to nigt eating variations affecting te volume and tis frequency of resonance. It as been presented above tat our universe is not likely a closed system, and terefore, it is quite possibly te fields connected wit te field generating particles are not isolated from te same energy tat created te Big Bang in te very beginning. Peraps negative energy and dark matter arises from 'sed' least quantum angular velocity from least quantum angular momentum tat in effect creates mass wic later on acquires energy by being jostled about by oter mass and energy. In closing, I prefer to tink of "static", wen applied to fields, as being at te very least an unfortunate limiting concept to wat is actually occurring. Reference: ttp:// Addendum #1 In Te Beginning... by Jerry E. Bayles Marc, 008 In te following addendum to te above paper, te beginning of te universe is examined in terms of te least Plank time and Plank radius. It will be developed troug use of te Heisenberg uncertainty equation tat te beginning energy is far less tan wat exists in te present day universe. As a result, te conclusion is tat te energy input to te universe is ongoing and as been since te very beginning. Tis meses wit te concept of field growt potential of unterminated carges yielding potentially an infinite energy increase over time. Te universe terefore is continually seeding itself for unlimited growt.

11 10 Tis addendum analyzes te energy available in te primary wavelet associated wit te Big Bang. Te universal gravitational constant is stated as: G := newton m kg Next, te contemporary least quantum Plank time and radius is stated as: Plank radius: r plk := G r plk = m 1) π c Plank time: t plk := G t plk = sec ) π c Te Plank energy related to te Plank time above via Heisenberg's uncertainty is: E plk := E plk = joule Te Plank torus energy density is: 3) t plk E plkfld := 8 π 3 ε o r plk E plkfld joule = ) m 3 S plkfld := E plkfld c α S plkfld = kg sec -3 5) Te Poynting vector power S plkfld may also be stated in te conventional units as sown below: S plkfld watt = 6) m It is of interest tat te Plank Poynting vector power may be expressed as mass per time cubed. Tus, it can be surmised tat our tree dimensional distances in space can be expressed as (velocity x time) 3 wic means tat time is te fundamental parameter. Velocity ten relates to te offset angle per observation interval of time. Tis approac naturally includes Einstein's so-called fourt dimension of time as applied to is Special Teory of Relativity. Te final Plank rest mass energy is: E plkmass := S plkfld π r plk t plk E plkmass = joule 7) E Note te dimensional adjustment between te plk α 1 two energy fields involves te product of te E plkmass π = ) square of te fine structure constant α as well as te 1/(π ) related to torus structure. E plk = volt E plkmass = volt 9)

12 I consider te beginning of te universe very similar to te Dirac delta function werein at zero time, te amplitude is effectively infinite and at all oter time, te amplitude is effectively zero. However, in reality, te Dirac delta function may be tougt of as equal to te impulse function used in transient analysis. Te weigted impulse function is similar to te Dirac delta function but does not involve infinities or zeros wile still approacing te limits of zero time and infinite amplitude. As suc, it is a practical engineering approac to real boundary limits. Te narrow time and ig energy of te beginning may be viewed as a wavelet tat spreads over time and tus reduces also in amplitude correspondingly. Te age of te universe is set close to 13 billion years as sown below and ten te total number of seconds is calculated based on tat time duration. Ten te age is sligtly adjusted to yield exact results below equal to te expected Higgs energy as well as te related exact energy of te electron. 11 U age := years Age in seconds: U agesec := U age 50) Te above age is very close to 13 billion years but is accurate to 11 significant figures. Te degraded wavelet energy related to time spread is: E plknow := E plk U agesec E plknow = joule 51) GeV := volt V plknow := E plknow V plknow = GeV 5) It is predicted tat te Higgs particle is very close to te above energy V plknow in electron volts. Please see te article, "Te Coming Revolutions In Particle Pysics", by Cris Quigg, p. 9 of Scientific American, Feb. 008, pp 6-53 inclusive for an excellent overview of te Higgs particle and its expected energy. Quote: "Electric carge: 0, Mass: Expected below 1 Tev, most likely between 11 and 19 GeV. Believed to endow W and Z bosons, quarks and leptons wit mass."-unquote. P. 8 of te above reference. Next, te energy of te electron in electron volts is arrived at by te same age reduction process applied to te field related plank energy in electron volts V plknow from above: MeV := volt V plknowmass := V plknow α 1 ( ) π For te electron rest mass energy: m e c V e := V plknowmass = MeV V e = MeV 53) 5) Te above strongly suggests tat te energy of te Higgs and te electron and even te proton is decreasing as te universe ages. Meanwile, te dark matter and negative energy is quite likely going to increase by te same time aging process. Te total number of seconds in te age of te universe is uge and we would likely not notice te cange of mass of te basic particles in te sort time of te existence of our quantum science. U agesec = seconds. A million years = 7.79 x of te value at te left. 55)

13 1 Tere is a geometric number based on powers of π tat wen multiplied by te mass of te electron yields te mass of te proton to a almost exact correspondence. Te accepted S.I. value of te mass of te proton is: ( ) m ep := m e 6 π 5 calculated mass = m p := kg m ep = kg 56) Te sligt difference may be in relativistic mass added to arrive at te measured value of m p. Since te original creation energy was muc greater tan te rest mass energy of a single proton, te rest of te missing energy may be found in te creation of similar particles, neglecting radiation energy from potons and te mass of electrons. Ten, rougly speaking, te total number of protons in te universe tat will add up to te total energy available during te creation event can be calculated as sown below. p tot := E plk m p c p tot = Te mass related to te number of protons above is: Total # of protons aving te rest mass sum total energy of te primordial creation event. 57) m ptot := p tot m p m ptot = kg 58) Obviously, tis is far less mass tan we observe to exist in te present universe! Terefore, it is suggested by te above result tat energy in te form of energy fields being converted to mass is and as been an ongoing process since te original creation event. Tis conforms wit te beginning analysis in te main body of tis paper werein it was proven tat a single carge-particle ad te potential ability to generate an unlimited amount of energy in te field in te case of being unterminated by a conjugate carge-particle. Terefore, te so-called creation event, also known as te Big Bang, was only a kick-start to begin te running up of te engine of creation as an ongoing process. Te geneses carge-particles were and are energy seeds tat generate more energy seeds and so on. In tis scenario, te universe is alive wit new energy from one second to te next and all particles in a olistic sense are connected togeter troug teir centers to eac oter all te way back to te creation event and instantly trougout time to eac oter troug te same effectively timeless and infinite energy space tat created te original geneses event. It is by means of tat structure tat te non-local electrogravitational action mecanism is also generated. Ten not only are te fields connected wit carge-particles not static, but te total energy in te universe is not static wit te passage of time. Te universe is growing, expanding outwards as a flower or tree does grow. At te eart of all pysical evolution is te natural number e as well as te golden ratio. Ten te structure of te Great Pyramid a Giza may represent ancient engineering tat tuned into tat universal dynamic quantum energy and during te process of resonance wit te basic frequencies of creation, generated energy tat was utilized by an entire world-wide system of pyramidal structures. It will be developed below tat te pyramids may ave been used to terraform te planet starting about 00 million years ago. Te basic particles vibrate in a quantum fasion in response to tis input energy from energy space and te carged ones generate a frequency at or near 7.83 Hz as te analysis in te main body of tis paper proved matematically. Tus, planets aving an atmospere of carged particles would collectively and sympatetically generate 7.83 Hz as a result of te basic quantum energy associated wit te open field structure of magnetic and electrogravitational field actions. Tere can be developed equations wic fit te geometry of te Great Pyramid tat involve te proton mass and carge as well as te square root of te golden ratio and te natural number e. Tis is presented beginning on te following page. Not only is te Great Pyramid at Giza of interest regarding te golden ratio but also te Partenon in Atens is said to incorporate te golden ratio in its design.

14 A new fundamental equation tat establises te velocity of sound at sea level and at 70 degrees Fareneit is presented below as relevant to te acoustic velocity as well as quantum parameters of ydrogen: 1 First, te Bor radius of te n1 energy sell is stated as: R n1 := ( π m e c α) 13 v air := were, v air = m 59) π ( e) ( π R n1) sec m p or, v air ft = 60) sec Te wavelengt based on te least quantum electrogravitational frequency f LM, R n1, e and proton rest mass m p is given as: λ air := were, 18 f LM R n1 e λ air = m 61) m p or, λ air = ft 6) Note tat f LM is te least quantum electrogravitational frequency related to field of te electron and proton carge and is terefore applicable to bot particles. In te air, te carge coupled electron follows te vibration of te proton since te proton is muc more massive. Te frequency related to te velocity and wavelengt above is: f air := were, f air = Hz 8R n1 λ air e m p 63) Te King's Camber frequency in te Great Pyramid at Giza as been measured by several people as aving an acoustic resonance of 38.3 Hz. However, te above frequency may be te original designed frequency. f air π e π = Hz were, f Srad = Hz 6) (From eq. 38 above.) Te above is a remarkable result since te equation results of (35) and (38) above agree almost exactly wit te equation result above. Note tat: π e π simplifies to 6 π e = ) It is of interest tat powers of appear repeatedly wen various acoustic equation solutions are formulated. For example, in te below form of velocity of te air solution: v' air := were, ( 10) R n1 λ air e 3 v' air ft = 66) sec m p f LM were, 10 = and, v air ft = 67) sec

15 1 Terefore, let: λ gallery := v air f Srad or, and λ gallery = m λ gallery = ft 68) 69) Te above result is extremely close to te distance from te first to te last "resonator" bank in te Grand Gallery. Te total lengt of te Grand Gallery is very close to feet. "See te below figure tat is from te book: "Te Giza Powerplant - Tecnologies of Ancient Egypt," by Cristoper Dunn, copyrigt 1998 by Cristoper Dunn, Publised by Bear & Company, p. 168, Figure. Te distance of 1.6 feet in te below figure 1 is arrived at by ratio and proportion wit a mecanical divider and a ruler. Figure #1 If we allocate a multiplier of two to eac resonator bank, te total amplification of 7 resonator banks is: A gain := 7 A gain = ) Te acoustic force acting on a proton at 7 degrees Fareneit and at sea level at te velocity of sound and at te wavelengt λ air above is: m p v air Fair p := λ air Fair p = newton 71) Te above force parameters are demonstrated below to ave an electrogravitational V LM connection as well as a connection to electron mass and te radius of te n1 Bor energy level of te H1 atom.

16 Te force relative to te least quantum magnetic force at te n1 energy sell of te ydrogen H1 atom is calculated next as: 15 m e V LM F Rn1 := F Rn1 = newton 7) R n1 Te ratio of te two forces above is: Fair p Ratio F1_ := or, Ratio F1_ F Rn1 = (Equal to exactly.) 73) Ten eac interaction of te acoustic (air) motion of te proton in te H1 Hydrogen atom wit its associated electron in and about tat atom will allow for an equalized force gain equal to exactly two times te least quantum magnetic force wic is also equal to one of te parameter elements of te total electrogravitational force interaction. See te below electrogravitational equation for clarification. Te acoustic wavelengt (λ air ) is te key to syncronizing te electron and proton force parameters for te energy boost sown above. Te electrogravitational force between two electrons at te R n1 radius is: F EG := F Rn1 µ o F Rn1 F EG = newton enry newton 7) m Tis provides te mecanism for amplification of te least quantum electrogravitational energy up troug te resonator banks to te desired level. It turns out tat te quantum energy result yields an energy tat as a frequency very close to te yperfine frequency of te H1 atom wic is ubiquitous in te universe. E LMgain := f LM A gain E LMgain = joule 75) Te quantum frequency associated wit te above energy is: E LMgain f LMgain := f LMgain = Hz 76) Te empirical value of te ydrogen yperfine frequency is: f H1 := Hz and f H1 f LMgain = Hz 77) Note: Please refer to my previous work at ttp://electrogravity.com/dualfreqeg/a_frequency.pdf, equation (9) on page 6, werein A' dbf = MHz. Te location of te Great Pyramid is of interest from te standpoint tat it was a carefully cosen site tat would arness te inertial as well as gravitational energy potential of te Eart. Te energy was ten imparted to te proton in te ydrogen H1 atom, quite possibly being freely available in te Grand Gallery. Te slope of te Grand Gallery may ave also played an important part in arriving at te most efficient energy capture. Te following parameters are necessary for te calculation of te related Eart energies as described above. Te radius of te Eart is: R E := m Te circumference of te Eart is: C E := π R E Te actual Grand Gallery slope as been measured as: slope GG := deg

17 Te latitude of te Great Pyramid is very close to 9 degrees, 58 minutes, 51 seconds Nort. Terefore, te longitudinal velocity of rotation may be calculated as sown below. Great Pyramid latitude: Lat := deg 16 C E V pyr := cos( Lat) or, V pyr ft = 78) 3600 sec sec Te inertial force acting on a proton of te H1 atom 90 degrees to te Eart's axis of rotation is calculated as: m p V pyr F r := or, F r = newton 79) R E cos( Lat) Te additional product of te cosine of te latitude (Lat) in te denominator is used to arrive at te actual radius of rotation wit respect to te ortogonal axis of rotation of te Eart. Next, te gravitational force acting on te proton is calculated relative to te radius of te Eart from te center of mass of te Eart. Te mass of te Eart is: M E := kg Ten te force of gravity at te radius of te Eart on a proton is: GM E m p F Gp := F Gp = newton 80) R E Te total vector energy is arrived at by multiplying te force times a distance of action wic is very close to te total distance of te Grand Gallery lengt wic is feet. First, te net vectored force related to te gravitational force is found. Te result is te force parallel to te disk of rotation at te latitude of te great Pyramid. F' Gp := F Gp e i Lat F' Gp = i 10 7 newton 81) F tot := F' Gp F r F tot = newton 8) Te slope of te Grand Gallery nearly coincides wit te plane of rotation at te latitude of te Great Pyramid. Te difference is taken into account to allow for a sligt adjustment of te force and ten te work or energy is arrived at by multiplying te net force by te adjusted working distance as sown below. F' tot := cos( Lat slope GG ) F tot F' tot = newton 83) E Gp := F' tot ( ft) E Gp = joule 8) Below is te yperfine H1 energy divided by te above net energy E Gp imparted to a proton in te Grand Gallery by te gravitational field of te Eart as well as te inertial energy in te plane of rotation. f H1 = were, E Gp π = arg f H1 E Gp = 0 deg Viola! 85)

18 Te /π ratio is te result and may be taken as significant since it is very close to being equal to te square root of te Golden Ratio. Te kinetic quantum energy of te proton in te H1 atom due to te rotational velocity of te Eart at 30 degrees latitude Nort is: 17 m p V pyr E r := E r = joule 86) E r Te quantum frequency is: f r := were, f r = Hz 87) Te energy of interaction is te energy in te field. Te average is obtained by dividing by. Te frequency is very close to te mean of te Cosmic Background Radiation frequency of 3 x Hz wic is associated wit te background energy left over after te Big Bang event. Next, te geograpically related rotation velocity of te proton is used to calculate te force on a proton at a given magnetic field strengt of te Eart ortogonal to te direction of te velocity related to te carge on te proton. It is desired to arrive at a wavelengt related to te energy of rotation E r above tat will yield a wavelengt equal to te electron's least quantum electrogravitational wavelengt. Let: B E E r ( sin( slope GG )) := tesla λ BE := λ BE = m 88) V pyr B E π And: Anoter way of calculating te velocity of te Great Pyramid is sown below. Te standard equation for te linear velocity of a particle experiencing angular rotation is: 1 V = π f r were: f = and T is time in seconds. T λ LM := ( Equals electrogravitational wavelengt.) λ LM = m 89) m e V LM Tis may cause a resonance of standing wave nature between te proton and te electron wit te quantum electrogravitational wavelengt λ LM as sown above. T rot := 3600 sec r T rot = sec (3600 seconds/our times ours in 1 day of rotation.) r Ten te velocity at te latitude of te Great Pyramid is: 91) (From above) π ( R E cos( Lat) ) V1 pyr := or, V1 pyr ft = V pyr ft = T rot sec sec Wat would be te cange in latitude to arrive at a cange in radius equal to te least quantum electrogravitational radius R LM? were, λ LM λ LM = m R LM := R LM = m 9) π 90)

19 18 Te statement tat addresses te above question is: ( ) ( R E cos( Lat) ) R LM = R E cos( Lat) 93) were Lat is te required latitude to produce te electrogravitational wavelengt along te plane of rotation. Utilizing Matcad's symbolic equation solver we solve for te unknown Lat as follows: ( ) ( R E cos( Lat) ) R LM = R E cos( Lat) Lat = deg as solution(s) ( ) R LM R E cos( Lat) R LM R E cos( Lat) π acos or, Lat := π acos 9) R E R E Lat = deg ceck: R E cos( Lat) ( ) ( R E cos( Lat) ) ( ) = m 95) Te cange in circumferential distance due to te cange in latitude is equal to π times te cange in radius R LM wic is equal to te least quantum electrogravitational wavelengt λ LM. ( ) ( R E cos( Lat) ) Ten: Cir := π R E cos( Lat) or, Cir = m 96) Were te least quantum electrogravitational wavelengt is: λ LM = m Distance equals velocity times time. T := Cir T = sec f := V pyr T f = Hz 97) π f = Hz 98) Note: f α π = Hz (See eq. 63 above.) 99) Te ultrasonic frequency f may exist in matter suc as granite and marble. Tis would explain te use of te 38.3 Hz frequency above in te Grand Gallery. Finding te rigt frequency of acoustic and electrical pulsation may syncronize te pulsating energy coming from te Eart so tat over time te energy would build to a ig level suitable for levitation and energy extraction by resonator mecanisms. For instance, te magnetic energy coming from te Eart may be considered as not moving wit te rotation of te Eart. Terefore, an object moving wit te surface of te Eart would be cutting te magnetic lines of flux continually. Also, adjusting an acoustic frequency of pulsation properly will allow for resonance wit te pulsations of te Eart so tat as te object rotates around a circle of te Eart from East to West, te bands of magnetic and acoustic pulsation would syncronize wit te pulsations of te object suc as te Grand Gallery Hydrogen atoms and energy would continue to build in te Hydrogen gas as a result. Tis principle would also work in te case of te famous 'stone levitators' werein levitation is accomplised by acoustic means suc as drums and gongs played near te stone at a certain beat rate. See: ttp://www3.sympatico.ca/slavek.krepelka/ideas/stolev.tm ttp:// "Stone Levitation" "Tibetan Sound Levitation Of Large Stones Witnessed By Scientist"

20 19) Tere are also lines of flux corresponding to so called 'Ley lines" tat can be detected by some people. Next, we examine te corrected air velocity relative to te sligt difference of te slope of te Grand Gallery wit te latitude of te Great Pyramid and find te vector slope in degrees relative te velocity of te pyramid wit te corrected air velocity in te Grand Gallery. Tis will lead to a calculation of wen te Great Pyramid at Giza may ave been built. Lat = deg slope GG = deg angle := Lat slope GG angle = deg 100) V pyr ft = v air ft = sec sec v air := cos( angle) vair v air ft = 101) sec atan V pyr v air = deg atan π = deg 10) It may be tat te angles above agreed in te beginning wen te Great Pyramid was first built. It terefore may be possible tat wen te Great Pyramid was built, te velocity of te Eart was faster resulting in a sorter period of rotation. Using Matcad's symbolic processor, we can solve for te old or original Eart rotation velocity at te latitude of te Great Pyramid tat will yield te angle equal to te atan of /π. atan V pyr v air = atan Solving for V pyr as solution(s) 103) π π v air or, V pyrold := π v air V pyrold ft = 10) sec Ten te actual seconds per rotation of te Eart wen te Great Pyramid may ave been built is: V pyr 3600 sec r = sec were, V pyrold r V pyr = ) V pyrold Te seconds per rotation now (year 008) is: 3600 sec r r = sec 106) Te total time te Eart as slowed since te Great Pyramid was built is: T total 3600 sec r V pyr := 3600 sec r T total = sec 107) r V pyrold r T total = r Define: cen := 100 yr cen = sec 108)

21 0) Te Eart is measured by scientists of today as slowing an average of seconds per century. Extrapolating backwards in time, te age of te Great Pyramid is calculated as: sec = sec ( n cen) cen were n is te number of centuries. 109) Solving for (n) as te number of centuries from te above equation: sec = sec n cen cen = Solving for n: = total number of centuries. 110) 111) cen = yr = million years. 11) It is suggested by te above result tat te Great Pyramid may ave been used as a power generator to cause global terraforming. Cristoper Dunn, in is notable book, "Te Great Powerplant At Giza", suggested tat microwave energy was beamed upwards into space troug waveguides sloaping upwards from te King's Camber. Te microwaves were possibly generated via a maser process involving resonance wit te Hyperfine frequency of te ydrogen atoms wic were cemically generated in te Queen's Camber and fed upwards to te King's Camber via te Grand Gallery sown above. Te same may be said of te pyramids on Mars. Tis is not unreasonable since te age of te Eart is measured as about to.5 billion years old. Peraps terraforming killed te dinosaurs in te process. Equations 71, 7 and 73 from above ave a correlation wit te results of one of my previous papers. In tat paper, equations 9 and 10 on page of, "Te Derivation Of Te Least Quantum Electrogravitational Frequency", werein equation 9 yielded a quantum frequency equal to exactly twice tat of equation 10. Tese frequencies ave an energy equivalent by E = f.. See: ttp:// Eq. 9: d V e t e λ e f LMneg = dλ e E v_m α new f LMneg := i Hz 113) Quote: "Wat does te above formula mean? It is stating tat an extremely low spin related frequency is generated by te cange of one Compton wavelengt in obtaining te cubic form of te magnetic field related to te electron Compton area in te numerator. Tis frequency is furter brougt about by te transfer mecanism of te special case fine structure constant in (meter per second ) terms. Tat is, wen te plane area canges to a volume in te magnetic field. Te positive result for f LM is stated below in terms of a cange of one Compton wavelengt related to te E field of te electron. Te derivative wit respect to one Compton wavelengt yields a potential witout te reference "per meter" since te derivative raises te power to zero of te wavelengt." --Unquote. Eq. 10: d T vsec_m f LMpos = dλ e 1 V e λ e α new f LMpos := i Hz 11)

APPENDIXES. Let the following constants be established for those using the active Mathcad

APPENDIXES. Let the following constants be established for those using the active Mathcad 3 APPENDIXES Let te following constants be establised for tose using te active Matcad form of tis book: m.. e 9.09389700 0 3 kg Electron rest mass. q.. o.6077330 0 9 coul Electron quantum carge. µ... o.5663706

More information

A Reconsideration of Matter Waves

A Reconsideration of Matter Waves A Reconsideration of Matter Waves by Roger Ellman Abstract Matter waves were discovered in te early 20t century from teir wavelengt, predicted by DeBroglie, Planck's constant divided by te particle's momentum,

More information

Problem Solving. Problem Solving Process

Problem Solving. Problem Solving Process Problem Solving One of te primary tasks for engineers is often solving problems. It is wat tey are, or sould be, good at. Solving engineering problems requires more tan just learning new terms, ideas and

More information

The structure of the atoms

The structure of the atoms Te structure of te atoms Atomos = indivisible University of Pécs, Medical Scool, Dept. Biopysics All tat exists are atoms and empty space; everyting else is merely tougt to exist. Democritus, 415 B.C.

More information

A = h w (1) Error Analysis Physics 141

A = h w (1) Error Analysis Physics 141 Introduction In all brances of pysical science and engineering one deals constantly wit numbers wic results more or less directly from experimental observations. Experimental observations always ave inaccuracies.

More information

Work and Energy. Introduction. Work. PHY energy - J. Hedberg

Work and Energy. Introduction. Work. PHY energy - J. Hedberg Work and Energy PHY 207 - energy - J. Hedberg - 2017 1. Introduction 2. Work 3. Kinetic Energy 4. Potential Energy 5. Conservation of Mecanical Energy 6. Ex: Te Loop te Loop 7. Conservative and Non-conservative

More information

M12/4/PHYSI/HPM/ENG/TZ1/XX. Physics Higher level Paper 1. Thursday 10 May 2012 (afternoon) 1 hour INSTRUCTIONS TO CANDIDATES

M12/4/PHYSI/HPM/ENG/TZ1/XX. Physics Higher level Paper 1. Thursday 10 May 2012 (afternoon) 1 hour INSTRUCTIONS TO CANDIDATES M12/4/PHYSI/HPM/ENG/TZ1/XX 22126507 Pysics Higer level Paper 1 Tursday 10 May 2012 (afternoon) 1 our INSTRUCTIONS TO CANDIDATES Do not open tis examination paper until instructed to do so. Answer all te

More information

Notes on wavefunctions II: momentum wavefunctions

Notes on wavefunctions II: momentum wavefunctions Notes on wavefunctions II: momentum wavefunctions and uncertainty Te state of a particle at any time is described by a wavefunction ψ(x). Tese wavefunction must cange wit time, since we know tat particles

More information

Extracting Atomic and Molecular Parameters From the de Broglie Bohr Model of the Atom

Extracting Atomic and Molecular Parameters From the de Broglie Bohr Model of the Atom Extracting Atomic and Molecular Parameters From te de Broglie Bor Model of te Atom Frank ioux Te 93 Bor model of te ydrogen atom was replaced by Scrödingerʹs wave mecanical model in 96. However, Borʹs

More information

1 Power is transferred through a machine as shown. power input P I machine. power output P O. power loss P L. What is the efficiency of the machine?

1 Power is transferred through a machine as shown. power input P I machine. power output P O. power loss P L. What is the efficiency of the machine? 1 1 Power is transferred troug a macine as sown. power input P I macine power output P O power loss P L Wat is te efficiency of te macine? P I P L P P P O + P L I O P L P O P I 2 ir in a bicycle pump is

More information

1watt=1W=1kg m 2 /s 3

1watt=1W=1kg m 2 /s 3 Appendix A Matematics Appendix A.1 Units To measure a pysical quantity, you need a standard. Eac pysical quantity as certain units. A unit is just a standard we use to compare, e.g. a ruler. In tis laboratory

More information

Quantum Numbers and Rules

Quantum Numbers and Rules OpenStax-CNX module: m42614 1 Quantum Numbers and Rules OpenStax College Tis work is produced by OpenStax-CNX and licensed under te Creative Commons Attribution License 3.0 Abstract Dene quantum number.

More information

Derivation Of The Schwarzschild Radius Without General Relativity

Derivation Of The Schwarzschild Radius Without General Relativity Derivation Of Te Scwarzscild Radius Witout General Relativity In tis paper I present an alternative metod of deriving te Scwarzscild radius of a black ole. Te metod uses tree of te Planck units formulas:

More information

HOMEWORK HELP 2 FOR MATH 151

HOMEWORK HELP 2 FOR MATH 151 HOMEWORK HELP 2 FOR MATH 151 Here we go; te second round of omework elp. If tere are oters you would like to see, let me know! 2.4, 43 and 44 At wat points are te functions f(x) and g(x) = xf(x)continuous,

More information

Derivatives. By: OpenStaxCollege

Derivatives. By: OpenStaxCollege By: OpenStaxCollege Te average teen in te United States opens a refrigerator door an estimated 25 times per day. Supposedly, tis average is up from 10 years ago wen te average teenager opened a refrigerator

More information

Part C : Quantum Physics

Part C : Quantum Physics Part C : Quantum Pysics 1 Particle-wave duality 1.1 Te Bor model for te atom We begin our discussion of quantum pysics by discussing an early idea for atomic structure, te Bor model. Wile tis relies on

More information

Graviton Induced Nuclear Fission through Electromagnetic Wave Flux Phil Russell, * Jerry Montgomery

Graviton Induced Nuclear Fission through Electromagnetic Wave Flux Phil Russell, * Jerry Montgomery Graviton Induced Nuclear Fission troug Electromagnetic Wave Flux Pil Russell, * Jerry Montgomery Nort Carolina Central University, Duram, NC 27707 Willowstick Tecnologies LLC, Draper, UT 84020 (Dated:

More information

Problem Set 4 Solutions

Problem Set 4 Solutions University of Alabama Department of Pysics and Astronomy PH 253 / LeClair Spring 2010 Problem Set 4 Solutions 1. Group velocity of a wave. For a free relativistic quantum particle moving wit speed v, te

More information

The derivative function

The derivative function Roberto s Notes on Differential Calculus Capter : Definition of derivative Section Te derivative function Wat you need to know already: f is at a point on its grap and ow to compute it. Wat te derivative

More information

nucleus orbital electron wave 2/27/2008 Quantum ( F.Robilliard) 1

nucleus orbital electron wave 2/27/2008 Quantum ( F.Robilliard) 1 r nucleus orbital electron wave λ /7/008 Quantum ( F.Robilliard) 1 Wat is a Quantum? A quantum is a discrete amount of some quantity. For example, an atom is a mass quantum of a cemical element te mass

More information

Practice Problem Solutions: Exam 1

Practice Problem Solutions: Exam 1 Practice Problem Solutions: Exam 1 1. (a) Algebraic Solution: Te largest term in te numerator is 3x 2, wile te largest term in te denominator is 5x 2 3x 2 + 5. Tus lim x 5x 2 2x 3x 2 x 5x 2 = 3 5 Numerical

More information

Chemistry. Slide 1 / 63 Slide 2 / 63. Slide 4 / 63. Slide 3 / 63. Slide 6 / 63. Slide 5 / 63. Optional Review Light and Matter.

Chemistry. Slide 1 / 63 Slide 2 / 63. Slide 4 / 63. Slide 3 / 63. Slide 6 / 63. Slide 5 / 63. Optional Review Light and Matter. Slide 1 / 63 Slide 2 / 63 emistry Optional Review Ligt and Matter 2015-10-27 www.njctl.org Slide 3 / 63 Slide 4 / 63 Ligt and Sound Ligt and Sound In 1905 Einstein derived an equation relating mass and

More information

Why gravity is not an entropic force

Why gravity is not an entropic force Wy gravity is not an entropic force San Gao Unit for History and Pilosopy of Science & Centre for Time, SOPHI, University of Sydney Email: sgao7319@uni.sydney.edu.au Te remarkable connections between gravity

More information

Problem Set 4: Whither, thou turbid wave SOLUTIONS

Problem Set 4: Whither, thou turbid wave SOLUTIONS PH 253 / LeClair Spring 2013 Problem Set 4: Witer, tou turbid wave SOLUTIONS Question zero is probably were te name of te problem set came from: Witer, tou turbid wave? It is from a Longfellow poem, Te

More information

Time (hours) Morphine sulfate (mg)

Time (hours) Morphine sulfate (mg) Mat Xa Fall 2002 Review Notes Limits and Definition of Derivative Important Information: 1 According to te most recent information from te Registrar, te Xa final exam will be eld from 9:15 am to 12:15

More information

MAT 145. Type of Calculator Used TI-89 Titanium 100 points Score 100 possible points

MAT 145. Type of Calculator Used TI-89 Titanium 100 points Score 100 possible points MAT 15 Test #2 Name Solution Guide Type of Calculator Used TI-89 Titanium 100 points Score 100 possible points Use te grap of a function sown ere as you respond to questions 1 to 8. 1. lim f (x) 0 2. lim

More information

2.8 The Derivative as a Function

2.8 The Derivative as a Function .8 Te Derivative as a Function Typically, we can find te derivative of a function f at many points of its domain: Definition. Suppose tat f is a function wic is differentiable at every point of an open

More information

The Electron in a Potential

The Electron in a Potential Te Electron in a Potential Edwin F. Taylor July, 2000 1. Stopwatc rotation for an electron in a potential For a poton we found tat te and of te quantum stopwatc rotates wit frequency f given by te equation:

More information

Combining functions: algebraic methods

Combining functions: algebraic methods Combining functions: algebraic metods Functions can be added, subtracted, multiplied, divided, and raised to a power, just like numbers or algebra expressions. If f(x) = x 2 and g(x) = x + 2, clearly f(x)

More information

Numerical Differentiation

Numerical Differentiation Numerical Differentiation Finite Difference Formulas for te first derivative (Using Taylor Expansion tecnique) (section 8.3.) Suppose tat f() = g() is a function of te variable, and tat as 0 te function

More information

INTRODUCTION AND MATHEMATICAL CONCEPTS

INTRODUCTION AND MATHEMATICAL CONCEPTS INTODUCTION ND MTHEMTICL CONCEPTS PEVIEW Tis capter introduces you to te basic matematical tools for doing pysics. You will study units and converting between units, te trigonometric relationsips of sine,

More information

, meant to remind us of the definition of f (x) as the limit of difference quotients: = lim

, meant to remind us of the definition of f (x) as the limit of difference quotients: = lim Mat 132 Differentiation Formulas Stewart 2.3 So far, we ave seen ow various real-world problems rate of cange and geometric problems tangent lines lead to derivatives. In tis section, we will see ow to

More information

Math 102 TEST CHAPTERS 3 & 4 Solutions & Comments Fall 2006

Math 102 TEST CHAPTERS 3 & 4 Solutions & Comments Fall 2006 Mat 102 TEST CHAPTERS 3 & 4 Solutions & Comments Fall 2006 f(x+) f(x) 10 1. For f(x) = x 2 + 2x 5, find ))))))))) and simplify completely. NOTE: **f(x+) is NOT f(x)+! f(x+) f(x) (x+) 2 + 2(x+) 5 ( x 2

More information

CHAPTER 4 QUANTUM PHYSICS

CHAPTER 4 QUANTUM PHYSICS CHAPTER 4 QUANTUM PHYSICS INTRODUCTION Newton s corpuscular teory of ligt fails to explain te penomena like interference, diffraction, polarization etc. Te wave teory of ligt wic was proposed by Huygen

More information

INTRODUCTION AND MATHEMATICAL CONCEPTS

INTRODUCTION AND MATHEMATICAL CONCEPTS Capter 1 INTRODUCTION ND MTHEMTICL CONCEPTS PREVIEW Tis capter introduces you to te basic matematical tools for doing pysics. You will study units and converting between units, te trigonometric relationsips

More information

Grade: 11 International Physics Olympiad Qualifier Set: 2

Grade: 11 International Physics Olympiad Qualifier Set: 2 Grade: 11 International Pysics Olympiad Qualifier Set: 2 --------------------------------------------------------------------------------------------------------------- Max Marks: 60 Test ID: 12111 Time

More information

Material for Difference Quotient

Material for Difference Quotient Material for Difference Quotient Prepared by Stepanie Quintal, graduate student and Marvin Stick, professor Dept. of Matematical Sciences, UMass Lowell Summer 05 Preface Te following difference quotient

More information

Introduction. Learning Objectives. On completion of this chapter you will be able to:

Introduction. Learning Objectives. On completion of this chapter you will be able to: Introduction Learning Objectives On completion of tis capter you will be able to: 1. Define Compton Effect. 2. Derive te sift in incident ligt wavelengt and Compton wavelengt. 3. Explain ow te Compton

More information

SECTION 1.10: DIFFERENCE QUOTIENTS LEARNING OBJECTIVES

SECTION 1.10: DIFFERENCE QUOTIENTS LEARNING OBJECTIVES (Section.0: Difference Quotients).0. SECTION.0: DIFFERENCE QUOTIENTS LEARNING OBJECTIVES Define average rate of cange (and average velocity) algebraically and grapically. Be able to identify, construct,

More information

Quantum Mechanics Chapter 1.5: An illustration using measurements of particle spin.

Quantum Mechanics Chapter 1.5: An illustration using measurements of particle spin. I Introduction. Quantum Mecanics Capter.5: An illustration using measurements of particle spin. Quantum mecanics is a teory of pysics tat as been very successful in explaining and predicting many pysical

More information

Finding and Using Derivative The shortcuts

Finding and Using Derivative The shortcuts Calculus 1 Lia Vas Finding and Using Derivative Te sortcuts We ave seen tat te formula f f(x+) f(x) (x) = lim 0 is manageable for relatively simple functions like a linear or quadratic. For more complex

More information

1 The concept of limits (p.217 p.229, p.242 p.249, p.255 p.256) 1.1 Limits Consider the function determined by the formula 3. x since at this point

1 The concept of limits (p.217 p.229, p.242 p.249, p.255 p.256) 1.1 Limits Consider the function determined by the formula 3. x since at this point MA00 Capter 6 Calculus and Basic Linear Algebra I Limits, Continuity and Differentiability Te concept of its (p.7 p.9, p.4 p.49, p.55 p.56). Limits Consider te function determined by te formula f Note

More information

UNIVERSITY OF SASKATCHEWAN Department of Physics and Engineering Physics

UNIVERSITY OF SASKATCHEWAN Department of Physics and Engineering Physics UNIVERSITY O SASKATCHEWAN Department of Pysics and Engineering Pysics Pysics 117.3 MIDTERM EXAM Regular Sitting NAME: (Last) Please Print (Given) Time: 90 minutes STUDENT NO.: LECTURE SECTION (please ceck):

More information

6.4: THE WAVE BEHAVIOR OF MATTER

6.4: THE WAVE BEHAVIOR OF MATTER 6.4: THE WAVE BEHAVIOR OF MATTER SKILLS TO DEVELOP To understand te wave particle duality of matter. Einstein s potons of ligt were individual packets of energy aving many of te caracteristics of particles.

More information

PHYSICS 1050 Mid-term Test 1 University of Wyoming 15 February 2005

PHYSICS 1050 Mid-term Test 1 University of Wyoming 15 February 2005 Name: (4 points) PHYSICS 1050 Mid-term Test 1 University of Wyoming 15 February 2005 Tis test is open-note and open-book. Tis means tat any reference material is permitted during te test. Calculators also

More information

Exam 1 Review Solutions

Exam 1 Review Solutions Exam Review Solutions Please also review te old quizzes, and be sure tat you understand te omework problems. General notes: () Always give an algebraic reason for your answer (graps are not sufficient),

More information

Homework 1. L φ = mωr 2 = mυr, (1)

Homework 1. L φ = mωr 2 = mυr, (1) Homework 1 1. Problem: Streetman, Sixt Ed., Problem 2.2: Sow tat te tird Bor postulate, Eq. (2-5) (tat is, tat te angular momentum p θ around te polar axis is an integer multiple of te reduced Planck constant,

More information

2.11 That s So Derivative

2.11 That s So Derivative 2.11 Tat s So Derivative Introduction to Differential Calculus Just as one defines instantaneous velocity in terms of average velocity, we now define te instantaneous rate of cange of a function at a point

More information

Pre-Calculus Review Preemptive Strike

Pre-Calculus Review Preemptive Strike Pre-Calculus Review Preemptive Strike Attaced are some notes and one assignment wit tree parts. Tese are due on te day tat we start te pre-calculus review. I strongly suggest reading troug te notes torougly

More information

Krazy Katt, the mechanical cat

Krazy Katt, the mechanical cat Krazy Katt, te mecanical cat Te cat rigting relex is a cat's innate ability to orient itsel as it alls in order to land on its eet. Te rigting relex begins to appear at 3 4 weeks o age, and is perected

More information

Quantum Theory of the Atomic Nucleus

Quantum Theory of the Atomic Nucleus G. Gamow, ZP, 51, 204 1928 Quantum Teory of te tomic Nucleus G. Gamow (Received 1928) It as often been suggested tat non Coulomb attractive forces play a very important role inside atomic nuclei. We can

More information

Pre-lab Quiz/PHYS 224 Earth s Magnetic Field. Your name Lab section

Pre-lab Quiz/PHYS 224 Earth s Magnetic Field. Your name Lab section Pre-lab Quiz/PHYS 4 Eart s Magnetic Field Your name Lab section 1. Wat do you investigate in tis lab?. For a pair of Helmoltz coils described in tis manual and sown in Figure, r=.15 m, N=13, I =.4 A, wat

More information

5.1 We will begin this section with the definition of a rational expression. We

5.1 We will begin this section with the definition of a rational expression. We Basic Properties and Reducing to Lowest Terms 5.1 We will begin tis section wit te definition of a rational epression. We will ten state te two basic properties associated wit rational epressions and go

More information

MATH 1020 Answer Key TEST 2 VERSION B Fall Printed Name: Section #: Instructor:

MATH 1020 Answer Key TEST 2 VERSION B Fall Printed Name: Section #: Instructor: Printed Name: Section #: Instructor: Please do not ask questions during tis exam. If you consider a question to be ambiguous, state your assumptions in te margin and do te best you can to provide te correct

More information

How to Find the Derivative of a Function: Calculus 1

How to Find the Derivative of a Function: Calculus 1 Introduction How to Find te Derivative of a Function: Calculus 1 Calculus is not an easy matematics course Te fact tat you ave enrolled in suc a difficult subject indicates tat you are interested in te

More information

Printed Name: Section #: Instructor:

Printed Name: Section #: Instructor: Printed Name: Section #: Instructor: Please do not ask questions during tis exam. If you consider a question to be ambiguous, state your assumptions in te margin and do te best you can to provide te correct

More information

2.1 THE DEFINITION OF DERIVATIVE

2.1 THE DEFINITION OF DERIVATIVE 2.1 Te Derivative Contemporary Calculus 2.1 THE DEFINITION OF DERIVATIVE 1 Te grapical idea of a slope of a tangent line is very useful, but for some uses we need a more algebraic definition of te derivative

More information

Test on Nuclear Physics

Test on Nuclear Physics Test on Nuclear Pysics Examination Time - 40 minutes Answer all questions in te spaces provided Tis wole test involves an imaginary element called Bedlum wic as te isotope notation sown below: 47 11 Bd

More information

Continuity and Differentiability Worksheet

Continuity and Differentiability Worksheet Continuity and Differentiability Workseet (Be sure tat you can also do te grapical eercises from te tet- Tese were not included below! Typical problems are like problems -3, p. 6; -3, p. 7; 33-34, p. 7;

More information

Introduction to Derivatives

Introduction to Derivatives Introduction to Derivatives 5-Minute Review: Instantaneous Rates and Tangent Slope Recall te analogy tat we developed earlier First we saw tat te secant slope of te line troug te two points (a, f (a))

More information

Week #15 - Word Problems & Differential Equations Section 8.2

Week #15 - Word Problems & Differential Equations Section 8.2 Week #1 - Word Problems & Differential Equations Section 8. From Calculus, Single Variable by Huges-Hallett, Gleason, McCallum et. al. Copyrigt 00 by Jon Wiley & Sons, Inc. Tis material is used by permission

More information

Exercises Copyright Houghton Mifflin Company. All rights reserved. EXERCISES {x 0 x < 6} 3. {x x 2} 2

Exercises Copyright Houghton Mifflin Company. All rights reserved. EXERCISES {x 0 x < 6} 3. {x x 2} 2 Eercises. CHAPTER Functions EXERCISES.. { 0 < 6}. a. Since and m, ten y, te cange in y, is y m. { } 7. For (, ) and (, ), te slope is Since and m, ten y, te cange in y, is y m 0 9. For (, 0) and (, ),

More information

Sin, Cos and All That

Sin, Cos and All That Sin, Cos and All Tat James K. Peterson Department of Biological Sciences and Department of Matematical Sciences Clemson University Marc 9, 2017 Outline Sin, Cos and all tat! A New Power Rule Derivatives

More information

Solutions Manual for Precalculus An Investigation of Functions

Solutions Manual for Precalculus An Investigation of Functions Solutions Manual for Precalculus An Investigation of Functions David Lippman, Melonie Rasmussen 1 st Edition Solutions created at Te Evergreen State College and Soreline Community College 1.1 Solutions

More information

Exercises for numerical differentiation. Øyvind Ryan

Exercises for numerical differentiation. Øyvind Ryan Exercises for numerical differentiation Øyvind Ryan February 25, 2013 1. Mark eac of te following statements as true or false. a. Wen we use te approximation f (a) (f (a +) f (a))/ on a computer, we can

More information

4. The slope of the line 2x 7y = 8 is (a) 2/7 (b) 7/2 (c) 2 (d) 2/7 (e) None of these.

4. The slope of the line 2x 7y = 8 is (a) 2/7 (b) 7/2 (c) 2 (d) 2/7 (e) None of these. Mat 11. Test Form N Fall 016 Name. Instructions. Te first eleven problems are wort points eac. Te last six problems are wort 5 points eac. For te last six problems, you must use relevant metods of algebra

More information

Cubic Functions: Local Analysis

Cubic Functions: Local Analysis Cubic function cubing coefficient Capter 13 Cubic Functions: Local Analysis Input-Output Pairs, 378 Normalized Input-Output Rule, 380 Local I-O Rule Near, 382 Local Grap Near, 384 Types of Local Graps

More information

Some Review Problems for First Midterm Mathematics 1300, Calculus 1

Some Review Problems for First Midterm Mathematics 1300, Calculus 1 Some Review Problems for First Midterm Matematics 00, Calculus. Consider te trigonometric function f(t) wose grap is sown below. Write down a possible formula for f(t). Tis function appears to be an odd,

More information

Math 262 Exam 1 - Practice Problems. 1. Find the area between the given curves:

Math 262 Exam 1 - Practice Problems. 1. Find the area between the given curves: Mat 6 Exam - Practice Problems. Find te area between te given curves: (a) = x + and = x First notice tat tese curves intersect wen x + = x, or wen x x+ =. Tat is, wen (x )(x ) =, or wen x = and x =. Next,

More information

Average Rate of Change

Average Rate of Change Te Derivative Tis can be tougt of as an attempt to draw a parallel (pysically and metaporically) between a line and a curve, applying te concept of slope to someting tat isn't actually straigt. Te slope

More information

Analysis: The speed of the proton is much less than light speed, so we can use the

Analysis: The speed of the proton is much less than light speed, so we can use the Section 1.3: Wave Proerties of Classical Particles Tutorial 1 Practice, age 634 1. Given: 1.8! 10 "5 kg # m/s; 6.63! 10 "34 J #s Analysis: Use te de Broglie relation, λ. Solution:! 6.63 " 10#34 kg $ m

More information

Consider a function f we ll specify which assumptions we need to make about it in a minute. Let us reformulate the integral. 1 f(x) dx.

Consider a function f we ll specify which assumptions we need to make about it in a minute. Let us reformulate the integral. 1 f(x) dx. Capter 2 Integrals as sums and derivatives as differences We now switc to te simplest metods for integrating or differentiating a function from its function samples. A careful study of Taylor expansions

More information

Quaternion Dynamics, Part 1 Functions, Derivatives, and Integrals. Gary D. Simpson. rev 01 Aug 08, 2016.

Quaternion Dynamics, Part 1 Functions, Derivatives, and Integrals. Gary D. Simpson. rev 01 Aug 08, 2016. Quaternion Dynamics, Part 1 Functions, Derivatives, and Integrals Gary D. Simpson gsim1887@aol.com rev 1 Aug 8, 216 Summary Definitions are presented for "quaternion functions" of a quaternion. Polynomial

More information

Part 2: Introduction to Open-Channel Flow SPRING 2005

Part 2: Introduction to Open-Channel Flow SPRING 2005 Part : Introduction to Open-Cannel Flow SPRING 005. Te Froude number. Total ead and specific energy 3. Hydraulic jump. Te Froude Number Te main caracteristics of flows in open cannels are tat: tere is

More information

Chapter 2 Limits and Continuity

Chapter 2 Limits and Continuity 4 Section. Capter Limits and Continuity Section. Rates of Cange and Limits (pp. 6) Quick Review.. f () ( ) () 4 0. f () 4( ) 4. f () sin sin 0 4. f (). 4 4 4 6. c c c 7. 8. c d d c d d c d c 9. 8 ( )(

More information

SECTION 3.2: DERIVATIVE FUNCTIONS and DIFFERENTIABILITY

SECTION 3.2: DERIVATIVE FUNCTIONS and DIFFERENTIABILITY (Section 3.2: Derivative Functions and Differentiability) 3.2.1 SECTION 3.2: DERIVATIVE FUNCTIONS and DIFFERENTIABILITY LEARNING OBJECTIVES Know, understand, and apply te Limit Definition of te Derivative

More information

Notes: Most of the material in this chapter is taken from Young and Freedman, Chap. 12.

Notes: Most of the material in this chapter is taken from Young and Freedman, Chap. 12. Capter 6. Fluid Mecanics Notes: Most of te material in tis capter is taken from Young and Freedman, Cap. 12. 6.1 Fluid Statics Fluids, i.e., substances tat can flow, are te subjects of tis capter. But

More information

Differential Calculus (The basics) Prepared by Mr. C. Hull

Differential Calculus (The basics) Prepared by Mr. C. Hull Differential Calculus Te basics) A : Limits In tis work on limits, we will deal only wit functions i.e. tose relationsips in wic an input variable ) defines a unique output variable y). Wen we work wit

More information

Preview from Notesale.co.uk Page 2 of 42

Preview from Notesale.co.uk Page 2 of 42 1 PHYSICAL CHEMISTRY Dalton (1805) Tomson (1896) - Positive and negative carges Ruterford (1909) - Te Nucleus Bor (1913) - Energy levels Atomic Model : Timeline CATHODE RAYS THE DISCOVERY OF ELECTRON Scrödinger

More information

Phy 231 Sp 02 Homework #6 Page 1 of 4

Phy 231 Sp 02 Homework #6 Page 1 of 4 Py 231 Sp 02 Homework #6 Page 1 of 4 6-1A. Te force sown in te force-time diagram at te rigt versus time acts on a 2 kg mass. Wat is te impulse of te force on te mass from 0 to 5 sec? (a) 9 N-s (b) 6 N-s

More information

1. Consider the trigonometric function f(t) whose graph is shown below. Write down a possible formula for f(t).

1. Consider the trigonometric function f(t) whose graph is shown below. Write down a possible formula for f(t). . Consider te trigonometric function f(t) wose grap is sown below. Write down a possible formula for f(t). Tis function appears to be an odd, periodic function tat as been sifted upwards, so we will use

More information

Quantization of electrical conductance

Quantization of electrical conductance 1 Introduction Quantization of electrical conductance Te resistance of a wire in te classical Drude model of metal conduction is given by RR = ρρρρ AA, were ρρ, AA and ll are te conductivity of te material,

More information

3.4 Worksheet: Proof of the Chain Rule NAME

3.4 Worksheet: Proof of the Chain Rule NAME Mat 1170 3.4 Workseet: Proof of te Cain Rule NAME Te Cain Rule So far we are able to differentiate all types of functions. For example: polynomials, rational, root, and trigonometric functions. We are

More information

CHAPTER (A) When x = 2, y = 6, so f( 2) = 6. (B) When y = 4, x can equal 6, 2, or 4.

CHAPTER (A) When x = 2, y = 6, so f( 2) = 6. (B) When y = 4, x can equal 6, 2, or 4. SECTION 3-1 101 CHAPTER 3 Section 3-1 1. No. A correspondence between two sets is a function only if eactly one element of te second set corresponds to eac element of te first set. 3. Te domain of a function

More information

Higher Derivatives. Differentiable Functions

Higher Derivatives. Differentiable Functions Calculus 1 Lia Vas Higer Derivatives. Differentiable Functions Te second derivative. Te derivative itself can be considered as a function. Te instantaneous rate of cange of tis function is te second derivative.

More information

(a) At what number x = a does f have a removable discontinuity? What value f(a) should be assigned to f at x = a in order to make f continuous at a?

(a) At what number x = a does f have a removable discontinuity? What value f(a) should be assigned to f at x = a in order to make f continuous at a? Solutions to Test 1 Fall 016 1pt 1. Te grap of a function f(x) is sown at rigt below. Part I. State te value of eac limit. If a limit is infinite, state weter it is or. If a limit does not exist (but is

More information

2.3 More Differentiation Patterns

2.3 More Differentiation Patterns 144 te derivative 2.3 More Differentiation Patterns Polynomials are very useful, but tey are not te only functions we need. Tis section uses te ideas of te two previous sections to develop tecniques for

More information

E p = mgh (if h i=0) E k = ½ mv 2 Ek is measured in Joules (J); m is measured in kg; v is measured in m/s. Energy Continued (E)

E p = mgh (if h i=0) E k = ½ mv 2 Ek is measured in Joules (J); m is measured in kg; v is measured in m/s. Energy Continued (E) nergy Continued () Gravitational Potential nergy: - e energy stored in an object due to its distance above te surface of te art. - e energy stored depends on te mass of te object, te eigt above te surface,

More information

MATH Fall 08. y f(x) Review Problems for the Midterm Examination Covers [1.1, 4.3] in Stewart

MATH Fall 08. y f(x) Review Problems for the Midterm Examination Covers [1.1, 4.3] in Stewart MATH 121 - Fall 08 Review Problems for te Midterm Eamination Covers [1.1, 4.3] in Stewart 1. (a) Use te definition of te derivative to find f (3) wen f() = π 1 2. (b) Find an equation of te tangent line

More information

Quantum Mechanics and Atomic Theory

Quantum Mechanics and Atomic Theory A. Electromagnetic Radiation Quantum Mecanics and Atomic Teory 1. Ligt: consists of waves of oscillating electric field ( E ) and magnetic field ( B ) tat are perpendicular to eac oter and to te direction

More information

Function Composition and Chain Rules

Function Composition and Chain Rules Function Composition and s James K. Peterson Department of Biological Sciences and Department of Matematical Sciences Clemson University Marc 8, 2017 Outline 1 Function Composition and Continuity 2 Function

More information

Outline. MS121: IT Mathematics. Limits & Continuity Rates of Change & Tangents. Is there a limit to how fast a man can run?

Outline. MS121: IT Mathematics. Limits & Continuity Rates of Change & Tangents. Is there a limit to how fast a man can run? Outline MS11: IT Matematics Limits & Continuity & 1 Limits: Atletics Perspective Jon Carroll Scool of Matematical Sciences Dublin City University 3 Atletics Atletics Outline Is tere a limit to ow fast

More information

Lecture: Experimental Solid State Physics Today s Outline

Lecture: Experimental Solid State Physics Today s Outline Lecture: Experimental Solid State Pysics Today s Outline Te quantum caracter of particles : Wave-Particles dualism Heisenberg s uncertainty relation Te quantum structure of electrons in atoms Wave-particle

More information

The Verlet Algorithm for Molecular Dynamics Simulations

The Verlet Algorithm for Molecular Dynamics Simulations Cemistry 380.37 Fall 2015 Dr. Jean M. Standard November 9, 2015 Te Verlet Algoritm for Molecular Dynamics Simulations Equations of motion For a many-body system consisting of N particles, Newton's classical

More information

232 Calculus and Structures

232 Calculus and Structures 3 Calculus and Structures CHAPTER 17 JUSTIFICATION OF THE AREA AND SLOPE METHODS FOR EVALUATING BEAMS Calculus and Structures 33 Copyrigt Capter 17 JUSTIFICATION OF THE AREA AND SLOPE METHODS 17.1 THE

More information

Excerpt from "Calculus" 2013 AoPS Inc.

Excerpt from Calculus 2013 AoPS Inc. Excerpt from "Calculus" 03 AoPS Inc. Te term related rates refers to two quantities tat are dependent on eac oter and tat are canging over time. We can use te dependent relationsip between te quantities

More information

Recall from our discussion of continuity in lecture a function is continuous at a point x = a if and only if

Recall from our discussion of continuity in lecture a function is continuous at a point x = a if and only if Computational Aspects of its. Keeping te simple simple. Recall by elementary functions we mean :Polynomials (including linear and quadratic equations) Eponentials Logaritms Trig Functions Rational Functions

More information

Simulation and verification of a plate heat exchanger with a built-in tap water accumulator

Simulation and verification of a plate heat exchanger with a built-in tap water accumulator Simulation and verification of a plate eat excanger wit a built-in tap water accumulator Anders Eriksson Abstract In order to test and verify a compact brazed eat excanger (CBE wit a built-in accumulation

More information

LIMITS AND DERIVATIVES CONDITIONS FOR THE EXISTENCE OF A LIMIT

LIMITS AND DERIVATIVES CONDITIONS FOR THE EXISTENCE OF A LIMIT LIMITS AND DERIVATIVES Te limit of a function is defined as te value of y tat te curve approaces, as x approaces a particular value. Te limit of f (x) as x approaces a is written as f (x) approaces, as

More information

Symmetry Labeling of Molecular Energies

Symmetry Labeling of Molecular Energies Capter 7. Symmetry Labeling of Molecular Energies Notes: Most of te material presented in tis capter is taken from Bunker and Jensen 1998, Cap. 6, and Bunker and Jensen 2005, Cap. 7. 7.1 Hamiltonian Symmetry

More information