Bestand wählen
Merken

# Special Relativity | Lecture 5

Embed Code
Video lizenzieren

#### Automatisierte Medienanalyse

Beta
Erkannte Entitäten
Sprachtranskript
rule on Steiger University of Romario
questions I can answer 5 minutes a what-if about as look Who's depends on which weighs gone our before going that way and your moving with it and the wavelength looks long refused stand there were few moving the other way it looks about what was soft and hard remit career-high you worker you could work it out this geometry from geometry Lorentz transformations are desert you do 1st of all still the usual drawing of took orders frames 1 of them is arrest but your frame and my frame is moving from right From fraying it looks like there's carts draw more more afraid withdrawn the speed of light a draft final imagine that they woes a
wait and think about braves and the wave is moving to the right as it was right recurring follow the peaks of the way of bright points of the way of our withdrawal Amin red there also grew more with the speed of light so little quite says Norris as you know pop as as time goes forward as time goes forward the way we moved to the right answer right moving way of laying the In new or frame of reference is this distance over here the distance between successor Our Maxim work 3 week wavelength In my frame the wavelength and my frame is this Mr. bug this this parents as measured using she wrote using Lorentz
transformation of our words this distance here is the distance between the 2 lack at what you call an instant of foreign terror wave waved to me is the distance between the 2 maxima what I call them instant the time saw if you work out the Lorentz transformations and you look along here and ask What's the distance between these 2 points in your own mind reference Barry and compare with the distance along here you'll find the different that would give you the publisher you have the tools to do that you could do at you know Lorentz transformations you can figure out what the quarter if let's say the waiver wealthiest some way some particular wavelength you could figure out what the point is you could figure out what the point is and you could ask What's c X crying distance between X Prime is my coordinate axes your the X distance between news Islam and you could figure out what the X Prize distances strode forward to calculate busy publisher so I recommend as an exercise and arm and working with Lawrence has relations young off last yes courses next fall yet level is not sure of the
plant gave us a lot quicker broke assure Jr for drinks was possibly Jarreau committee possibly what quantum mechanics normally College of quantum mechanics was reasonably deep but not very Braun having not gotten through a harmonic oscillator that's a bad thing so I think before we study quantum field theory we do have a study of more or a karaoke bar there are natural which way to go next quarter ahead all us now it is dead Of back in red all Wall who met with the general refers to gravitational issue we studied gravity that's an issue combination of quantum mechanics and gravity you may know enough about corner character was poor amount of program here the sale the last where do you mean the goes on August and normal just over the bar for your half to do Lieurance French version right column right Prairie do it let's make this wavelength warned just a simplified so this 0 Miss 1 was to store for right now are like OK with disappointed could be important for me to locate where that point pointers this point is X Prize equal 0 that's same Prada's Richards repair feet equals V. X still firm that the surface a prime cause auroral is a same he equaled the extra wife our lines transformation halls keep crime is equal to T minus V X over square 1 I might as was with feet prime equals you're always a prime equals 0 that's what I call on the incident happened to be the instantly practical 0 and that's clearly teen equals V. X. This is equally X 1 over here for 1st or was a story up I XA JET moose white Siegel's warned that checks equals was a former broker Mark T.J I actually most people's right a or at the intersection of X equals people swarmed and vehicles VX Forbes work at our 5 pursue what that point this X equals teachers which is the X plus warned X 101 miners equals warned or X Is won all of 1 mind 3 firstly XO pouring nor about the sea of this scoring well we know the acts of this spawned or say free of EXO this point we can read off saying is equal tour V although 1 Marguez right now we would like to find out what about the simplest thing to do now is the figure out we know TV and we know X what I'd really like to know all I really like the point here I'd like to know X prime our would like to know the distance between successive Nunavut in my friend prime frame so we gotta Brian frames X Prize was is equal to X Miners VTE or square worn a square now I just plug-in for X and they were our fear guru over this year's ex prime equals X 104 Missouri minus V crimes feed sorrows NYTimes these squared or want miners very all divided by by square root of 1 mind as a square right we do it right example OK sought to now where 1 might be square water minors square provided by 1 minus V and also in the denominator Syria's square root of 1 square 1 might be watch our one-liners square is was 1 might firms warning possibly right it's 1 minus squared as 1 might be warned caused 3 of these canceled Our wrote what else can I do with well I can put this in the square root just by
Violet my way he might not be waived Harlow said effect emotional particle I would not get a test of the dynamics of fine now we're going to ask for those grungy and for the particle misses particle Lagrangian sorts remember From where is begin or if begin whether contains minors and by the square root of warn Martinez's X stock squared what was that this was a terminal a grand June correspondent joint action which just ended up old proper times all we moved segments of pack multiply by my an army in the role of us will grungy our integral to seal off the action integral to this and this is the same as mine an transient the will of the proper times 1 and the trajectory toward back-to-back RFC or Nehru around you for particle but that doesn't have anything in it in which the field affects the other particles let's put something for the hero a simple thing the simplest and think of symbols and I can't think of all this a little bit and broke plus foggy not fires as a function of position and time in general a function of X and bar allegedly for the moment normal functioning of the coordinates of the park broke find the same thing again 1 minus square that's a log on June 4 particle I could work out or the Lagrange equations and find out how far effects emotional particle I could do it but I'm gonna do it In this simplified approximation with the particle moves very slowly just as illustration I don't relativistic square root of 1 might be square and so forth this of course is square and we could be thinking about particle we don't need widens EU those just Exs enough to restrict the point article moving in one-dimensional when they put back the speed of light it's useful to put back the speed of light in order to see what the nonrelativistic limit is the nonrelativistic limit is basically the limit in which the speed of light goes to infinity or speed of light is bigger than any other velocity the problems of the nonrelativistic limit are the Newtonian limit Granzien has units of energy does not have units of energy on lesser speed 1 but the speed of light is not 1 we put my yeah M C squared now units of energy find going to leave that worry field is defined all let's talk constantly at astrology in here this constant is a constant called or coupling constant measures the strength by which they field affects the motion of the particles forwards Fujian here he could me anything for the moment we don't know what it is and might have receded my privacy and important at the moment which is G plans far-right fight plans are excellent now let's fix our and there are of course the square root of 1 minus X . squared overseas square vessel a graduate of all the speed of light enquiries and then let's go the nonrelativistic limit the nonrelativistic limit when gets very large we expand the square root will expand the square root using the binomial expansion 1 2 minus a small number of mightiest one-half power were factors becomes 1st thing becomes warned minors X squared Over twice she squares twice columns 1 half columns because this is something of a 1 have power using that 1 plus Solana warned minus steps or this case however 1 half is just 1 minus 1 half epsilon plus things which are really very much smaller because they will involve higher powers of the race TOO velocity velocity right so this is what we get plus tiny tiny things such racists system right warned that stuff square over to see square no best how we tried flying the biggest pieces the pieces which are almost important when the speed of light gets a la the Army's 1st pieces and square MC Square Warner but and she squared was the number it doesn't do anything when you start different it's a Lagrangian with talking about the or differentiating it will respect the position of Australia doesn't do anything just number so that is of no interest bidding constant grungy and doesn't have any effect on the equation of motion so they see square times 1 is of no interest the next germs has the speed of light canceling altogether so it's part of the limit in which the speed of light goes to infinity and survive survive that limit so the next turn is the product of NEC squared punch extort square to see familiar that X stop squared and over to the squares of canceled and that's just good old 1 here FMV square the nonrelativistic kinetic energy I want about Mr. G. flight times 1 that has no speeds of white so it clearly survived so we're minuses a minus sign here miners GE fly are X and that's all is when the park was moving slowly emerges compared wrapped With you know the old-fashioned nonrelativistic Labonte years no fashion 1 motorist circle around June was Connecticut Energy minus potential energy this is clearly kinetic energy and that identifies forest G. Kimes far as the potential energy of the particle potential energy of a particle in a field is a constant butt In for example electromagnetism and the strength of the coupling on a field of particles just electric charge the bigger electric charge the bigger before summer particle in a given field so it's natural that maybe some constant multiplying something about the field will come backers but July In this among wrote only reason I do the nonrelativistic limit is so we after Mark around always square roots of a former squared just illustrate a point this is effect this year represents the effect of the field on particle OK now about the effect of a particle airfield important thing to understand is doctors only 1 fracture total actually altogether and Republic actions and close action from the field and the actions of particle physics problem with trying to do could be if we wish we should drop picture was trying to do where a regional space in
car and prime space and we have a field in it right of a few euros a bunch of red Wash there's the field 1st of all what scientists are now have a field day that involves knowing although grungy and for the field and minimizing liberal Lagrangian for a field and in addition the mechanical system of the system or studying also happens particle moving freely store were trying to do a study of combined system of field particle moving through the field were trying to to minimize the action not just with respect to the motion of a particle with respect to the motion of a particle and promotion of a feel for how a few behaves in for that very simple we simply explore around changing the field letting it will in different ways change the particle trajectory until we minimize the action and that will give us trajectory field correspond to a principal with faction OK so let's go and write a whole actions including the field and particle will 1st of all we need an action for field we discussed his last time is going to write down what we wrote where starring are a simple case of it we're grows in the field we rode Elsa better since the Lagrangian from a field that would erupt down Western or are well the action the action itself before action integral but D 4 X before acts means and the X the also Foreign Region X Y and Z and he said we wrote our Western was the integral of one-half partial afire with respect to teach squared Martinez 1 have partial fired with respect X squared fish was freed white for the moment don't care about them anymore I just want to keep track of them in this little exercise over here but we want to keep track of the speed of light think it would be a oversee squared here this is a you feel that it doesn't bother particle but now I have all reaction was reaction Ferraro from the euro for the field mallets also interact reaction particle theory is right over here I'm going to take a very special simple case you could work out a more complicated case but I'm going to work out a case where the particle is at rest but put the particle arrests were starlets assume for a moment that there is a solution to the equations with the particles just rest the situations when electron might be addressed after and asked what the additional termed the actions we could make electron move it doesn't do we will but there are about 1st simplicity let's just take electron arrest not moving faster trajectory are more the action and that case it right over here X 5 0 which is 1 square by itself was no interest and so little of bungee and for a particle or contributions programs from this piece over here is just the integral of foggy fire exit goal or where what kind of integral persimmon dual right With particles our fields and will and integral along the world line article but it involves 5 balls the reason the particles responded to the field was because the Lagrangian for the particle involves finding the reason the field response of a particle was because that's a Lagrangian also has fine and the positions of the particles so while how how cold air this turn here I wanna it in a way that corresponds to an integral over space and part How could I turned this the willing to roll over space and time role sorry for bags that go back respect West put particle for convenience and simplicity just put it at x equals 0 it's particle position x equals 0 tomorrow it put in here what we put in the position of the particle effects which is 0 5 0 and in the integral now this term here is just an integral over the are in this town here is an integral all space time and factor where I wrote it spaces only one-dimensional so this will really be only a two-dimensional integral but OK can I write there's so that it looks like an integral overall space time just put in the 1 terms for the is simple the answer is you used the Dirac delta function the use of Dirac delta function all you do it so I wanna get cake 5 0 the time was not port here and they right In the following wary an integral call space of fire prices a Dirac delta function the pro-independence Islam port here just girls along for a ride to recognize this When you integrate the Dirac filter function Dirac delta function is a very high spike of a function at a high spike of a function of 0 everywhere except at x equals 0 so when you integrate the field with the delta function you just get the field at the origin and you feel that the origin or the field at the place where X will follow in the delta function has expired a Cinderella volume if this was a one-dimensional axis who just right DX works take a case of our workers are a review our guest I really need to tell you what built a fortune really means what I've written as just of X feel more precise score was dealt a freer vexed 2 3 X Delta free of X means the product of 3 delta function of X novo Delta where is functioning nonzero it's not uh 0 when the ball all 3 of these delta functions on on 0 the only 1 place for all 3 of them 0 x equals 0 1 0 0 equals so what is it is a function of free coordinate stop ex-wife Missouri a function of a freak space but it only has a value at 1 point of space and at that point the space it's enormously big where you integrate the field was
right about refresh our way detail integral fire Our ex-wife Missouri I won't bother writing your car independence Delta of X over a white grows it XTY busy bus robbery here at a meeting of the symbol here tribal belt function and integrating all wall space I had either the role was not very hot integral to do remember when any integrate a function with the delta function it gives you the value of a function at the origin or a place with Oprah function is big but has spiked with just using the equation and Demirel F other X belt of X eagles at 4 0 0 0 defining property of the dope function picks out for you the value of a function at the place will don't perform has a spike Our acts of I do these integral over extra 1st this 1 tells me cassette X equal to 0 visited meters that White equals 0 listened to local news of 0 so the whole thing is a fires 0 so I've written here right but Of course not less I also in order the calculate the action I also have to the integrated Over time we are actually was integral on safari over time so the actions we have the integral all coordinates due for X means BXQ argues EDT until felt killed the wrecks was Dr. killed the X is doing nothing but picking out the value of fiery at the spatial position of the particle board doing truck just a right this thing here it's a trick full writing the value of a field of a point as an integral 1 0 do it well I just wanna do it cause I would like it here plus the list something Mr. Judy very likely because of this yacht Ma yes GE Delta killed other ex wife of X and now the whole thing is just 1 big integral or all space and all current areas and this thing beside the big square bracket is is thing inside the big square this the action equals the action now an integral or space and time were getting good field Lagrangian good although field actually and this turn we hear represents the effect of a particle back on the field is just firm over here for the special situation where the particle happens to be arrest we could do a jazz laughter of particle moves as fire we could do that by making a delta function move around with time possible to do but that's not important right now OK now let's go take a on June and work out the equations of motion for inside the bracket is low Grosjean work out the equations of motion 1st of all we have to do it 0 Grosjean with respect through partial afire with respect to tea IDT that pressure state or by defied go ID to justify right so was what really give us a 2nd derivative of fiery with respect he squares some like an acceleration Ave a kinetic energy fight about squared we work out the equation emotion we have something that looks like an acceleration of party now are other terms coming from defined X squared exactly the same sort of racist just go for it and might these 2nd fiery by BX square there if that's although I was a research that equaled and this would be a good old-fashioned wave equation and warned them into case of rural they had more dimensions we where miners 2nd followed by a BY squared minus 2nd fire you nor was set that equal to 0 that would be the wave equation describing waves Of this far but now we have more fine dependency below Grosjean depends on in another way what happens if we differentiate the Lagrangian will respect the fiery was also put Iran right here are right side is now 0 was supposed to put the derivative of Lagrangian with respect the fight that's the other side of ranchers equations of the side of orange equations currency from here with the Renault the L. respect the fight exit is just a GE times the delta function which just G and Dokic you Chad Our equation is just this is equal to g Prime's dealt Q other books you recognize this equation you may have seen before a minus sign was a minus sign yes was a modest this is not equation awaiting equation except on the right-hand side at 1 point in only 1 point something different something going on at 1 point which is different from the rest of space what's going on is a foreign court it's affecting the equation of motion only 1 port let's just to try and make you little bit familiar what's supposed there were looking for solutions of the equation after a particle standing still I wish example of the particle standing still there might be a chance that as a solution of the equations with the field itself doesn't change with time just be effective standing still particle affecting the field is likely to make a field which also doesn't vary with tar and so we might look for a solution equation in which 5 High Noon depending news not here then again chase started of everything costs plus plus remote if you recognizes equation passed on accord was passed on equation for example for the electrostatic potential energy electrostatic potential was sometimes rewritten as Dell squared fine using corporal source on the right hand side charge density and this case a charge density is just the delta function of rural high a sharp spike so you see no not really doing electrodynamics at the moment we're doing no sort of
simplification involving the a scalar field Mr. electromagnetic fields but to see how works the same turned or grungy which tells the particle To move as if there was a potential energy G flight which exert forces on a particle also exchange terms where you was for the equation emotional find fueled cars or I feel has a source these are not independent thing the fact that the field affects the particle Carlos particle effects of field and with a specific simple example it even tells us exactly how the same parameter GE the parameter gene determines how strongly effect particle on the field is also tells us how strong the effect of the field on a particle so we have a nice story their fields of particles affect each other commenter Lagrangian or era How might we are right so let's just explore how how we might change is a bit less tobacco a one-dimensional case just arrest 1 space dimensions make it 1 space and this just becomes the 2nd a river of fire reversible equations 2nd the river fire with respect to x squares Okojie times of sex simply suppose it I want to put the particle someplace else I wanna put at the I want to put a X equals or 18 what I do fear however changes X minus a doctor of X minus a note X minus it has it's spike when Exs equal pay now supposing the particle moving positions work AT of tea will you would prove his right don't X minors a year for a year what this would tell you it it is a source of the field the source of fuel was moving at any given time it's a position of Sicily can accommodate a moving particle we can accommodate more particle the only thing is if we did have the a particle moving we wouldn't expect the field to be kind depends if particle moving around obviously the field will depend on at 4 Source the particles moving around so we will have to put back we'll turn he minus 2nd fiber IDQ squares there's no way we could get a solution with kind independent fly ball with the right hand side does depend on time the only way to make it consistent is to put back the time-dependent so Moly particle accelerator particle or migrating particle will car effective field give a time dependence horrible will radiate waves but were at the moment we're not going to solve wave equations woes somebody else would promote kind of just talking about the location of relativity rotation mean the mathematical rotation we used to make equations look simple and pretty and not have the right to every day and time equations 2nd derivatives part X squared sectors y square LeBlanc but have a need protection notation was partly in housekeeper partly but I young visitors classical and in that way by them it is what it 5 None of this crew convinced that is quite a while a bomb is what we really want to do Real monolog Mr. Blue Water keep of the overseas nor do we would not want to broker a number of Mr. Quayle but even nonrelativistic limit the right hand side this time so it will create few create ways by mushers accurate arrest if where road where if for tea now the tomb of formal request to their side of the equation to square yard form for a he yacht Japan's fires to be called G F well let's say emitting figure out the units how we knew you for your appears Pulitzer fire happened the units of energy and that means a unit of G R dimensionless numbers to depends on the dimensions face that's not always true Burke nurtured from all this knuckle port for 5 hours cars just reviewing for some of you will be reviewing for some we're learning Standard Notation for Roth of vectors for America's Taylor's and you make we talk about last time our back Amex galas ever had a mixed day was elected as directors of Skelos and how you condemned nor patients of the things we meet short of Austrian province but remember where next CEO of course stands for the 4 corners of space time and space time and which TXY owns 8 usually this is called 1 2 1 3 innings was called naught effects on portal mule runs over 4 notice now I'm going to pay pensions outerwear repetition next I've put the index up on top above but who need something for Strawberry the penumbra because think but this quantity of thought as a displacement from the origin is a former rector of 4 vectors a statement about the way it transformed transforms our transforms on orange transformation anything that transforms the same way on tourist cancelations ex ante is called for vector I want to write it down for a moment format there has 4 components same back 3 of the components of equal 0 for example that might be true ballads that would mean that it was a displacement that Our only involve free of accord fourth-quarter but it's not an invariant statement observer 1 you only Barbara was displacement from object which was purely time white rulers move in space I will say it did move varies so the statement let sector has 3 components equal to 0 may be true in some frame of reference but not invariant statement that everybody will agree upon other stated that all 4 components of 0 that everybody will agree on a solid says it is about yeah it is 0 no displacement that all nor agreed last year over there was a 0 fact 0 transforms itself armor Lieurance transformation of all 4 corners 0 before a transformation the Zero after Seoul on that's motion of a format vector
Germany for vectors differences 4 sectors of 4 sectors sold differential displacements of 4 sectors and so for our format that we can make a scalar of a scalar meaning thing which doesn't transform which is to say that friend I know we've done this before going over all material the West but Allah Ave displacement we can construct a proper time to Bacau squared million or about it is that he squared minus the X squared square feet square and that the property that it is the same in every reference frame it doesn't have components only has 1 component 1 value and it's a or course for vector scalar now the Belo is equal to 0 in 1 frame of reference it's 0 on every frame of reference y because the scalar was the same as a thing which is the same in every frame of reference for some frame of reference a scalar whatever practically definition of scalar which is every frame of reference 0 1 frame 0 Anthony framed sought variants statement to say scalar Our is equal to 0 in here we have an example of how you make a scalar for that's very general if you'll have a former rector of any kind scored a new say let of 4 vector means it transformed the same way under Lorentz transformations then the difference of the square of a kind components of space components are scalar Barber writing at the difference 8 times square mile space squared are a scalar is a scalar it's the same in every reference price 1 my sweet right that I don't wanna have a right AT Times Square Marseille square Mercer why square murders scored for Matt Williams may matrix the matrix name called the Matrix In special relativity is usually denoted a dash Peters Greek letter are with the Greek letter a tea a hillside bat here is a matrix is a Greek letter which varies from critics and the Matrix has tho indices new and but it's a very simple matrix it's practically the identity matrix but not quite it has 3 a elements which are equal the 1 on the diagonal society unit matrix for the 4th element was minus 1 of off element corresponds to Prime minus 1 0 0 0 0 0 plus 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 hour to make this plus warned it it would just be a good old unit matrix but is not now let's take this matrix we're going to represent vectors by columns column vector for example a new will consist of 4 entries time component and free space component X warrior Missouri must the Matrix and multiplied by the wreck created look by matrix multiplication all we can say with who are of course a new with Susan Newell he knew runs also from 1 before of multiply the matrix private sector we can represent back a them you know a rule some you that's 1 way to write it just this warning are just matrix multiplication but it can be represented by summing 801 warned Monday 1 8 0 1 tool a tool and so forth and this represents a nor object which will call 80 With a lower index here let's see what to a figure out what is what a indexes it's not Buergel if this really were the unit matrix and you multiply vector you would get back exactly the same thing but not quite reminder sworn here means that when you multiply aided by a what happens it changes the signing of a 1st plan Everything else so I can write down immediately without me the work last 82 for mule you would know it doesn't matter who represents the column vector of vector Martinez AT T A X K Y a hazy another words when I perform this operation I simply change the sign of the times component parts about 1 piece on dislocation this is killed a patient of ice is no real content the 2nd occasional the rice which no doubt you're also somewhat familiar with the Einstein summation convey the Einstein summation conventions when every you see a downstairs index and upstairs index was seen in this case no budget automatically the rule that you samples were you don't need to write summation every time Aurelius repeated the index as long as 1 of them is upstairs and 1 of them is downstairs don't Barber riding summation signed its imports and plus and the conventions of the notation Blake why of this operation here just has the effect of changing the sign of the times know now I should warn you depending on who was riding the equations this can sometimes be plus 1 minus 1 minus 1 minus 1 for minus 1 plus 1 plus 1 plus were different convention I happened like this 1 almost everybody else I know what the other 1 from West there wrote to study general Pavle like the convention are used our sold it's 1 of these conventions and never quite settled down and so for Republicans and Democrats Jews and Muslims they hate each other like all hell there are those who use 1 conventional also use of convention where were born a group of people or use 1 when I would be of a group of people or use other so far have been killed about over progress will use is 1 null 1st gaffes are duty his arrest location just gives you a year more blankets multiple crises multiplication really means multiply this by this process by this course the artists forces progress on this case it's just trivial or more generally Ada was not the Unity matrix was somebody else Guatemala matrix John Major at will be this time this close this time there's plus this this force turned back the other would be a summation we're but I suddenly got fed up riding summations mold realized every in every equation was writing Baron
solutions which we allow to be complex complex functions and later Picarille parts or your example your our cut solutions look for your look for solutions which 1st oscillate can remember warm off from writers here but this was it raises he reminded I major TV that's oscillation with time with frequency omega and no room for solutions which are so even space which means there needs to be IKX of some pride he really RIA idea KX Pires X plus KY times y cases he can't be what is entering numbers kayaks just 3 numbers recalled wave numbers for the wave vector archaic sphere warring easy and on was a former year the solution I will look for maybe there is no solution workers were Jack 5 our 1st of all we can rewrite this McCartney way rewrite it as he earlier on May opens you with X at come from well if I take if I think of Katie as a former rector with components my guess Omega my assuming thinks KXK wine is but Everything cover for components are very complex of things barring Omega KXK winding Kinsey grouped together was formed here call figure them for a moment that may not really be they are actually a year before think of them as the a former them what company X here is just King KUX prime component Omega primes T X component of Katie projects Sosa nicer location but will likely use of marriages Carita were discovered using form he West see what happens if we try solve this equation what is the 2nd derivative of fire with respect he squared well what happens when you differentiate eagerly RIO with respect to carve scored twice over multiply what you have here by minus Omega square so where's scoreless services misses are guests at 5 what happens if we differentiate twice restricted we William minus Omega squared terms flight what happens this if we differentiate with respect acts we get kayak square with respect the white twice KY squared so altogether would you mind this Omega square plus KX quoted gross KY square plus squared or furnished we still have cost and Square here stock rose plus and square but folly Fuqua 0 well it's very easy now defined the solution you simply say homemaker must equal the square root of kayak square plus KY square close Casey scored a and square and our words you just at work in a bracket here equal to 0 and you have a solution this is just tell you what the frequency years In terms of a wave number frequency square root of Kerik square pose KY square puts Casey squared blossoms square that's a solution of acquiring or equation there are 2 possible solutions a square work like this commit plus or minus little the equation where you put plus some may go minus on radio and cost for every KXK wine case there are 2 solutions with fossil my little girl for summer route of the square root are Bozo solutions of a corn Gordon liquid their of we're back with me many cartoons quantum a or misses a classical field version of an equation which describes a quantum mechanical particle with mayors an energy Omega momentum Katie Mr. each equal square of key square for some squatters Clark's constant Mary argue about the Solheim on use of these equations aha a simpler linear equations you solution at the end of the the day after you've constructed solution In this way you could take a real part of it and the real part of the deal was supposed to be real there was a real number you just real part folk any questions getting guys along narrow with talking about for a rooftop we are aware of the plan were but he is but he's on the the glass as a little larger I may merger if you look for a local this is pretty much the unique will run which would lead the linear equation a call will come back so why we're looking for linear equations for the moment this is just an example Martha at an example or new a set of bras by order of Minnesota next to the jury is at the you're out the Granzien with respect to a thief IDT best justified new different shape that with respect the card your same thing he said Porsche with respect now they remember remember what the rules are OK what's right back the auto arrives equation for a field Debye theory Ericsson where now right and reform IDX mu of partial of L with respect to defile a body DAX mule that means some Lemuel equals or minor um miners quash or row with respect of 5 musical 2 0 talk theme IDX Newell This is a kind of cold areas index of downstairs index is denominated here you have 5 IDX knew that makes upstairs acts but progress it is of the form of a lawyer index plans opera index Our there you could see that this equation has some advice relativistic structure operator seasonal or indices contracted Narrandera Vera everything is a scalar so this could
1 day electrons become worse that would be a disaster for Adams because size of Adam was governed by the inverse of mass of the electron and the massive electoral were grow 0 the Adam would become infinitely begin that would not be good for you say Da protons and neutrons would be virtually unaffected very very weakly effect on March the art well I mean it is it is true that a world without a hint of would be a better world figures many many things will go wrong as a set up a waste of which would be the electron would become vessels would sold alone Adams mergers Her have Noah nor callers nor were they like Chara become completely on localized for more please
visit us at Stamford . EDU
Wellenlänge
Großtransformator
Bildfrequenz
Handwagen
Lichtgeschwindigkeit
Besprechung/Interview
Lineal
Desertation
Tiefgang
Minute
Wellenlänge
Kaltumformen
Waffentechnik
Längenmessung
Dunkelheit
Energieniveau
Woche
Werkzeug
Amplitudenumtastung
Großtransformator
Bildfrequenz
Mikrowelle
Abhörgerät
Herbst
Amplitude
Magnetisches Dipolmoment
Gagat
Mechanikerin
Erwärmung <Meteorologie>
Lichtgeschwindigkeit
Fehlprägung
Regelstrecke
Transistor
Bildfrequenz
Fahrgeschwindigkeit
Wellenlänge
Kaltumformen
Rollsteig
Längenmessung
Gigant <Bagger>
Standardzelle
Rootsgebläse
Hall-Effekt
Übungsmunition
Magnetische Kraft
Elementarteilchenphysik
Werkzeug
Weiß
Jahr
Analogsignal
Gewicht
Klassische Elektronentheorie
Direkte Messung
Verbunddampfmaschine
Bergmann
Fahrzeug
Hobel
Armbanduhr
Reaktionsprinzip
Teilchen
Polarlicht
Fuß <Maßeinheit>
Klangeffekt
Kombinationskraftwerk
Elektrizität
Rauschzahl
Ausschuss <Technik>
Druckkraft
Strahlungsnebel
Schiffsklassifikation
Nassdampfturbine
Gleichstrom
Großtransformator
Mikrowelle
Modellbauer
Magnetspule
Juli
Greiffinger
Schaft <Waffe>
Messung
Drehen
Magnetisches Dipolmoment
Summer
Kugelblitz
Lichtgeschwindigkeit
Leistungssteuerung
Schlauchkupplung
Juni
Violett
Fahrgeschwindigkeit
Klemmverbindung
Wasserfahrzeug
Feldstärke
Licht
Körner <Metallbearbeitung>
Rootsgebläse
Übungsmunition
Nebel <Astronomie>
Gleiskette
Elementarteilchenphysik
Weiß
Atmosphäre
Newtonsche Flüssigkeit
Jahr
Fliegen
Direkte Messung
Gelenk <Technik>
Kraftfahrzeugexport
Bergmann
Irrlicht
Begrenzerschaltung
Mallet-Lokomotive
Reaktionsprinzip
Teilchen
Kopfstütze
Klangeffekt
Relativistische Mechanik
Behälter
Entfernung
Windrose
Tag
Elektronik
Trajektorie <Meteorologie>
Raumfahrtzentrum
Koffer
Rucksack
Magnetspule
Ersatzteil
Wärmeaustauscher
Drehen
Bombe
Magnetisches Dipolmoment
Kugelblitz
A6M Zero-Sen
Spannungsabhängigkeit
Rotationszustand
Bildfrequenz
Array
Wrack
Kaltumformen
Elektronisches Bauelement
Teilchenbeschleunigung
Formationsflug
Übungsmunition
Panzerung
Angeregtes Atom
Weiß
Satzspiegel
Römischer Kalender
Flüssiger Brennstoff
Bestrahlungsstärke
Druckfeld
Jahr
Elektrostatik
Lineal
Dotant
Drosselklappe
Fliegen
Schubumkehr
Klassische Elektronentheorie
Penumbra
Sternsystem
Direkte Messung
Kraftfahrzeugexport
Sensor
Bergmann
Auslenkung
Begrenzerschaltung
Jacht
Nanometerbereich
Teilchen
Textilfaser
Gasturbine
Kopfstütze
Brechzahl
Treibriemen
Klangeffekt
Stunde
Schutz <Elektrotechnik>
Stutzuhr
Strangpressplatte
Omnibus
Tag
Rauschzahl
Raumfahrtzentrum
Druckkraft
Ziegelherstellung
Lastkraftwagen
Fernbedienung
Großtransformator
Source <Elektronik>
Mikrowelle
Tonkassette
Ersatzteil
Greiffinger
Schwimmer <Technik>
Summer
Abformung
Regelstrecke
Prozessleittechnik
Bildfrequenz
Montag
Passfeder
Vorlesung/Konferenz
Multiplizität
Array
Wrack
Kaltumformen
Elektronisches Bauelement
Fahrerassistenzsystem
Kardieren
Formationsflug
Übungsmunition
Gleiskette
Werkzeug
Matrize <Drucktechnik>
Jahr
Lineal
Ringgeflecht
Astronomisches Fenster
Supernova
Gewicht
Direkte Messung
Maßstab <Messtechnik>
Irrlicht
Wetter
Auslenkung
Armaturenbrett
Satteldecke
Eis
Videokassette
Fuß <Maßeinheit>
Brechzahl
Stufenversetzung
Kontraktion
Bahnelement
Klangeffekt
Stunde
Tastverhältnis
Waffentechnik
Mark <Maßeinheit>
Gruppenlaufzeit
Eisenbahnbetrieb
Rauschzahl
Druckkraft
Gleitsichtglas
Schiffsklassifikation
Ziegelherstellung
Heft
Großtransformator
Ersatzteil
Material
Feinkohle
Fahrzeugsitz
Manipulator
Direkte Messung
Magnetisches Dipolmoment
Bombe
Sternsystem
Sensor
Mechanikerin
Bergmann
Maßstab <Messtechnik>
Lichtgeschwindigkeit
Fehlprägung
Auslenkung
Begrenzerschaltung
Chirp
Nacht
Satz <Drucktechnik>
Minute
Teilchen
Rotationszustand
Bildfrequenz
Gasturbine
Montag
Fuß <Maßeinheit>
Unterbrechungsfreie Stromversorgung
Brechzahl
Reisewagen
Ausgleichsgetriebe
Stunde
Array
Tastverhältnis
Kaltumformen
Relativistische Mechanik
Behälter
Elektronisches Bauelement
Diesellokomotive Baureihe 219
Rootsgebläse
Band <Textilien>
Gleiskette
Formationsflug
Druckkraft
Raumfahrtzentrum
Übungsmunition
Gleitsichtglas
Tonbandgerät
Weiß
Ziegelherstellung
Großtransformator
Fernbedienung
Source <Elektronik>
Jahr
Lineal
Schaft <Waffe>
Greiffinger
Messung
Drehen
Summer
Magnetisches Dipolmoment
Geokorona
Satz <Drucktechnik>
Passfeder
Fahrgeschwindigkeit
Kristallgitter
Einbandmaterial
Vorlesung/Konferenz
GIRL <Weltraumteleskop>
Ausgleichsgetriebe
Kaltumformen
Elektronisches Bauelement
Kardieren
Rootsgebläse
Übungsmunition
Werkzeug
Railway Industry Association
Jahreszeit
Jahr
Lineal
Analogsignal
Fliegen
Drosselklappe
Kraftfahrzeugexport
Mechanik
Bergmann
Tonfrequenz
Dienstag
Tiefgang
Feldeffekttransistor
Teilchen
Gasturbine
Edelsteinschliff
Brechzahl
Speckle-Interferometrie
Steckverbinder
Gasdichte
Stutzuhr
Windrose
Kombinationskraftwerk
Eisenbahnbetrieb
Glasherstellung
Maxwellsche Theorie
Rauschzahl
Schmalspurlokomotive
Gas
Telefon
Gleitsichtglas
Negativer Widerstand
Nassdampfturbine
Telefonaktiebolaget L. M. Ericsson
Gummifeder
Mikrowelle
Sprechfunkgerät
Ersatzteil
Drehen
Masse <Physik>
Kugelblitz
Hitzeschild
Dienstag
Chirp
Neutron
Steckkarte
Teilchen
Schlauchkupplung
Textilfaser
Gasturbine
Klangeffekt
Jukos
Kaltumformen
Feldstärke
Waffentechnik
Antiquark
Hafnerzunft
Elektrizität
Antineutrino
Tag
Rotverschiebung
Rauschzahl
Elektronik
Rootsgebläse
Magnetische Kraft
Gleiskette
Übungsmunition
Satzspiegel
Supraleitung
Lineal
Leistungsanpassung
Masse <Physik>
Mark <Maßeinheit>
Konfektionsgröße
Tag
Wasserfahrzeug
Neutron
Rauschzahl
Elektronik
Klangeffekt
Computeranimation