Good afternoon and welcome to today's energy seminar. We have a very exciting talk on a very big science achievement today by Chris Young from Lawrence Livermore Labs and introduce him. I've asked Mark Capelli from the Department of Mechanical Engineering here who was in fact Chris's thesis advisor not long ago to give a personal introduction to him. So Mark. Well, welcome everybody. So my name is Mark Capelli and as John said, I'm in the Department of Mechanical Engineering and it's a pleasure for me to introduce Chris. Chris received his bachelor's degree, master's degree and PhD degree at Stanford. So he's cardinal all the way. Can you hear me? Yeah. His bachelor's degree was in engineering physics and his master's and PhD was in mechanical engineering. So his tenure here is close to 10 years.  So his research and his PhD involved studying the intricacies of plasma accelerators that are now widely used for space propulsion and much of what we learned about how those engines perform came from some of his research. Now he's a target designer in the inertial confinement fusion program and the high energy density physics program at Lawrence Livermore National Labs. He started that in 2016 and now has co-led and supported a variety of the campaigns there. Using the experiments with very large multi-physics simulations, he's going to show you some of the outcomes of those simulations here. And if there's anyone here, any students here that are involved in SIMPS, Chris is a graduate of SIMPS. SIMPS is the improv group here at Stanford. I would say that next to plasma physics, SIMPS was his second passion here at Stanford. So Chris, I'm going to turn it over to you. Thank you, Mark. Appreciate it. Thank you so much for the introduction. It's always great to be back on campus. And I'm excited to share a bit of context into the really awesome results that we've gotten recently at the National Initiative Facility. Has anyone seen the 60 Minutes episode that came out in January? A little bit awesome, or some of the news articles that came out. Great, if not, that's okay. I'll hopefully tell you all you need to know, but provide some context around what was going on. And let's figure out where I'm standing here. We've gone through the introduction, but just again, my name is Chris Young. I spent a whole lot of time here and really, really great to be back. And I've been about six and a half years at the lab. So just starting to really get a handle of what's going on in my job there. So I'm one person standing up here, representing the work of a cast of thousands. Many people have spent their whole careers, decades working on the problem of fusion. And this work is a collaboration between industry, academia, and the national labs. And I'll give a special thanks to Dr. Nathan Meezan who helped put this talk together with me. We actually gave a joint version of it in this building upstairs a couple months ago. And he is another predecessor of mine through both Mark's lab and then into the initial confinement fusion program. So an outline of the talk today, give you a bit of a history of the fusion program at the lab and a physics introduction to the physics of ICF and what we're trying to do. Talk a bit about the NIF facility itself. It's the world's largest most energetic laser. And the targets we shoot are some of the most pristine, perfect materials that we can manufacture. We'll go through the program as a whole and some major turning points, which led to the big breakthroughs that we've had recently and talk a bit about what we're doing next. So as I'm sure you're all aware, fusion is what powers the stars. And if we can bring this power and harness that for energy production on earth, that would really be a game-changing technology for humanity. If you're not familiar, fusion is the process of combining light elements into slightly heavier ones and releasing the binding energy of the nucleus in the process. It distinguishes it from fission, which is the splitting apart of large atoms into mid-sized atoms. All of the elements are produced either directly via fusion or indirectly through supernova explosions in the universe. And all of the energy radiated towards earth from the sun, sustaining life, was originally produced through fusion. And there is a lot of naturally occurring deuterium in seawater. So if we were able to collect that and harness, release that energy through fusion reactions, we're basically looking at a limitless source of energy. So the potential use of this is enormous for humanity. When we turn to fusion research in particular, there's many different approaches with lots of wide-ranging applications. Looking at the top left here in terms of confining a hot fusion plasma, you basically have three options. One, you have to be the size of a star and confine under your own gravity. Since we're trying to do this in the laboratory in a building, that's not going to work for us. So we have to either use strong magnetic fields, which is like the Eater machine that they're building in France, or a Tokamak, if you've heard those types of names. Or you can use lasers to deliver a whole bunch of energy and squish this fusion fuel under its own inertia, and that's called inertial confinement. And that's what we do at Livermore. Over here is a few different types of the inertial confinement fusion architectures. You can hit the fuel directly with the laser. We call that direct drive. We can put the fuel inside of an x-ray oven called a horom. We'll be talking a lot about that today, and we call that indirect drive. That's mainly what we do at Livermore. And you can also use a magnetic field to basically have store a bunch of energy capacitors and dump it down a wire really fast in what's called a Z-pinch configuration, and you can confine a cylindrical plasma that way. And there's of course many applications for this work in clean energy research to fundamental science studying the universe, the interiors of stars, basically materials at very extreme densities, temperatures, and pressures, and the conditions that occur in nuclear weapons explosions, which is why we, the Defense Lab, are also involved in this work. So to give you a bit of history as to how we got here, the quest for inertial fusion ignition has been a 60-year journey, and it really started the idea for ICF is as old as the idea of the laser itself. And very soon after the laser was invented, people realized that this could be a mechanism of delivering a lot of energy into a small volume to initiate fusion reactions. And there was a seminal paper by John Knuckles, one of our early lab directors, which was kind of outlined this process in the open literature for the first time. And since that time, NIF has been home to an array of ever-increasing lasers in size and scale and energy, culminating with the National Ignition Facility 1.9 megajoule laser, which is the largest most energetic laser in the world right now. And you'll notice that NIF came around, right around, and kind of the concept of NIF in the beginnings can be traced right around the same time when we stopped underground nuclear testing and switched as a nation to the science-based stockpile stewardship model for ensuring the safety and security of our nuclear stockpile. And NIF is one of the only facilities in the world able to make these super-high energy density states. It plays a very important role in that. And so there was this study by the National Academy of Sciences in the 1990 timeframe that said, you know, we need to achieve fusion ignition and go after, you know, very high gains, right? We're going to come back to the concept of gain. That's just the fusion energy liberated through the process compared to the energy that we had to invest to initiate those reactions. So a gain of one is energy break-even. And we would like to be operating somewhere around, you know, gain of 100 if you were going to say build a power plant based on this technology. We've spent a whole lot of time trying to pass gain of one for right now. But rather than go after that gain of 100 huge facility first in the 90s, it made more sense to take a smaller step and go after a small to mid-sized gains of 2 to 10. So that was built into the idea of NIF. Demonstrating ignition should be our highest priority. And it took a little bit longer than we were hoping, but as of December of last year, we've finally gotten there. And another point is that, you know, the only possible technology in the 90s when this was being conceived that we could kind of set up at scale at that time was to use a glass laser. So if we were going to rebuild NIF today, there are many more efficient laser architectures that we could use, but at the time it was amplified neodymium glass lasers. And we could take advantage of all the existing experience infrastructure at Livermore. So that was the natural place to site this facility. And they recommended that we recognize ICF primarily as a defense program, not as an energy program, again, for the reasons that I mentioned. So the rough timeline is, you know, in 1990 we have the idea we need to build NIF. In 93 we got the commitment from the Department of Energy to build it. We broke ground in 97. We commissioned the laser with the first experiments in 2009. Really in 2011 is when we first introduced the deuterium and tritium fusion fuel and started trying to make neutrons. So we've had a robust program over the last 12 years. So I've said the words inertial confinement fusion a lot now. So what are we actually doing? The main idea is that we're going to implode a sphere. And that lets you take, you know, something and increase its density and pressure many, many times over to reach the very extreme temperatures, density and pressure that we need in order to initiate fusion reactions. And here's the title of that seminal paper in nature in 1972 that laid this out. And actually George Zimmerman is still working at the lab. He's still developing the code that he started way back then to model these systems. And he remains an amazing contributor. I kind of mentioned this before, but the direct drive concept we're just going to put the fuel inside this shell. We call it the ablator because its main purpose is to absorb that laser energy and just blow up. And you get an equal and opposite reaction inward that's going to compress that fuel as it travels in. You can think of it like a spherical rocket where the rocket exhaust is pointed outwards and the rocket's traveling in towards the origin. But as you can imagine with direct drive, say you're taking a water balloon and you're trying to squeeze it in between your fingers and you're trying to keep it exactly spherical, what's going to happen? You're going to get some bumps that's going to squeeze out in between all the parts where you can't exactly balance everything. And there's a lot of laser plasma interactions going on around this that is preventing you from getting that laser light exactly where you need it to balance it perfectly. So it turns out that maintaining your spherical symmetry is pretty challenging in direct drive. But we do have a facility, the University of Rochester, the laboratory for laser energetics or LLE has a smaller laser that studies this extensively. But at Livermore we go more with the indirect drive approach and that adds this HORAM or a radiation cavity, it's a German word, you can think of it just as an x-ray oven whose job is to take that laser light coming in and make a bath of x-rays and because ovens are very good at redistributing the heat in a nice even way, it's much easier to push on this thing evenly from all directions and maintain a spherical implosion. So we usually use cylindrical HORAMs here, we have two holes on the top and bottom which are pretty stupidly named, the laser entrance holes or LEHs. X-rays bounce around in here again with the capsule, we have our deuterium tritium fuel. In this particular scheme we'll have gas in the middle and a shell of cryogenically frozen DT ice and the goal is to get that thing going at about 400 kilometers per second to give it the kinetic energy it needs to spark and kind of start off the fusion reactions when this thing implodes to about 30 or 40 times smaller in radius than it started. And that's exactly what's illustrated here. So we illuminate this with x-rays, the ablator blows up, we get that equal and opposite force inward, it accelerates, it decelerates, when it stops we call that stagnation and that's kind of when the fusion burn starts happening and then the name of the game is try to keep it there cooking as long as possible before it's going to blow up and dissipate all that energy and then we stop making fusion reactions. And so the picture right when we're burning is we're trying to go for a hot spot that's formed from the DT gas that's kind of lower energy or sorry lower density but very high temperature 5 keV is you know a few million degrees Celsius on the robustly burning ones we might be five times the temperature of the sun when the fusion reactions are going and then that's going to be surrounded by the shell of colder DT that's been compressed up to about a kilogram per centimeter cubed of density. All right so there are tons of physics challenges this is why it's a very hard problem but also very rewarding physics problem to work on. We have to bring together these big multidisciplinary teams of people with expertise kind of all over the map. So on the whole ROM side and this is where I live most of the day you know we're worried about how we couple the the laser energy into this system and use that energy efficiently and and also the the HOROM dynamics kind of determine how well we can press on this capsule symmetrically throughout the whole implosion. On the capsule side there's a lot of hydrodynamic stability that you have to worry about these are classically Rayleigh Taylor unstable systems as you have shocks and accelerating decelerating interfaces of different density kind of all mixing together and then as I was mentioning you know trying to figure out that yeah we want to add as much energy as we can and we want to minimize all the losses as much as possible to get the favorable energy balance and lots of fusion reactions. So to achieve ignition really extreme conditions are required and this puts some numbers to that. I kind of mentioned this already we used deuterium tritium because that is the easiest fusion reaction to to conduct it has the highest cross section for a given temperature. The goal here the product of that is a fast neutron that basically goes out and leaves the system and this slower alpha particle or a helium nucleus and the name of the game is trying to get enough stuff there to stop those alpha particles and get them to re-deposit their energy in the hot spot thereby continuing to add heat and heating everything up more and we have this nice feedback where the cross section for DT actually goes up as it gets hotter so as you get hotter it's easier to fuse more fuel and so on. But it takes really really high densities like 100 grams per centimeter cubed peak central density really high pressures 400 gigabar like we're talking interior of planets type pressures. Mark mentioned I was going to show you a simulation right these experiments are very expensive we don't have many opportunities to take them on the facility so we do quite a lot of modeling work with large multi physics simulations that run on the supercomputers at Livermore and we can model all parts of this the radiation transport of the lasers coming in the response of the horror materials blowing up the x-rays kind of making this rocket effect happen and then all the nuclear reactions and things that happen on the inside of the capsule this is a bit of an older design with a longer pulse but it kind of illustrates the point we put in a little bit of energy in the beginning and that kind of launches some shock waves and sets up the implosion the way we want it and later on in time we come along with this kind of the full power of the laser really hit it and that's when the radius really starts to go down and compress very quickly and you can see here the time scale we're talking about is 10 20 nanoseconds the billions of a second so very very short time and the fusion reactions are cooking in there for about 100 picoseconds so very very fast so you can imagine energy is lost kind of in many different ways along the way we start with a two megajoule laser and it would be great if we could get all that energy into you know directly helping to start those fusion reactions but this is kind of the reality of what we have to deal with some of that energy gets just scattered right back out of the Whoram and it goes into producing hot electrons and low density plasma kind of this you know as we see from the simulation of how stuff is going to come in and the the wall is going to blow up and start filling the Whoram and block the laser beams so but the majority of that energy goes into x-rays and a lot of that has to go to heating the Whoram walls right it's getting the oven hot enough so that it can cook our capsule and some of those x-rays are going to escape through those holes which we need to let the lasers in and unfortunately they also work as they work against us in that way so of the two megajoules that come in we get about 150 to 250 kilojoules onto the capsule a lot of that goes into ablating those outer layers and only maybe 10 or 15 kilojoules actually gets converted into kinetic energy of that DT shell that's going to start moving inward and do the compressive work on the fusion fuel on the inside and we would love to convert all of that into the internal energy of the fuel except as I was mentioning with the the you know asymmetry keeping it spherical is really hard so if we don't push exactly the same on all sides and we squish it a little bit you know like a pancake or a little bit like a sausage those are the words we use of that residual kinetic energy is going to rob your hotspot so that's that arrow and then finally you have some hotspot and internal energy you're going to make some fusion reactions now it's a competition between everything that's going to the things that are adding energy to the system which is the fusion reactions and the alpha particles that you're able to stop and that's in competition with all the loss mechanisms of radiation conduction and the mechanical work of this thing is going to start expanding and it's going to cool when it starts expanding so after all that energy flow we're going to get some amount of energy back out through the fusion reactions and for a long time this number was a lot less than two megajoules but in august of 2021 we had our first very near ignition shot 1.3 megajoules came back we were very excited to see that and then in december of last year we hit 3.1 which was for the first time you know a net energy gain compared to those two megajoules of laser energy that we put in now it's not quite that easy because 320 megajoules came out of the grid to charge the capacitors that we need to run the laser so from a power plant perspective three is still less than 320 but nif is not meant to be a fusion reactor it was meant to be the demonstration technology kind of that there's no fundamental physical process that is impeding this from working and now there's actually several startup companies out there that are pursuing different types of architectures some in the inertial confinement fusion space trying to get this number up and make that yeah make that math work for power production and because we have all these different energy exchanges I'm just noting that there are many different kind of efficiencies or gains that you can describe and so don't get confused if you hear people talk about well I got more energy out of the capsule than I put into it or I got more energy out of the fuel than I put into it but you know we passed gain of one on these much earlier than we did on on the full target all right so now we'll talk a bit about the facility itself primarily or primary experimental facility for doing ICF is at Lawrence Livermore it's in a very large building it's about the size of three football fields and as I mentioned 320 megajoules of electrical energy from the grid get converted to 1.9 megajoules of UV light and at the end on one side there there's this nice 10 meter diameter target chamber that's under vacuum and so all those 192 beams that you know all through this part of the facility all kind of come up and plug in on all sides and there's some really cool movies on YouTube you can check out like how this works sitting in the very middle of this on a little arm is our whole room and it needs a whole bunch of hardware to keep it cold I mentioned it's cryogenic frozen DT ice on the inside there so that starts at 18 Kelvin and takes a whole lot of you know thermal management to grow a nice ice layer and keep it all uniform and so that's about one centimeter large and then on in the inside of this thing that the shell of material that's actually holding the fusion fuel is only about two millimeters in diameter so crazy span of length scales here right takes this giant facility to deliver that energy into basically the something the size of a peppercorn the each laser beam travels about one and a half kilometers through the facility and it gets amplified through flash lamp pumped amplifiers four times before it gets diverted into the target chamber and so there's that picture of all the beams kind of coming in and we have a bunch of diagnostics and looks really cool it looks so cool that they filmed part of a Star Trek movie in there and called it the the reactor core or something here's when the target chamber was installed in 1999 I think a bunch of people came on site on Saturday to come see it and it weighed a whole lot a bit more on the whole room it's about one centimeter long and six millimeters in diameter and we cool it to about 18 kelvin before our really high performance shots here it is in the the cryo shroud here that's kind of keeping everything cold and then you know in the last minute before the shot it just kind of opens and then we go really fast we need heaters there's some windows in here so that we can get x-rays out to our diagnostics to see what's going on and we need all these dimpled shields here to keep the laser safe because we have the world's largest laser if that reflects off something in the chamber and goes back up a different laser beam or laser pathway that's going to be really bad news so we spend a whole lot of time making sure each shot shot is safe for the laser the capsule itself that's going to sit basically in between two pieces of saran wrap but it's very very thin it's about 20 nanometers we call that the support tent that's holding the capsule in the middle there we make this out of CVD diamond or ch plastic or a summer beryllium but mostly what we do right now is diamond here's to scale what it looks like to start at two millimeters and compress down you know 30 or 40 times and basically this x-ray emitting part in the middle of the hot spot is something less than the diameter of the human hair it's it's around 100 microns we need to get the fusion fuel in here somehow and we've used a series of continually decreasing fill tube sizes so this guy right here is five microns it used to be 20 and then 10 and five and now we can do two sometimes although those break a lot more often but just incredible engineering to you know laser drill this five micron hole and get a tube in there to fill it with the dt but all of these engineering features kind of we can't get away from them we have to engineer around them and these all provide some issues for us later in disrupting the the perfect 1d spherical implosion that we're going after now here's a picture inside the target chamber the hole ROM is inside this white circle and we have some other diagnostic arms there around it and what happens after the shot I think on the ignition one we burned it you know way back here and that was the first time we've done that one slide on diagnostics this is a huge area of research and amazing engineering but they've provided really key insight into our experiments as you can imagine it's very hard to make any progress if you cannot see what you are doing and see what happened so we have things like x-ray cameras and neutron cameras in some case time resolved so we can get a bit of a movie as to what's happening and we have different platforms that look at different parts of the implosion so we can make sure we're doing what we're intending to early on all the way to the the full up integrated test we can look at gammas and different self-emission and backlit radiography okay so here's one slide you all have my job for a second you are the ICF designer what kind of parameters do you think you can vary in your next experiment here in the whole room and the capsule just throw it out there dimensions yes well your laser is staying the same size so you don't have a lot of play in how big you can make it materials of all these components yet the particular way you you put the the diamond on here or what material you make the whole amount of any other guesses thoughts no wrong answer all right point is there's a lot on the whole on side everything about the laser where you point these beams what particular pulse shape you put on it the wavelength of the laser as we mentioned all the materials and their their dimensions there is a whole bunch of kind of derived metrics we call these like the rocket quantities about how fast you're going to go the different growth factors leading to what your hydrodynamic stability is going to be so this is a huge huge parameter space and as I'll show you very few experimental opportunities to test these things so we really have to to use codes and simulations and theory and try to put it all together to to make progress without testing every idea on the facility so now I'm going to do a little bit of a time series here to identify major turning points in the program that kind of led to the most recent results and we're going to plot this as thermonuclear yield versus year on the facility each one of these is a particular experiment so in 2011 when we started the DT experiments we were using plastic capsules we had a fantastic design that everyone had been working on for years just on paper and in one dimensional and two dimensional codes and turns out three dimensional reality was a bit different and there were some other things we hadn't thought of and weren't modeling taking into account and so by 2013 though by changing the design we call low foot to high foot we demonstrated alpha heating in the capsule for the first time and really the the gain from these early years was we were figuring out how this was working and we were developing all the experimental platforms we needed to again see what we were doing and get the diagnostic suite all set up and so just kind of getting all these gamma spectroscopy x-ray emission of the core going but at the end of this era we're still a factor of a thousand away from fusion ignition where we need to be in 2013 the laser pulse shape was changed to help mitigate some of the hydrodynamic instabilities and we go from something that looks like this remember we're trying to keep this a smooth sphere coming in evenly from all sides to something more like this that had much fewer hydrodynamic instabilities also going to a shorter pulse that helps it be less unstable and also helps just kind of get everything done quicker before the whole ROM has a lot of time to fill up with plasma and start blocking the laser beams and stuff and these are three-dimensional simulations in the code hydra that we run and actually so these features right here this x those are the impact of the capsule support tent that saran wrap that's holding the capsule where it touches kind of launches these jets and then also this is the fill tube here which kind of on every experiment like oh yep there's the fill tube jet we can't get away from it we then moved to diamond and this really helped we were able to use a lower-horon gas pressure which cut down on a lot of the laser plasma interactions that were happening in the whole ROM and that helped things get a lot more repeatable and able to be modeled well in the codes it's also improved our symmetry control with the shorter laser pulse again kind of getting everything done before the whole ROM shuts down in 2017 we again first demonstrated alpha heating with diamond and I've mostly said all this already but again using all of our different diagnostics to be able to check that we are remaining round at all times throughout the implosion and that was a key advance in this period so then we made the shells bigger and that might seem like an obvious thing to do but again the size of our laser is not getting bigger with us so that means learning to be more efficient and getting finer and finer engineering control kind of over all aspects of the process and so we increased the capsule size relative to the whole ROM so we're using our energy more efficiency implosion symmetry gets harder now but we were able to build in some new tricks to maintain that and by 2020 we achieved the burning plasma state and so physically it's starting to be a different regime it's where you're starting to make enough alpha particles that the heat and energy from alpha particle deposition is starting to change what's going on so in the analogy of say trying to light a wood fire right you're providing a spark for a long time spark went away nothing happened here in the burning plasma regime you're you're catching a little bit of the fuel and you're you're getting enough heat back kind of about equal to the amount of energy you're putting into it in in terms of just the sparks that would be like the kinetic energy of that shell in the the big long flow chart of energy that we were talking about anyway it's the the sign that you know something is fundamentally about to change and we're getting very close to that cliff where ignition is going to really take it off and the thermal instability is going to kick in and so how do we do that we scaled up the capsule more than the whole ROM so that makes it more efficient but we had to improve our control over symmetry and so one thing that we use a lot now is called cross beam energy transfer but you can take advantage of a particular laser matter interaction where by changing the laser wavelength subtly like an angstrom or less you can kind of pass energy between laser beams using the plasma as a mediator and that kind of helps you take energy away from from certain parts of the whole ROM and put it say in in other parts of the whole ROM where you want it and getting good engineering control over that was really important and also from our target fabrication side you know throughout this whole process every year the quality of the targets are getting better the the number of defects are going down with time and that definitely helps things so finally in right August of 2021 I kind of mentioned this before we had this guy right here the gain 0.7 and loss in greater than one so there's something in fusion called the loss in criteria it basically says you need to get something hot enough and dense enough for a long enough period of time in order to get a fusion ignition it's pretty easy to get two out of the three of those things it's very hard to get all three at the same time and keep this plasma confined that just wants to you know blow itself up so we we got to a part where you know we had so much alpha depth alpha production and fusion reactions that we exceeded loss in criteria for the first time that was the first time any experimental facility in any type of fusion configuration achieved that but in terms of the gain and that national academies report definition of ignition which said gain one we still haven't surpassed that milestone yet and we did a few repeats actually this is probably the most repeated experiment that we've ever done trying to do exactly the same thing and you can see they all fell short kind of illustrating the extreme sensitivity right at the beginning of the yield cliff where any little thing can kind of drag you back down and that competition between the energy gains and losses the losses win but how did we get to the threshold of ignition and the the loss and greater than one shot we made these holes smaller again that might sound like a really obvious thing hey you're losing energy out these holes how about you plug them up a little bit but again we need to be able to get the lasers inside and symmetry control basically by restricting where you can point the laser beam all gets really hard when you do that we also changed the laser pulse to not push quite as hard but push longer and experimentally the system really seemed to respond well to that and that August shot had the most pristine capsule with the fewest defects fielded to date and we have not actually been able to recreate a capsule that pristine again so it was really a unique and exciting time and then what happened we upgraded the laser a bit we had 1.9 megajoules to play with and then we got to 2.05 just for these couple shots and doing so allowed for slightly thicker capsules to be used and if you can go thicker that provides a little bit more margin and robustness to all these little defects in the fill tube in the tent and everything that's trying to kind of rip this apart as you are imploding it at 400 kilometers per second and so here in December for the first time we surpassed the amount of laser energy that we put in and got gained 1.5 and again with a capsule that was not quite as good as that other one and so the fact that we were still able to do that means all right we're getting better in a more robust regime and as I mentioned the things that enabled that we got a little bit more laser energy we thickened the capsule we continued on this track of pushing longer not necessarily harder and keeping everything around and in doing so we we finally surpassed the ignition threshold so what are we doing right now our current campaigns are trying to improve performance and robustness and it's kind of a funny business to get in you know we've spent decades and decades and decades trying to just get ignition with kind of single-mindedly focusing on this goal and you do that and the next day the goalpost just moves down like great you got one give me 10 so 10 megajoules is kind of the next level that we're aiming for we're going to continue optimizing around the the design that gave us ignition playing with those rocket parameters like the velocity and the different thicknesses and the laser pulse we have campaigns further increasing efficiency right we have a playbook that kind of worked how far can we push this this is one of my campaigns this year just trying to make the cylinder and the laser entrance holes smaller and deal with the symmetry challenges that come with that we have some different ideas about how to increase compression right you can increase the the confining material around the hot spot by shocking it up to even higher and higher densities but there's trade-offs associated with that and we also have some different horon geometries like this guy called the frustrum which can use less surface areas so be be more efficient again trading off with other challenges like symmetry so summarize the outlook for fusion research is bright right now it took longer to reach fusion ignition on nif than we originally envisioned in 1990 but we can say now we've kind of attained the the original goal for both the performance and then now all the scientific inquiry that can follow again nif is not a fusion reactor we're not going to plug it into the grid and start producing power but by attaining this ignition and demonstrating this you know we've kind of shown that there's nothing physically that's fundamentally holding us back from going to higher and higher gains the physics works it's it's going to turn more into an engineering type of problem to get a fusion reactor going and many startup companies are trying to do exactly that hope i was able to show you that you know there wasn't one Eureka moment that just kind of suddenly everything worked right it was a very long slog performance made in incremental steps continually improving all parts of the process continuing to improve our diagnostics and engineering control and our simulation capabilities getting larger and larger computers being able to go from 2d to more routine 3d simulations things like that so far we have not been able to repeat the the ignition experiment we actually haven't had a real attempt at it yet but in may next month we'll hopefully get a second one on the board and start assessing how what the variability and robustness is in this new regime and as i mentioned we have several new campaigns that are continuing to try to push forward and get to the 10 megajoule limit the next few years nif will be upgraded either in power and energy and that should enable you know quite a bit more more work and more margin to to continue improving this so so with that i will say we are hiring if this sounds exciting to you i'd be happy to talk to you and there's you know besides just fusion livermore is an amazing place to work with a whole bunch of very very smart people across all different fields so highly recommended but thank you for inviting me on such an exciting and spell-binding story that you shared with us today thank you so much we now have time for any questions from students or others if we have some tactical experts up here in the front anybody want to see a hand back there yeah i'm i'm not exactly sure about that but it's public it's a line item on the congressional budget every year so you can look it up probably in a hundred million range but that's spread among a whole lot of different pieces okay hello is this working okay um yeah so i was just curious i know you said that this is still like initial phases and you guys are not trying to build a power generator or anything but with that experimental setup is there any mechanism to introduce like more fuel into the reaction or are you constrained by whatever fits into the initial package right yeah it's basically the size of the laser driver is going to set kind of this size of your target and how much fuel so we we can still go a little bit bigger um and the trade-off there is with more fuel and or like a larger size we won't be able to push it as fast so then you're getting into all those engineering trade-offs with kind of the rocket equation stuff that you know great we might have more fuel to burn but if you give it a smaller spark it won't uh it might not take off and you won't hit that ignition threshold we got one in the back um sounds like you spend a lot of time trying to maintain that symmetry and sometimes symmetries introduce instabilities which i think is what you saw in the direct um configuration and so i was wondering has anyone ever considered um introducing some asymmetry to the problem which um can can uh i guess cause more predictive can result in more predictability um sometimes that sort of technique is used in different physics problems um and yeah because of the predictability might be able to increase your yield because you'll better understand the result so yeah sure yeah there are there are a few papers on that topic um you know one idea is to compensate for known asymmetries in the drive so like if you know you're gonna end up squishing this way you can start with a a a spherical capsule to start to start with so basically it'll compensate out um also yes exactly there's uh you know we can kind of induce whatever asymmetric uh trajectory we want by how we partition the the laser energy in the different beams um so far the majority of the implosions really have been focused on trying to stay uh round kind of throughout the the whole implosion but i think there's there's some interesting things to look at in that space yeah so presumably you you have um um well you have this huge simulation capability and you also have all the observational capability um you didn't show any comparisons of the of the the shapes of things with the simulations and and so on but i'm sure you i'm sure you're doing them and so my real question is with this a parameter space as big as you have obviously the simulation side is going to be the way you explore portions of that are you comfortable enough that you that simulation capability is is good enough that that you really can optimize the exploration with a small number of shots or is there more to do on the simulation side yeah absolutely and there's a whole chunk of our program devoted to just improving predictive capability of the Whoram you know that's really where uh a lot of the the challenge comes in um i'd say right now we're kind of in a hybrid mode where the simulations are good at certain things especially early time before the Whoram fills up with plasma and you really start getting these complicated uh laser matter interactions uh and then for the later part in time we have different analytical or semi empirical models kind of informed by our previous experiments say since you mentioned shape like that's that's a huge one we spend uh you know for every high yield dt experiment that we go for there's usually at least one or two symmetry tuning experiments to make sure it's round in in those conditions um and that is something that we've as the continual battle to kind of massage the code into agreement with with that last question over there if you don't mind i'll put you in an awkward spot you did a nice overview of laser confinement magnetic confinement etc if somebody was going to work wanted to work on or invest in the most likely first to produce commercial power right i'll ask you to although i know you're coming from the laser side yep what's what's on track to get there right um our lab director answered this question at the doe panel after the the press conference but it yeah magnetic fusion has had much more investment into the engineering side and uh turning it into a practical reactor than icf has um however we haven't demonstrated a burning plasma or fusion ignition conditions in a magnetic uh device yet so on the inertial side we have i'd say some catching up to do on on the technology required to actually put this into a power producing configuration i'm not sure right now which one will get there first um but kind of uh you know we we should invest in all of them and uh a set of it in thank you one last time there is this kind of it to me it's interesting that as you mentioned a lot of companies startups are trying to actually do this right now so there may be some evidence there i can also say as a longtime observer and admirer of the office of science and do we there's probably a lot of technology that you're doing that is attracting commercial interest do you find that to be the case pieces of this different kinds of reactions and materials and geometries and whatnot that could be used in totally different ways than you're using them here uh right i mean there just in the past two years there's been an incredible infusion of private money and to to startups you plot that versus time it's like yeah taking off but yeah to that point like there are many uh common problems between the different schemes uh like the first wall of the reactor in a magnetic um configuration if somebody solves that that's going to kind of help all the other schemes uh and sharing sharing technology like that that said we're out of time and thanks again for sharing this amazing story with us thanks very much chris thank you for having me


Hey guys, welcome back to our channel. My name is Andrea with Food and Mentory Adventures in Food and I am here today with a haul from the 99 cents only store. I have not been to the 99 cent store since before the pandemic. So I was really excited to go there and see what they had and I found some really great deals and I wanted to share them with you. So if you guys like this video, please make sure you give us a thumbs up and hit that subscribe button. So I will start off over here. I got two boxes of the Perrier juice drink and this is pineapple mango and you get four cans for 99 cents and I did want to mention not everything at the 99 cent store is 99 cents but everything that I picked up today was. Another thing I picked up were four packages of top ramen noodle soup. So on one of our what's for dinner videos which comes out on Tuesday, I tried a new recipe which was ramen noodle agro bowls and they were really actually when I first made it I didn't care for it but then I had it as leftover and it was really good. So I'm kind of hooked on the ramen noodle agro dish. So I picked up some ramen noodles for that. I found this pumpkin spice latte by husband like stuff like this. So I got this for him and then I grabbed two packages of green onion dip mix. I have a lot of sour cream in my fridge and I plan on making some dips and we can have them have the dip chips with dip. Easy for me to say and then I found this instant nonfat milk. When the pandemic first started I was looking for a carnation instant milk and I could not find it anywhere. I did find this and I went ahead and grabbed it because I do have some recipes that call for nonfat milk so I just grabbed it. I also saw this Betty Crocker whipped vanilla frosting which is a good deal. I have cake mix in my pantry that I need to use because I'm kind of working on a pantry challenge so make some cupcakes or whatever else I can use with this frosting and then my husband has been working from home since the pandemic started. I saw these Lunchables. They had turkey and cheddar only but these are six inch sub sandwich kits and you get water, you get, what else do you get? Mayo, you get Hershey's Kisses, you get Kool-Aid Single. Let me see it's Oscar Mayer Turkey, Kraft Cheddar and then you also get some Pringles and I got him two boxes for lunch. You can't beat that for 99 cents so I got that and then I've been doing a lot of baking and my recipes called for graham cracker crumbs so I grabbed a box of that and then I was most excited about these Kraft Caramels because of the recipes that I've been seeing a lot of them have been asking for caramel and so I grabbed three packages of caramels for that, couldn't beat that and also couldn't beat this Baker's Dark Chocolate Chumps for baking as well, grab that and I also grabbed the Baker's Premium Bar. I mean incredible deals on that kind of stuff. I was excited to see this cracker barrel spreadable cheese. It is garlic and herb cheddar which sounds really good, probably have that on crackers. My husband is from Jamaica and he likes tropical fruit and stuff like that so he also likes coconut water so I grabbed him a can of coconut water, mango juice, drink and then soursop as well. Then I saw this Crusty's Buttermilk Baking Mix and I recently discovered Crusty's products and I really like them. I have made their cookies and Howard and I really like them and so this was just a buck and I decided to give it a try. Also been making pizzas a lot lately but not a lot but we've been making pizzas some and so I needed some more pizza sauce so I grabbed that. It's organic, classic code is a great deal and then also some salsa 16 ounces for 99 cents as well. I think I have bought these before and I think Howard and I both like them. Cheese that's snapped this is just the original cheese double cheese so I got that and then I haven't seen these before but I do like fruit chews and these are sun kissed and they are sour citrus so I grabbed two boxes sounded really good to me and then I also have some recipes that have called for self-rising cornmeal and I didn't have any so I grabbed this let me see two pound bag of white lily buttermilk cornmeal mix and then the last thing that I grabbed from the 99 cent store was this glue away. I have some stuff around the house that still has adhesive on it and I just couldn't get it out so this was 99 cents and I decided to give it a try so this is it this is all that I grabbed from the 99 cent store and my total was 27 dollars and some change but um as you can see I got some really really good deals so I hope you guys like this video again if you did please make sure to give us a thumbs up and thanks so much for watching and we'll see you guys next time


And now we move on to chapter 4 in Holland Johnson which is Hermitian and Symmetric matrices. So Hermitian and Symmetric matrices are special matrices which are, I mean I am distinguishing them only in terms of complex versus real, a Hermitian matrix is one where A conjugate transpose equals A and a symmetric matrix is a real valued matrix where A is equal to A transpose. But these arise naturally in a variety of applications. So for example, if you are looking at the, looking at a function f which is defined from some n dimensional real space which I call d to the real line then and suppose it is twice continuously differentiable. We define the Hessian matrix for this f as a matrix h of x which is an n cross n matrix whose entries are h ij of x and each h ij is the second partial derivative of this f with respect to xi and xj. So this is a function of n variables, you take the second partial derivative with respect to xi and xj and this is a matrix that will be in r to the n cross n. And a key property of, this is very important in optimization and especially in optimization differential equations and so on. But a key property of this partial derivatives is that dou square f over dou xi dou xj is equal to dou square f over dou xj dou xi, this is a fundamental property of these partial derivatives for every ij equal to 1, 2 up to n. And as a consequence h is equal to h transpose i e, it is symmetric. Similarly if we look at what is known as the quadratic form, so suppose we are given an a in r to the n cross n and x in r to the n, then consider q of x which could be one example of this function f here, q of x is equal to x transpose ax. Then so note that we can always write a as one half of a plus a transpose plus one half of a minus a transpose, this can always be written.  So this is always true. So if I substitute for this in here, that means that q of x is equal to half x transpose a plus a transpose x plus half x transpose a minus a transpose times x. And if a is a symmetric matrix, then actually a need not even be symmetric here. So if you just examine this x transpose a minus a transpose x, this is equal to x transpose ax minus x transpose a transpose x, but this is a scalar. So its transpose is equal to itself. So for any scalar, this is equal to scalar. And so the transpose of this is x transpose a transpose x. And so these two are always going to be equal. So this is equal to zero. And so this q of x is actually equal to one half x transpose a plus a transpose times x. Now what that means is that a and one half a plus a transpose generate the same quadratic form. So basically that means this is a very, very crucial point actually. So that means that if you are interested in studying quadratic forms, it is sufficient to study quadratic forms generated by symmetric matrices. So we will be studying quadratic forms generated by symmetric matrices. And finally, a third example is that I'm out of time, so I'll just very briefly mention it in graph theory. We define an adjacency matrix. A graph is defined by a set of vertices and edges. And an edge, so the set of vertices, you can denote it by one to n. So if there are n vertices in the graph, then you can label those vertices as one to n. And edges are pairs of indices, pairs of vertices, i, j, i1, j1, i2, j2, etc. And an edge exists if the two nodes i and j are connected to each other. And we define an adjacency matrix to be a matrix A which contains entries a, i, j where A, i, j equals 1 if there exists an edge between nodes i and j and 0 otherwise. And if the graph is undirected, undirected means the edges don't have a direction associated with them. So you have two nodes, you define an edge like this, not with an arrow going like this. It's an undirected graph, then undirected graphs. So these are three examples where symmetric matrices arise. And so with this background, we will actually start in the next class studying about Hermitian and symmetric matrices. They have lots of very intricate, very beautiful properties. And so we'll cover them.


I mean, I have dreams of me selling cocaine. Yeah, yeah. The police kicking the door in and me going back to prison and then all my friends were sitting there like, we knew you was coming back. Yeah, we on boss talk one on one, one on one. Yeah, we gonna talk. And people don't really think about it. You can't buy good health. Correct. It's something you gotta, you have to have peace. You think like when it comes to anger and all these different things, do you believe that? I don't really get angry. That's what I'm saying. You really ain't mad. It's like so many of the things that they say triggers it, but now stress and stress is something that you don't know. You can carry stress and don't know, right? Yeah.  Really? Yeah, during the daytime. Dreaming. You know, I have nightmares about being in prison. Yeah, yeah. And most of my dreams, I still do. I still have nightmares about prison. Really? You gotta realize he did 20 years. I know.  Subconscious mind is locked in your subconscious mind. That's why it's one of the most important things that we have to learn is to control our subconscious mind. How? Well, whatever we feed it is what it picks up on. Yeah, so we have to really just be conscious of everything that we're feeding our mind. Because most of my dreams are about business. Mm-hmm. And it's usually about business where I barely make it. It's like this real tense situation and we struggling and we struggling and I barely make it, you know. So I imagine. And that's your dream? Those are my dreams. Okay. Those are the things that I dream about. Like I don't dream about being on Fantasy Island or... No, because you say you dream a lot about being in prison too, but that's not a dream of in prison. No, no, no, no. It's either being in prison or some kind of business. You gotta understand, my friend, like I was locked up when he passed away. Mm-hmm. And that haunt you. Years, I used to dream about him until one time I dreamed and I told you about it. Yeah, last year. He died in my dream. Yeah. And I never have dreamed about him no more. I wish I could stop dreaming of dying. My prison assistant would die, but... Yeah. I just... I mean, I have dreams of me selling cocaine. Yeah, yeah. The police kicking the door in and me going back to prison and then all my friends are sitting there like, we knew you was coming back. I wonder if our subconscious is because that is a fear that we have subconsciously. I wonder if that's the reason why subconsciously you dream about certain things because it's fears. It is. It's a fear and it's also an addiction. You know, selling drugs is an addiction. Right. So, quite naturally, if you done it as long as I did it and enjoyed it as much as I enjoyed it... Because it was a challenge. It was a challenge.  Because one thing he's always told me is when we started out selling clothes and stuff like that, he would use it as like, that's the drug game because we will go hustle the clothes and I love the adrenaline rush of being on the street, selling the clothes and walking up to people and be like, hey, especially making a lot of money doing it and I can see how I can think about that as a drug game.  So that's why I'm like, it should be the same. Well, it is kind of, I mean, the games are the same but I haven't been able to take those nightmares out of my system to where I only think about the stuff that I'm doing now. Now, I don't have a prison dreams every night, you know?  Sometimes we don't remember. I can remember though, when I wake up, you know... For a moment and then you forget after a while. Wow, that was crazy. I'm glad that was a dream. Dream. That's real. It gets so vivid to where I would dream that I was, like, I would dream stuff and it would happen while I was locked up. I would dream it and... Oh, no question, dreams are a form of things to come. Yeah, yeah. Not all, but some. Not all, but most of them. Or Jamaicans would say, dream, don't walk straight. That's what Jamaicans would say, dreams don't walk straight because when I say that, some people can dream things and it happened and people would be like, that's deja vu but it's really something you dreamt and then it happened but some dreams, it's like you have to have it interpreted. You know what I mean? You dream about fish and then Jamaicans would say, oh, somebody's pregnant. Right, right, I heard that. Well, I believe that dreams are a form of you predicting what the future is. It's like a vision where you visualize something to come and that's how I see it is a vision. Where I can see myself in business. I mean, if you go back and you look at some of the articles that I did when I was in prison and I'm talking about these dreams and these articles and now all the things that I talked about in those dreams are manifesting themselves right now in the world. And dreams prepare you for different things because I can remember one dream in particular that I think that dream prepared me for my father's death because it was a dream that happened and it was so weird because of what I saw in the dream and then no later than probably a month later I got that call and I knew that that dream that's what that dream was. Yeah, wow. Yeah. Yeah, we on boss talk one on one. One on one.


A sitting of the House of Assembly is scheduled for Tuesday, November 10, 2020, with papers to be laid by the Honourable Prime Minister and Minister for Finance, Economic Growth, Job Creation, External Affairs and the Public Service, the Honourable Minister for Agriculture, Fisheries, Physical Planning, Natural Resources and Corporatives, the Honourable Minister for Commerce, Industry, Enterprises, Development and Consumer Affairs, and the Honourable Minister for Tourism, Information, Broadcasting, Culture and the Creative Industries. The Minister for Finance will seek Parliament Authorization to borrow an amount of US $30 million from the Caribbean Development Bank consisting of a special funds resources portion in the amount of US $10.8 million and an ordinary capital resources portion in the amount of US $19.2 million for the purpose of financing the implementation of policy reform initiatives designed to support St. Lucia's COVID-19 crisis response and achieving fiscal stability. Four bills are down for consideration including public finance management and tourism levy. The St. Lucia Tourism Authority, in collaboration with Events Company of St. Lucia Inc and guided by the Ministry of Health and Wellness, will welcome the Atlantic Rally for Cruisers, Arc and Arc Plus 2020 fleets to St. Lucia in early December. St. Lucia has been home to the nautical competition for over 30 years. This year's edition will be recognized as the first major event amid the coronavirus pandemic. A total of 26 vessels participating in the R Plus departed Las Palmas de Gran Canaria on Sunday, November 8, 2020. The Arc fleet of approximately 60 boats will start their Atlantic journey on Sunday, November 22, 2020. The event this year will take place within the realm of strict COVID-19 protocols, get towards ensuring the health and wellbeing of all staff, participants and stakeholders at the forefront of the event. All vessels were required to arrive in Las Palmas no less than one week before departure. Minor participants underwent the necessary COVID-19 PCR testing and remained in isolation until departure. They were also required to participate in mandatory virtual briefings on St. Lucia's strictly enforced protocols ahead of departure. St. Lucia Tourism Authority's Public Relations Manager, Geraint George's, indicated that once in St. Lucia, all protocols must be adhered to, including daily health screenings while in quarantine. There will be mandatory health screening upon arrival. There will be continued wearing of the wristbands to differentiate non-quarantine participants from those cleared for movement by Port Health. You will also have daily health screening of those in quarantine and additionally individuals aboard the vessels arriving in St. Lucia ahead of a 14-day period. So as much as they're quarantining from they get to Las Palmas, they're sailing down to St. Lucia. If they observe less than 14 days underwater, they're still required to quarantine in St. Lucia. So all efforts are to ensure that this is one of the safest events that we are hosting. Given the nature of the arc, participants often spend 14 days or more at sea and in such cases an additional quarantine period is not necessary on arrival to St. Lucia. George's explained that those arriving to St. Lucia ahead of a 14-day period from the point of departure in Las Palmas will be required to quarantine aboard their vessel on the designated quarantine dock at the IGI Rodney Bay Marina or at a COVID-19 certified facility until the mandated quarantine period has elapsed. Given that the quarantine period for arc events will be observed with a minimum of 14 days, COVID-19 PCR testing upon arrival in St. Lucia will be conducted as necessary by Port Health authorities. All arc participants must always wear face masks once outside the cabin of their vessel. George's added that social distancing measures will also be enforced and as a result, all vessels will be docked a minimum of 15 feet apart. Security will be available 24 hours daily to ensure all rules and protocols are followed. Participants were informed that violation of the quarantine act is an offense and punishable by fines or prison. When it comes to the protocols and the access at the marina, access remains the same. Of course, you would have had to be cleared by Port Health as a crew member to be able to access the regular operations at the marina just like the locals are able to access. So if you have not met the requirements by Port Health, you will remain in quarantine. That means you are on a dock that is secure, that is also monitored by security so that we you as someone who have not met the quarantine requirements are not able to access and mingle with anybody else. St. Lucia has been home to the Atlantic Rally for Cruisers for over 30 years. The Arc and Arc Plus Events 2020 edition will once again culminate at its home base for the 31st year at the IGY Rodney Bay marina. From the Government Information Service, I am Janelle Norville.


So, let me just remind you, right, so last time we talked about the function R in R is differentiable at some point, rather than jump today. If two things happen, P is in the interior, so it's not an isolated point, it's not on the edge, and also if you take the limit as divided by that distance minus, oh, sorry, there is a vector of that, the same a dotted with the unit vector x minus t, it's linked. And so here, that's the definition and in fact, a is the gradient of f, the vector, which is the vector of partial derivatives. This we did last time, so what this is really saying, oh, sorry, equals zero. So those two things tend to be the same. Somebody would say we don't have to divide by the links, but we do, because if you don't divide by the links, then you just have the limit of one thing going to zero minus another thing. Yeah, it's just you have zero minus zero. Of course it's always zero as long as that limit exists. Right, because we don't divide by these links, and this goes to zero, this goes to zero, there's nothing interesting going on, we can put any vector. But if you divide by the links, this is saying just like in the single variable case, if you look on, so this says on a really small scale, looks like in the very middle of the term, a constant that's this part, plus the projection onto some vector a, onto some vector a, that's what that's saying. So this business, so this gives you a unit vector, or this business and this business is really saying on a really small scale, because we blow it up to a normal size. So this is sort of the same as in the single variable case, if we, here's my analog of p, and I blow up this picture, so this distance is my x minus p twice with, and I blow up this picture, and then I blow up this picture again, and I keep blowing this picture up, and more and more it looks like a straight line, with slope, well in this case the slope is, well whatever, with this negative slope, it looks more and more like a straight line, where this slope here is the derivative. Now here we don't really have a slope, because my function is well in two variables, I have some surface, this is my surface here, my point p, and I zoom in, I blow this up so that it looks more and more flat, and this vector a is giving me this tangent point, well it's not really the tangent point, is giving me something about this slope, in fact I'm drawing the wrong vector, it's this way. It's actually where it tilts the most, and this is a projection, the change here is pretty much the direction, if I go in some direction like this, and I look at how much I blow in this direction, it gives me kind of the same thing. So this is, this is, this vector is playing the role of this slope. Yeah, that's just to rule out, so. Yeah, so p is a point where the function is defined, if it's not in the domain of the function, this doesn't make any sense, and not only does it have to be in the domain of the function, it can't be in the edge, so I can't, just like in this case, if this function ends here, I can't take the derivative at the edge, and similarly, if this thing has a hard edge, it can't take the derivative here, or if I have an isolated point here where it's defined, I can't take the derivative at that isolated point. So we have the same condition in one variable, we have to have a neighborhood or an interval around, around the point where we're taking the derivative, you can't take the derivative just at the edge of where it's defined. And the same thing here, we have to be able to take this limit in all directions, where we can't have a cut, because then weird stuff could happen. So we really need, in the domain of this p, we really need all around, so that we can come in from all the directions and look at that limit. Yeah? There's no idea of coming in from all sides, it's like analogous to, like, left sided and right sided limits. Yes, it's like we have left sided and right sided limits, so in one variable, there's only two ways you can come, you come from above or below. Here, I want to be able to come from above, below, right, left, up, down, part way through all the variations. I need space around it. On the line, right? On the line, I just need space around it, but there's only two ways to go. So this is the, this statement is the analog of saying I have both left and right limits. This is just saying I have left, right, and all combinations of up and down and all around. Yeah? What is the gradient of the unit? It's the vector that, it's the vector where the function increases the most. It's the direction in which the function is steepest, but we'll see that in a little while, but this, so it gives us a direction, right? The gradient that doesn't live, so I first drew this, and that's wrong, because the gradient doesn't live in that space, right? The gradient, I'll draw a picture of it.  I want to think of, so I take this plane and I lift it up there to get my surface. So this height, so this is, this is the land where x and y live, and this height is the f of x, y direction. And if I have some point, p, p lives here. p is a combination of x's and y's, right? And the gradient vector, so the gradient vector here lives here. The gradient vector is some relationship between x's and y's that tells me about the tangent plane. Well, it's a, it's a number, it's a vector. It's not a number, but it's a vector. It's constant. Just like the derivative at, so just like if f, if g of x is x squared plus 5x, g prime of 1 is 2. g prime of 2 is 2 times 2 plus 5 is 9. It's a number. And that is the slope that tells me how much g is changing at this point. Similarly, if f of x, y is x squared y plus 5x, there we go, then the gradient of f at some point 1, 2 is, so now I have to compute the partial with respect to x, so that gives me 2xy plus 5, and the partial with respect to y gives me x squared evaluated at 1, 2, which is 4 plus 5 is 9 and 1. So this guy, this vector of 9, 1 is telling me that if I go, if I increase 9 in the x and 1 in the y, that's like the slope. That's the place where the function increases the most. Yeah. Is something gradient technically defined for three variables? Any number you want. Any number? Yeah, this is a gradient too. It's a gradient of a function from R to R, so it's kind of, it's the vector 9, which is, right, and I can define the gradient for 23 variables, number 1. It doesn't matter. It's easiest to visualize in the case of either 2 or 3 variables because it's all the same. So one thing that you should keep in mind is that everything we do in this class should apply, if you interpret it correctly, two single variable calculus. So here when we have a gradient, we have a one vector, which is also a scalar. So the derivative is a gradient, which is not a very interesting vector. So you should always, when we talk about something, you should say, how does that compare to the thing I already know? How is it extending that other notion? Because that's really what the whole point of this is, is take the stuff that you worked out in one variable and extend it to more than one variable. See that it's the same thing if you think about it the right way. All right, so we did that, and I remember we did that. So this is, you know, this is a very important idea. Yeah. Is there a concept of gradient like for a vector value function where you have like R2 to R3? Yeah, that's what I want to get to today. Okay. It's called the derivative. Wow. So the gradient is like a special derivative, but we want to generalize that to functions from R2 to R3 or R1 to R3, all of those things, right? We're not quite there yet. We're kind of, I mean, if you think about it a little bit, you could probably figure out what it's going to be because it's kind of like the gradient. So we've already covered the case. So let me not tell you yet, but I hope we'll get there. We already talked about a function from R to Rn, and we, so we know what f prime of t means as a vector, and we've also talked about, I'll use g now, some function g from Rn to R, and we have this gradient object which is also a vector. And you want to sort of, maybe one of these should be m. You want to sort of put these two ideas together. If sort of one goes one way and one goes the other way, so it shouldn't be surprising if we have some h from Rn to Rm, it's going to be. These are both vectors, but they're sort of vectors measuring two opposite things. So this would be a matrix to count of h prime. Derivatives of h is a matrix. So this is a linear approximation of this function, and it has to be a vector. This is a linear approximation. This tells us something about the linear approximation to this, and it's a vector, but it's sort of a vector on the input side. This is an output vector for an input vector, and this we need to consider both how the inputs vary and how the outputs vary in the matrix. So I'll come back to that, I hope, in about half an hour, 20 minutes, something like that. I'm not quite ready to go along. Other questions or? No, you have a question. Yeah, but I forgot you long ago. Sorry, it's okay. So a, gradient vector, that assures us that we do that with a circle in this quickly arrow symbol of direction. It doesn't matter what direction x approaches from. A would be able to scope it out, whether it's limiting the problem. So, I mean, in this picture, there's going to be some vector a here. And we can approximate, so f is going to take this picture, and it's going to send it, well, to some height, or some width, whatever. Right? So think of it as just squishing. And we want to say, well, gee, how does f affect? I have the other color. How does f affect stuff along this line? So the gradient, figure out how much is going to change along that line, we can just sort of look at how it changes along a. And that gives us some idea of how f moves everywhere. So we can take any direction here and somehow see how f transforms these green things by just looking at the projection of that on how f will hit a. And the projection, all the lines should somehow match up. Yeah.  On a very small scale, this function f, maybe it stretches a lot here. Right? So, I mean, this picture isn't very good. But on a very small scale, maybe f grows a whole lot, grows a lot here, and only a little bit here. But on a very small scale, there's some constant rate of change. What it derivative does is it replaces something wrongly here with something linear. And our linear map is just a dot product in some vector. And so this a is the vector that we're taking the dot product with. This a is also called the gradient of f. The gradient of f is the vector on which the function acts like a dot product with. Remember a dot product, I have one vector and I have another vector. And if I take the dot product of these two, it's giving me something about the projection onto that. Well, in some sense, all linear maps from rn to r are just like a dot product. I'm just taking combinations of the x's and the y's. I mean, it's okay. This is a really important point. If you come in on a line orthogonal to the a gradient should a zero. So then the function should be in that small neighborhood constant. That's right. So, I mean, if you think about some tangent plane, because the thing is essentially a tangent plane. And I know that it's steepest this way. This is the direction. This is the direction. This is now thinking of it lifting this up. This is the place where it grows the most. That's the way to walk uphill with fastest. Well, if I go orthogonal to that, I'm not going uphill at all. These are the contour lines.  I don't go down. I just go opposite from up.  And if I go a little bit above, well then, my gain is just how much of that I actually used. So this, this thing is like having a little arrow that says this way up, you know, to the top. That's what that gives us. It tells us the direction in which the thing grows the most. Yeah. If you remember your exact statement was about dotting a vector with a gradient kind of difference. It gives you the amount that that vector sees of the gradient. If I have two vectors and I take their dot product, I'm essentially taking the projection of the one vector. It's the angle between them. So I'm taking the component of that vector in that direction, the reason for this is that the product was like projection times the absolute values. It is. So we're going to do the absolute values of the other two. Oh. Yeah. So I'm, I'm, I'm being a little fuzzy when I'm saying projection. Okay. It's just like a derivative is the projection onto the line that it started with, but then scaled by how much it stretches. So, so when you are looking at something or like something very small you're doing is taking a vector and like mostly we have a particular vector as a group base. I think a vector on the xy plane is not that would be brilliant and that would give me how much. Well, the gradient has to have something to do with the function. Right. I can't just take an arbitrary gradient. So, so, so let me, let me talk a little, let me say another thing and then we'll come back to our score. Okay. We're alright with this? I hope maybe. Alright. Somewhat kind of sort. Okay. So let me, I'll leave that. That picture has nothing to do with anything, but I'll just come. I'll leave this. Although I don't know why. Okay. So, this, this gradient function is a kind of a derivative, but it's not a derivative, but it says something about the derivative. And now let's, let's try, so we already had this idea of the partial of f with respect to some variable x. Right. So if I have f of x, y, I don't care, z, whatever. But this is the limit, let's say at some point b. As x goes to p, t goes to 0. Just to 0. Um, f of, f of x plus t, y, whatever, all of the variables. Let's just do x, y. Minus f of, oh, this is, yeah. Okay. Let's just do it that way. x, y over t. Right. This is our derivative in the x direction. This is a number. It's a scalar. It is not a vector. But we have several of these. We can fix any, any particular coordinate vector we like. We look at a limit like this. It gives us the slope, the amount of change in that direction. Well, we don't have to be limited to the coordinate directions to do this. We could, by analogy, say, suppose we take any vector v, which is not 0. We want the 0 vector. But we take some vector v. We could make a similar limit. So this is the derivative with respect to some vector, which we could take as the limit as t goes to 0 of f of x plus that vector minus f of x over, so we can certainly define this thing. Remember, x is a vector. t times v is a vector. So we can certainly do this. Whether it means anything or not is a different question. We can certainly do this. Right. So we can just take some vector and look at the derivative. Let's just define the derivative with respect to that vector, which is some combination of x and y as this. So as an example, I'm not saying what it means. It's just a thing. So we could take v to be the vector of 1, 2 and f of, let's just do two variables to be, I don't know, x y squared. Maybe x squared. Let's do x squared. Okay. And so now I can define the partial with respect to the vector of 1, 2. Derivative with respect to the vector of 1, 2. So this is sort of a direction, a direction, but it also has a length. It's defined to be the limit as t goes to 0 of, right now, x plus t plus 2t squared minus x cubed y squared. Right. So this is something. But in fact, so, but I don't want to do that. What would this thing mean? I mean, so you can do it out. You can take the limit, blah, blah, blah. Yeah, okay. I have a guess. The derivative in the x direction plus the derivative in the y direction. Okay. So the claim, the guess, let's call it a guess. Yes. This is, you're saying this is x 3x squared y squared times 2 plus x cubed 2y, oops, times 1 times 2. Is that what you're saying? Yeah. Okay. People think that's maybe true. I don't care to forget about what it means. It's just a limit. I think it's true. Why not? Sure. Let's just say it's 7. No, x is in y's here. So what is this? This is the gradient of f dotted with where you came up with 1, 2, blah, blah. Right. So this is the gradient of f dotted with p. Now, really, I've been just thinking about everything at a point, but fine. Think of x and y as constant here. So that's fine. So that's actually true. So this is just a definition that we can just define for the heck of it. But, so this is actually, I don't know, we call it a theorem. The fpv, this funny vector derivative, is in fact always gradient of f dotted with v. So if f is differentiable at the point where it matters in the neighborhood of that point, then that's true. Well, so if, yes, it's differentiable x.  So through that, although, I don't know, does this seem reasonable for people or does it just seem like I'm just writing garbage? Or maybe both. The combination of the both. Okay. So here, I'm differentiable in some vector x. I think you have x as fixed. Then the partial here fx is the gradient fx. So let's see why this is true. And then I actually only want to see what it means in a special case. So f is differentiable. So it's differentiable in some vector x. And so now I'm going to think of x as playing the role of p here in this definition. And I need some other letter to put for the x here. So let's call that s. So by the definition of what that means, that means if I write here, should I write here instead? It's too hard to see here. Okay. So since f is differentiable in s, I'll just copy that down. That's the limit. As my variable s tends to my fixed state x of f of s of the vector minus f of x of the vector over the distance between s and x minus the gradient of f evaluated at x dotted with s minus x over its length. So that's just rewriting the definition of, you know, I know a is a gradient. So let's just rewriting that here. So I'm going to use the substitution using s for the thing that moves and x for the thing it moves to. Right? Okay. So now in the special case, not s, if, so if we take s to be x moved a little bit in the v direction. Right? That's all that says. So just think of t as some small number and it just says, start at x and move away from x by t times my vector v. Then we can just rewrite this. Oh, this equals zero. I always predict it equals zero. So that equals zero. Let's write it here. Zero equals that. So that equals this. So I can rewrite this. This is the limit. Well, since s equals x plus tv, that's the same thing as saying tv goes to zero or t goes to zero. So I can just put a t goes to zero here.  That's just this part. And let's put them all over one direction. Minus gradient of f at x. And then s minus x is tv. And this is just left over nonsense. And then we divide everything by the length of s minus x. So that's the length of the vector t. So these are equal. This is zero. Yeah? Okay. So now what? I can cancel this t out. So t is going, since this is zero, I don't care about the sign of t. So if I change the sign, it's still going to zero. So that means that I can divide t by absolute t. And it's still going to zero. Right? So I can cancel out this t. I don't know why I put it together. I want to split it up. So this t and this t can cancel out. You want to do much to it. Except maybe let's put the v out in front. So let's actually write this. I'll do the same thing. So I have a fact, this is a number. Remember this whole thing equals zero. This is a number. So I can just multiply through by this number. This number isn't zero. So I just forget about the v on the bottom. And since this whole thing equals zero, I can just move it over there. So that's what I want. So I'm done. Yeah? Okay. So do you put that using the one definition of a gradient? I use the definition not really of the gradient, but of the derivative. But yeah. So, right? The function has a derivative or it's differentiable. If there is some vector, what do we have to call the gradient? For here we can call a whatever. There's some vector that we call grad a. So that this is true. The definition is a divided by b. I would call it a. Sure. I mean, this notation, we adopted as this vector. And then we showed that that would be the vector of partial. So if you remember last time, we just said there's some vector. Just it's a vector. Whatever. And I'm going to call that vector a with that higher gradient a. So I'm going to start writing this. And that vector, by the way, happens to be the vector consisting of the partial. So this is not a definition. Yes. It's in the reading of it. But this is not a definition. This is a fact. This falls out of this definition. So given that, so I didn't really use this part in there. Just did the thing using that definition, changing the letters around calling xs and then you'd be able to get that. Is there another question? Okay. So that means that whatever the heck this thing represents, it's easy to calculate. And it represents something which is something not actually very useful, but it's a thing. But in the special case it is useful. And that special case is the useful case. And then we can forget about the other part. If b is a unit vector, so let's call it u, then taking the derivative with respect to the unit vector, well the theorem tells us that this is the gradient of f dotted with that unit vector. This is called the directional derivative. Or if you prefer it's the derivative, the direction. So let's just do this same example again where we take u to be, well I don't want 1, 2 because the length of that vector is not 1. I want it to be a unit vector so I have to divide by root 3. Sorry, root 5. That's a unit vector. It goes in the direction parallel to 1, 2. And if I take, let me also write it, sometimes this notation is a big D with a real u. So it's d sub u of f. So the directional derivative of f in this u direction is whatever that number was, oh I didn't do it at a point. So let's do it at a point. At 1, 1 is, I've lost, what is f? Let's write what f is. f is x cubed, x cubed y squared. So f of x, y is x cubed y squared. So let's calculate the directional derivative of f at the point 1, 1. Well it's just take the partials, 3 x squared, 3 x squared, y squared, 2 x cubed y dotted with 1 over root 5, 2 over root 5 and evaluate this at 1, 1. So that's 3, 2 dotted with 1 over root 5, 2 over root 5 which is 3 over root 5 plus 2 over root 5 which is 7 over root 5. It doesn't matter if you evaluate it before. No, no because this is a vector. So then the first partials are just special cases of directionals. Exactly. So the first partial is just the directional derivative in the x direction and in the y direction. And this is, this is the rate of change in the u direction.  It's the rate of change in the v direction times the length of v, which is kind of, I don't know, not so useful. It would be like taking the derivative with respect to 3 x. Usually you don't take the derivative with respect to 3 x. So here, once we divide it and make it a normal length, then it's something kind of useful. And this is, this is exactly, I guess I'll come back over here.  My function looks something like that. This is in the u direction. And I'm just looking at, if I cut this thing open, I take out a saw and I cut it like this. I cut it along the u line. Then I have a graph here which corresponds to this slice. And then at this point, where my point is, here, this is the slope of that cut. That cut is not parallel to x. It's not parallel to y necessarily. It's just at some angle, some combination of x's and y's. And that just tells me the slope if I cut that thing open and look there. So I'm going to cut it going more y than x, twice as much y as x in this case. Increase y twice as much as x. And I get some slice like that. So this directional derivative gives me that kind of a measurement. So it's the slope of the tangent in the u direction. Any questions about that? So I'm sorry I don't do a whole lot of examples, but we only have a little bit of a problem. So we're all right. What are the axes on that thing that you just drew? Is one the vector? I'm sorry? What are the axes on the plane that you just drew? So the axes on this plane, so this picture is not the 1, 2 vector that I talked about because the axes in this picture typically would be like that. And so this is something like the vector minus 1, minus 1 plus 2. Something like that. Right? x is negative in this vector. I mean, you know where you have the vector on the up here? No, right next to it. Yeah, that one. Okay, what are the two axes? So what I did here to make this picture, I cut this thing open in this direction, and so cut it open in this direction. This surface cuts that plane in some direction, and my axes in that direction. So here this picture is I draw now this picture, the green, and I lay it down flat and I see that. So it's not the standard axes. It's I've rotated my axes so that my horizontal axis aligns with you. And that's the picture I'll see. Does that make sense? Yeah, I got it. So this, I mean this would enable us to, for example, take a derivative of a function with respect to negative x, because we look in the negative one direction, which will just flip it around and give us the negative slope below x. Okay, other questions on directional derivative? They're very easy to calculate. You just take the gradient and you dotted with the vector that you want, and that's what it means. It means the derivative in the slope in that particular direction. Okay, yeah, sure. Maybe you want to know, so maybe, so say you have a, you have a, suppose you have a function that describes, I don't know, the rate of increase of something. So I have a, I have a, suppose I, suppose it's an economics problem. So I have some, you know, I know that the revenue or the cost of producing something is, take the amount of, I don't know, free inputs, A, B, and C. So let me not, you know, you have sand, you have water, and you have gold. You're going to do something with sand, water, and gold. And the cost of it is the square of the amount of gold and the quarter of the amount of sand and the cosine of the amount of water. That makes no sense at all, but that's what it is. Okay? And so now I have, now I know that I'm always going to make something in the ratio of 312.  By doing it in the ratio of 312. And so I really want a directional derivative in the direction of 312. But of course, I don't want the 312 to count, so I want to divide by the length of, so what's that, 9, 10, 14. So I'll take the directional derivative of that cost function in the 312 direction. So if that will tell me how much if I increase a little bit, how much does my cost increase? Because both easier and essentially clearer. Yeah. It's just a way of saying I'm specializing to a specific line. Also, I mean, if you imagine that you have some hillside where you know if I have some surface like that and I have some curve that goes along that surface, how much is it increasing right now? Will that be the directional derivative in the direction of the tangent of that curve? That'll correspond exactly to the tangent vector of that curve. So maybe in the base I'm tracing out some spiral path and so on this hillside, how much do I go up or down in this case if I trace out that spiral path? I don't know what that path is. That's complicated. But it's easy to calculate because I can just say well at that point I just want to look at the directional derivative in that direction tangent to that curve and tell me how much I'm dropping along that path. Okay. Okay. I don't know that we're going to get there. I will try. Yeah, I guess I really kind of want to say this. So one other thing that I want to mention and I'm worried too much about it but I think it's worth mentioning, maybe you remember from single variable calculus there is this theorem of the mean value theorem that says that if I take, I have a nice function and I take two points A and B that there's a point somewhere in between where the tangent line is parallel to or has the same slope as the difference in the values. So the mean value theorem says if F is differentiable on some interval AB, then there exists some C in that interval so that F prime of C is the same as F is coming in minus F of A over B. So that's the one variable mean value theorem. And this has this, you know, when you do this in calculus one, people go yeah, why do I care?  But you need this to be able to prove a lot of things and in particular one consequence of this, of this theorem which seems kind of stupid but it's true is that if the derivative of a function is zero then that function is constant. So this implies if F prime of X is zero on some interval then FX is constant. So that's one use that follows from the mean value theorem is to be able to prove that fact. You prove early on that the derivative of a constant is zero but that's the only way you can get a zero derivative on an interval. Right? It's a singularity fact. So there is a mean value theorem for more than one variable. So but the mean value theorem is really like a line kind of thing. So the mean value theorem for say some function of R end of R. So suppose I have a line, I'm going to use gamma. Maybe I should use s. And it's all inside the domain of F. So F is defined in some open blob and there's a straight line connecting two points P and Q. Oh shoot, I changed my notation. Oh well. Then and F is differentiable around that line. Then there's some point C, there's some vector C on that line so that, and I guess I need to look there, the difference in the values, I guess the Q is the end, is the same as the gradient of F evaluated at C dotted with the vector Q minus P. So this is sort of a directional derivative version of the mean value theorem. But this is the mean value theorem interpreted in the context of a directional theorem. But that means, so let me not prove the mean value theorem, there's a proof of it in the book. It works exactly as the proof of the mean value theorem in one variable except you have to make sure everything's a vector where it needs to be a vector. So this means that if the gradient of F, a constant sum open is zero, the zero vector on some open set called a blob, so I'm connected in the domain of F, then F is a constant. So again, exactly by analogy with the one variable proof, it generalizes the two variable. Now it needs to be connected, right? If I have a function where the gradient is zero here, and even if it sort of sneaks inside, and then the gradient is zero here, that doesn't mean that the function is constant because these could be different constants. In the same way with the single variable, if I know the derivative is zero here and the derivative is zero here, it doesn't mean the function is constant. Okay, so I just want to mention that. It makes me feel bad if I don't at least mention this. Any questions at all, man? So I mean this is really just the mean value theorem in one variable, but you don't want to align that maybe it's just a ratio of x to y. It's really just the mean value theorem. So I guess I did say I would get to something, so let's get there. So now let's move up some dimensions. Let's consider general functions. So now I'm going to consider f taking rn to rn. So this function takes in an n vector and gives out an m vector. And so associated to this, we don't have a derivative vector. So what we want, we want df or f prime derivative. Well, the derivative is always the best linear approximation. Well, linear maps are matrices. A linear map from rn to rm is an n, an n by n matrix. It's just a combination of all of the variables in each of the variables. So if I take an n by n matrix and I multiply it by, so if I take 1, 2, 3, 4, so they're from 2 to 2 times x, y, this gives me x plus 2y is the new x variable and 3y plus, sorry, 3x plus 4y, the new y variable. So the linear map in each of the components. And I have some mapping f. Let's just do r, I don't know, rg, r2. Let's just have an example. So this would be something like f of x, y, z is, well, I get 3 coordinate functions. I get say x squared plus 2z, xyz, z plus x plus y. And then to stop at 2, okay. So I put in 3, I'll come here, right? Yeah, otherwise it would be r3, r3. So I want to approximate this by linear, some matrix thing. Well, but if you think about it, this is just, I have some function f1 of x, y, z here and f2 of x, y, z here. And I already know how to calculate derivatives with respect to a function from r3, r1. Gradient tells me I know how to calculate from r3 to r1. So each of these coordinates I can view as separate functions from rn into r. Or I could cut it the other way and I could look at the column vectors, except I have far too many. I could also look at the other way. Okay, so that means what I want to do is some combination of gradients. So that means that what I can do to find derivative matrix, well, I just take a partial of f1 with respect to x and the partial, what do we call it, x1? Partial of f1 with respect to x2. And let's see, how far across do I go, m or n? Partial of f1 with respect to xn. And then down this way I take partial of each component function until I get to the last one. That's an n. That's an n. Right? So I just take all of the, so these are the, these are the gradients of each of the component functions, but I'm not adding them. I'm just listing them. These are gradient vectors. So I'm not dotting them. And these, these guys are actually the tangent vectors with respect to a given code. So, right, the columns are tangent vectors and the rows are gradient vectors. Do an example or two? I guess I'm moving this way. I should probably be speaking in Hebrew or Arabic. I don't know this way. So maybe an example, let's say f of x, well I wrote one over there. Let's see if that is it. f of x, y, z is, I can write it this way, x squared plus 2z x, y, z. So then the derivative matrix is going to be, well it's going to be 3 by 2. So I'm going to have the partial of this guy with respect to x, the partial of this guy, wait, the partial of this guy with respect to y, the partial of this guy with respect to z, the partial of this guy with respect to x, the partial of this guy with respect to y, the partial of this guy with respect to z. And at a given point, so let's say you just plug in 2, 0, 2, 6 and this tells us that it tells us something how this function of three variables changes when I vary those three variables. So this tells us that near the point, near 1, 2, 3, the function of x squared plus 2z x, y, z, 1, 2, 3. What is it? So 1, 2, 3 is 1 plus 6 is 7 and 6. It's a lot like 7, 6 plus 2, 0, 2, 6, 3, 2 times, let's call it u, x, y, z. Let's call it x, y, z. So this is saying, so that is 7 plus 2 delta x plus 2 delta z, 6 plus 6 delta x plus 3 delta y plus 2 delta z. Again, thinking delta x delta z, these are small quantities. So we could also write 7 plus 2 times x minus 1, 2 times z minus 3, 6 plus 6 times x minus 1, so 3 times y minus 2 plus 2 times z minus. This is just sort of putting together all of the stuff that we've already done where we're just looking at how it varies in each of the output components gives us a derivative matrix. So maybe another way to say the same statement that's left is so we have some function f from Rn to Rm. Then I have a tangent approximation, so I have some point near some point b in Rn. Then the tangent approximation to my vector x is going to be my function evaluated at the point plus my derivative matrix evaluated at the point dotted with the distance between the x vector and the point where I think of this as a column because this is a matrix and the dimensions will be wrong but I don't write that as a column. My tangent approximation is the value plus how far away I moved in the various input directions times the derivative matrix. That sort of puts everything together for a general function from Rn to Rm. I'll do a couple more examples next time and then we'll move on to the vector field because you're still the same kind of thing for any program. Okay? I will update the homework.