How's it going everybody? DatoDoy here and it is finally time after what feels like months of waiting.  Now I say out like that because really he's only out on the Nintendo Switch version which I don't currently have on me right now and for some reason because of the update that comes along with Goku GT you can also see him in some arcade mode teams. Luckily for us being in arcade mode means that we can give him a try as long as we find him and use Captain Ginyu's body switch on him.  Nintendo Switch users like I said can just go ahead and buy the character and play him right away and you are going to want to do that because this character is so much fun. So this is going to be a basic overview of the character we're not going to get into the nitty gritty just some fine details of the character show off some of his supers and stuff how they work what you can do with them just a little but a more in-depth guide with combos and stuff will probably come at a later date. Alright so let's get started with two questions that we all have about almost every character that comes out in this game. Does he have a low 2L? The answer is yes he does have a low 2L. Now he is a short character so it is a little bit stubby but of course having one is already a thousand times better than not having one. So Goku GT can go ahead and get a checkmark in that category. The other question I always see people asking is what is this character's assist? For some reason Arxis always holds this info close to the chest but Goku GT's assist is actually a beam which is super good. Think of it like the opposite of Cell. You know Cell comes out and jumps up and fires a beam downwards. Goku GT runs out a little bit and then fires a beam that goes up. I like this a lot better than Cell's assist because one it covers a more useful area of the screen in my opinion. It might be a little worse in the corner though but it goes full screen it acts like a beam it pretty much has all the useful qualities of it. I didn't know it's not as good as Super Saiyan Goku's beam or Goku Black's beam which are just straight on amazing assist probably the best in the game but this one is going to be super easy to slide into teams you know you never have a hard time getting a beam assist character on a team in fact they're usually pretty easy. All that matters after that point is if the rest of the character is good and I think it's very early but I think Goku GT could be exactly that. Another quick thing I should mention is that Goku GT does not have a standard down heavy in the air so you won't be able to do those standard extensions like you would normally expect with most of the cast. I think someone like Ryu Zamasu you know how if you do it down heavy with him he just kind of does his normal heavy but with a little bit of extra flip I don't know how else to explain that. You know what I'm saying if you've ever played Zamasu. The same goes for Goku GT here so you're gonna need different air routes in order to fully maximize what you can do with this guy. Moving on to the supers we've pretty much seen everything this character has to offer. His level one is a normal super Kamehameha and he'll go up to Super Saiyan in order to use it. Interestingly enough I don't know if this is going to be like this when he actually comes out or if this is like an arcade room thing but if you kill somebody with his level one close enough to when they get knocked out it actually is a destructive finish. I don't know if that's on purpose like I said but if it is I guess that's a cool little feature. Not really too important though. If a character is knocked out he will go up a level it's a Super Saiyan 3 and you'll gain a little bit more damage on that level one super. Still a level one though so that's always preferred having to spend two more bars. And now as far as level threes go we have a lot to talk about here. He has a lot of options. For starters he has this galactic spirit bomb we've all seen this we've been looking at it and yeah he can act directly out of after it which means he falls to the ground and you can act out of it but what you actually want to do with this thing is you can punish a super dash with this if you see him coming and you happen to hit spirit bomb probably not the most consistent thing but if you can get it down it's really good. Then they'll fly right into you get caught by your galactic spirit bomb and you can actually combo after the fact with this thing by switching out into another character. That is super crazy base Goku cannot do that. Already it's looking like this spirit bomb is superior. Now as other level threes involve the Super Saiyan 4 transformation you can see it right here after you land a hit. It plays a little cutscene just like Super Saiyan Goku's with Super Saiyan 3 and he powers up to Super Saiyan 4 does a dragon fist does immense amount of damage and then instantly turns back into Kid Goku. I know a lot of people were hoping even with what they've shown off that somehow you'd stay afterwards that is not the case here he just transformed back to a kid instantly. Although interestingly enough because of this update people found voice files with Goku GT Super Saiyan 4 talking about Kamehameha 10 times Kamehameha and it's really interesting. I don't know why they did that maybe it's just like future proofing like all right we got we got Sean in the booth might as well take advantage of his talents in case we ever do want to do something else with this character but people have also pointed out that it's very familiar to Super Saiyan Goku's level 3 so maybe there's something going on there I don't want to talk about that too long because I have no idea what that is but it is very cool. Okay well I got off track there Goku also has a normal level 3 that you can use when a teammate isn't defeated you'll see here it resembles when he took down Demon King Piccolo he just launches in the air punches straight through you and then decides to finish you off with a beam down very cool I actually kind of like this more than the Super Saiyan 4 Goku don't get me wrong Dragon Fist is sick but I don't know there's something so simplistic about this that I just can't help but love all right now let's move on to the moves here you can see Goku GT's entire move list I just got that up on the screen for you we'll go through a few of them this first one is just a simple charge you're gonna be seeing this in a lot of corner combos that's just the way Dragon Ball Fighters work if you got a charge that keeps them in the air a little bit longer you're probably gonna be seeing this in the corner now the medium version of this is actually kind of interesting because it plays a little cutscene if it connects I never got the heavy one to connect though the AI was just not having it for whatever reason but if the medium one connects this does play out and that's going to require you know more testing to see what that really does for you now one of my favorite things about Goku GT is the fake fake out Kamehameha this dude is a mix up monster okay it's going to take some practice against real opponents to really test all this out but man I was putting the moves on this bot look at these fake out Kamehamehas I was all over the place you can follow that up with a beam uh and not to mention we'll talk about it later but the power pull stuff is very good for mix ups as well this character is so much fun I honest opinion guys maybe the most fun DLC character ever I don't know it sucks because you know we're all aware of the stigma around this character another Goku didn't get the install that people wanted but I gotta keep it like it is dude it's a good movement tech right here I like this with that of course he does have a normal beam super useful just like any other beam the game gets us some good vanish stuff but man what a character guys now you'll notice here on the move list it doesn't say anything about his power pole and that's because the power pole is actually tied to the special button for Goku GT he doesn't have any normal keyblast as far as I'm aware you just click the special button and the power pole will come out in a straight line taking up a huge amount of screen space by the way but the power pole also has other applications like launching Goku GT in the air you can use down s to fall on them with the stick actually I really see a lot of potential in this move catching people off guard going overhead and then dropping on them with the power pole and he falls a little quicker unless I'm imagining something uh that so that in itself is already super useful but unfortunately testing mix-up stuff like this on computers is pretty much pointless you know it's pretty much I'm just running algorithms to see if they get hit uh so whether or not any of this stuff will be super useful remains to be known uh but hopefully it is because this dude is crazy thank you guys so much for watching I know this was a brief overview of the character but hopefully it got you more excited for him or maybe you still don't like the character uh but anyway I just wanted to get this out to you because I've been talking about how much I'm looking forward to this character and I wanted to get a video out right away regardless if you enjoyed this video and you're looking forward to the week of gohu GT ranked here on the channel whenever I can get that going uh probably right away and make sure to hit the like button and subscribe to the channel I also got some videos up on your screen that you can feel free to check out make sure to share this video with anybody looking forward to playing gohu GT or maybe share it with someone that was like eh I think he's going to be wack because uh who knows maybe seeing this will turn them around other than that though I have been Datsudoya thank you so much for checking out this video and I will see you in the next one


So, in the last lecture we were looking at how to analyze convergence of non-linear procedures for solving non-linear algebraic equations, iterative procedures. And we said that in general we could write any iterative method for solving non-linear algebraic equations as a one equation. See I want to solve for f of x equal to 0, x belongs to R n and f is a n cross 1 vector, this is the n cross 1 function vector. Any iterative method to solve this problem numerically can be written as x k plus 1 is equal to g of x k. Well it can be, so the old guess generates a new guess and this process is continued till difference between two successive solutions become negligible or norm of f of x goes close to 0. If you look carefully this is a non-linear difference equation, it is a non-linear difference equation. The index here is iteration index k. So, the new guess generated from the old guess g is the transformation. I showed you that all the methods that we are looking at iterative methods can be expressed in this form. Now, just like we had conditions for analyzing linear difference equations, earlier we had looked at equations of this type is equal to b x k and for this particular case we had derived necessary and sufficient condition for norm of x k to go to 0 as k goes to infinity. In this case we had a very very powerful result that is spectral radius of b is strictly less than 1. This was the situation for the linear difference equation. We had got this kind of a generic form while analyzing iterative methods for solving linear algebraic equations and this we could derive a very very powerful result here based on the Eigen values of matrix b. We wanted all Eigen values of matrix b to be inside the unit circle. Now, coming to non-linear equations, it is not possible to prove a so strong result. We can only give sufficient conditions. It is not possible to come up with necessary and sufficient condition for a general non-linear difference equation of that form. We have to come up with some kind of local condition. So, these local conditions I described through contraction mapping theorem or contraction mapping principle which forms the foundation of analyzing iterative schemes. And one special lemma that we saw was the operator g or which maps. So, g is something g maps a unit not a unit ball a ball around x naught of radius r 2 where r was a special radius which should be greater than or equal to a certain number that we had defined yesterday. If g is a mapping which maps a ball of radius r into itself then and if g is a contraction map. So, one simple way of finding out whether g is contraction map or u was to see whether dou g by dou x was strictly less than 1 or less than or equal to theta which is less than 1 for all x in. If the partial derivative of g with respect to x as a norm any induced norm strictly less than 1 everywhere then we know that map g is a contraction. If map g is a contraction then in neighborhood of x naught of radius r we were assured of existence of a solution we are assured that any sequence starting from any point in this region would converge to the solution. So, solution of this problem is x star is equal to g of x star x star is the solution and if this condition is met everywhere in this ball then it is a sufficient condition to say that any sequence generated by this difference equation will converge to this solution.  We had a sufficient condition here that if norm of b is strictly less than 1 then also this condition holds that is x k goes to 0 as k goes to infinity norm x k goes to 0 as k goes to infinity. So, we said this is a weaker condition than this necessary and sufficient condition, but this condition helped us to analyze to come up with diagonal dominance and all kinds of other things which were other theorems which were used to analyze iterative schemes. Likewise analogous to this when I come here this contraction mapping principle tells us very very important things one is that if g is a contraction map g is a contraction map if it is local derivative has see if you take g of x to be b x then local derivative of g with respect to x will be matrix b and any induced norm of matrix b being strictly less than 1 is the condition that we are looking for there. So, they coincide and then this particular equation well only difference there was the solution or the point where we wanted to reach was 0 0 0 origin in this case we want to reach a solution x star is equal to g of x star. It is possible to make everything in terms of 0 0 0 if you redefine or if you shift the origin to x star then you can make the two problems almost equivalent, but that is not so important it is just matter of shifting the origin what is important is that there is a analogous sufficient condition here for non-linear difference equations. It does not help us here to look at the spectral radius of this matrix we it does not help here for the reasons which are difficult to explain as a part of this course, but we have to use only norm and a norm if the induced norm if any induced norm is strictly less than 1 in some region then you are guaranteed that there exist a solution to this difference equation in that region. The solution is unique the third point which was very very important start from any initial guess you will converge to that solution start from any initial guess in that region you will converge to the solution x star is equal to g of x star. So, these are very very important findings of this particular theorem in general it is more difficult to apply this theorem for a complex real problem nevertheless it gives us some insights. It gives us insights for example you can try and make the sufficient conditions meet by ensuring that dou g by dou x has induced norm less than 1 you can try to do this if there is some problem in some solving some non-linear algebraic equations we can see these are sufficient conditions remember that if these conditions are violated even then the convergence can occur these are not necessary conditions, but if this happens convergence will occur just like in this case when we were talking about linear algebraic equations. If norm of b is less than 1 spectral radius is less than 1 it is a sufficient condition but if norm of b is greater than 1 even then convergence can occur because convergence depends upon the spectral radius, spectral radius can be less than 1. Similarly, contraction mapping principle gives us a sufficient condition for convergence it is not a necessary condition but if you meet the sufficient condition you are guaranteed to converge. So, this gives at least some handle to understand how the convergence occurs from that view point this is important. So, those of you who are solving large set of algebraic equations as a part of your research M Tech or PhD and hit into problems you should look at the norm of the Jacobian I mean at least that much you should remember look at the norm of the Jacobian that will give you some try to tweak try to see whether you can make the normal Jacobian less than 1 then you know you have good chances of convergence. Just to illustrate this idea of contraction map I will just give you one example here I want to solve simultaneously z plus these are two non-linear algebraic equations which I want to solve simultaneously. So, if I write this minus this equal to 0 and this minus this equal to 0 then this is f of x equal to 0.  I am formulating an iteration scheme here z k plus 1 is equal to 1 by 16 minus 1 by 4 y k square and I have just formed one iteration scheme this is not the only way to form iteration scheme I am just showing you one possible way of forming the iteration scheme. So, this is a Jacobi type iteration scheme what would be the Gauss-Seidel kind of iteration scheme if I were to use z k plus 1 here it will become Gauss-Seidel type iteration scheme this is a Jacobi type iteration scheme. Now, what I am going to do here is so I have this scheme which is y z is equal to g of y z where g is this right hand side function and then I am considering this unit ball I am considering unit ball let us say my x naught my initial guess is 0 1 and no no one minute my initial guess is x 0 is equal to 0 0 and I am considering this unit ball of radius 1 in the neighborhood of 0 0. So, I am looking at now what is what is this infinite norm g of I am taking some point x i and some point x j x here is x consist of y and g x is the vector consist of two elements y and g now I am looking at I am looking at this well what is the norm what is the infinite norm infinite norm is maximum of the absolute value of the two elements right see what I am doing is I am taking g x i minus g x j it has two elements I am just taking the maximum of these two absolute values will be the norm I am just using definition of infinite norm nothing else just this is just definition of infinite norm. So, you can show that this is less than or equal to max of I am skipping in between steps you should fill them up just go back and look at why this step comes from this you can prove these inequalities that is this particular difference is infinite norm of this difference is less than x i minus x j half times x i minus x j. So, actually the contraction constant is half see I just wanted to show that in this particular case you can show for this particular case you can show that well I am using here the fact that the elements are drawn from the unit ball elements are drawn from the unit ball. So, that is why this steps have been written and essentially using this inequalities what you can show is that g i minus g j using this inequalities you can also do analysis using the derivative of this and taking its infinite norm. You can also do analysis using derivative of this right hand side Jacobian matrix and infinite norm of the Jacobian matrix that analysis also possible. In this particular case we have found that if you apply g on any x i and x j then this inequality holds if this inequality holds what it means is that see this constant on the right hand side is less than 1. So, this is strictly less than 1. So, this g map is a contraction if g map is a contraction I am guaranteed that there exist a solution in this unit ball the solution is unique and starting anywhere in this unit ball unit ball this is with reference to the infinite norm. So, it will be a square not it will it will look like a square we have seen this how does the unit ball look like in different norms. So, starting from any initial guess within this the iterations will converge to the solution. So, this we are guaranteed because we are able to prove this inequality here for this particular x is equal to g of x. So, what is important here is that just looking at or just developing this inequality this is infinite and infinite norm just developing this inequality I am guaranteed that a solution exists in the ball I am guaranteed that I start from anywhere and I will reach the solution I will reach the solution and. So, this iteration scheme is going to work that is what I know from this analysis just right now just do not bother about this in between steps assume that this sequence is true because our aim is not to do this algebra you can work on this algebra later more important is that by doing this algebra I can show that infinite norm of g i minus g j divided by x i minus x j. So, for any i j I can prove this. So, I take any two points in this ball apply g on both the points the new points will have a distance which is closer than the original two points that is the main thing if that happens we are assured that a solution exist we are assured that starting from x naught will reach the solution moreover from any initial guess in this region if we start will still reach the solution that is the important point. It is difficult to do this analysis for a very large scale non-linear system. Nevertheless it is important to get this insight that how does one look at analysis of convergence of you know iterative schemes for solving non-linear algebraic equations because most of the times most of the times will be actually dealing with non-linear algebraic equations large scale in your work computational work because most of the chemical engineering problems 99.99 percent of them are non-linear problems reactions occur heat transfer occurs then turbulence and all these things will make the life very very complex. We have to work with set of non-linear algebraic equations what is it that governs the convergence we can get some clues if you look at if you can if you can show that the iteration scheme that you have formed actually is a contraction map difficult to show in general for a large scale system but this does give you insight which is very very important that is what you should carry. I want to stop here I do not want to get into too much details in the notes I have given some more detailed discussion on Newton's method. So there are special theorems for convergence of Newton's method and more than the proof and the theorem statement I have tried to give some qualitative insights as to how to interpret those theorems okay this go and I have not included the proof the proof can be found in any of the textbooks on non-linear systems like Reinhart and Ortega is one of the very well known textbooks. So you can find proofs there but the interpretation is quite important as to how do you make convergence occur. So typically if you have formed a iteration scheme well in this case I worked with directly worked I did not take a derivative but you could also try to see for this particular system you can work this out you can try to see whether dou G by dou x infinite norm if this is strictly less than 1 in the region of your in the region where you are trying to operate or trying to solve the problem or dou G by dou x 1 norm is strictly less than 1 okay. If these conditions are met then we are guaranteed that the solution exists and we will reach the solution okay. So these are some some why why why infinite norm and why one norm because they are easy to compute infinite norm and one norm are easy to compute okay other norms like two norms will require eigenvalue computation other than that one norm and infinite norm are easy to compute. So you can quickly make a judgment what is going wrong when you are solving the problem okay. So this brings us to an end of methods for solving non-linear algebraic equations we have looked at different concepts we have looked at how to solve them using different algorithms we just briefly test upon idea of condition number and also we very very briefly test upon the idea of convergence of iterative schemes we haven't gone deep into it but at least you know about you know what is the tool or what is the you know machinery that is used for actually looking at this problem. Let's move on let's move on to solving ordinary differential equations initial value problem. So now what I want to do next is before I proceed again we go back to our global diagram okay. So our global diagram was so we have this original problem then we use approximation theory to come up with transform problem. So we have been calling it transform computable forms and then we said there are four tools one is ax equal to 0. So this tool set we have which will be using and the other tool set was f of x equal to 0. So solving non-linear algebraic equations solving linear algebraic equations this is the second tool set that we have the third tool set that I am going to look at is ODE IVP because in many cases the transform problem is an ODE initial value problem okay even the I will talk about a method later on how do you transform a boundary value problem into initial value problem actually not just one initial value problem a series of initial value problems which are then solved iteratively and the third and the fourth tool is stochastic methods but we are not going to get into this. So right now we have done this how to solve ax equal to b we looked at many many many methods we looked at many issues that are associated with this we have looked at f of x equal to 0 and now I am moving to ODE IVP all this after all is going to give us approximate solution okay this is going to give us an approximate solution to the original problem. So moving on to solving ordinary differential equations initial value problem okay so general form of this the general the types of equation that I am going to look at is of this type dx by dt is equal to f of xt okay where f is the function vector and x belongs to Rn okay what I am what I am given is apart from this differential equation model I am also given initial condition I am given initial condition at time equal to 0 now before I move on let me explain one notational difference that we will have in this case if we are dealing with vectors we will have to deal with three different attached indices with the with the vector so suppose x is my vector okay here at element of the vector will be given by x i okay this notation we have been using even earlier bracket k will indicate kth iteration okay now additional complexity comes in we have time so time will come here so there are three things attached to the vector in some cases you will have ith component of the vector okay you will have time t appearing here okay and you may have kth iteration in some cases we do not need i and k we just might work with xt xt means vector x at time t okay vector x at time t so now a third dimension comes into picture here when you write in the notation okay sometimes there are schemes which are iterative and you need index sometimes you need to refer to ith component so you need x i and t is time okay now if now what kind of equations I am worried about what kind of equations that I am going to look at you might say that well what is written here is only a first order vector matrix equation right dx by dt is equal to f of x I am writing only a first order equation only first order derivatives and in your engineering problems you have often come across models which are second order third order fourth order and then when you did your first course in in the no differential equations you had nth order differential equations right and then you had methods of solving nth order differential equation so why am I doing things only for the first order differential equation though the difference here is a vector differential equation earlier you are looking at scalar differential equation what I am going to show that any nth order differential equation can be converted into n first order differential equations okay so this form which I have written here is very very generic very very generic okay so let us begin by looking at this conversion so let us say you have this let us say I have this differential equation in the scalar variable y so y is a scalar y is some mass fraction or y is some temperature or whatever whatever is the case you have some differential equation let us say this is nth order differential equation in general a non-linear differential equation we do not know I am just writing a generic form could be anything this is in one variable and independent variable is time okay so what I am going to do now I am going to define new state variables okay so my first state variable and when what I am given together here to solve this problem say initial value problem so what do I need to solve this problem I need a differential equation and I need the initial conditions initial conditions are given for y0 dy by dt at 0 so we are given initial condition we are given initial y0 initial derivatives up to order n-1 these are required to solve this differential equation with this differential equation together with this initial condition will be initial value problem right solving ordinary differential equation initial value problem this is this is what I get okay now what I am going to do now is what I am going to do now is to start defining a new state of variables okay so my new variable x1t is equal to yt okay x2t is equal to dy by dt okay x3t is equal to d2y by dt square up to x and t is equal to dn-1y by dt n-1th derivative I am defining new variables x1 to xn is equal okay now you can see that these variables are related through first order differential equations okay I can very easily say that dx1 by dt is equal to x2 tx2 by dt is equal to x3 so I have such n-1 equations so this is my equation number 1 equation number 2 and this is my equation number n-1 I have n-1 such one relationships between the variables okay all of them are first order differential equations okay the last one is now just the equation that we have okay so the last equation nth equation okay this is dxn by dt dxn by dt is this is nothing but d by dt of dn-1y by this is this is my definition this is my definition okay this is equal to fx1 x2 xnt I have an nth order differential equation which got converted into n first order differential equations this is my first equation second equation n-1th equation and the last equation came from the original nth order differential equation okay x1 x2 x3 xn are the new state variables that we have defined okay so what I have actually done is a scalar nth order differential equation I have converted into n first order differential equations in new variables okay so if I have nth order equation I can convert it into n first order equations okay if I have two simultaneous equations one nth order in one variable other mth order in other variable first one will give me n first order equations okay second will give me m first order equations you can stack them together into a bigger vector you will still get this form so this is a very very generic form I am not doing any compromise any nth order equation or any set of nth order equations nth mth order equations can be combined into finally this form this is a very very generic form so do not worry about why are we looking at only first order vector differential equation okay so all the books on advanced books on nonlinear differential equations will worry only about this generic form because anything can be converted to the generic form that is the first thing to understand so all the methods that we will develop are for this if you have nth order equations okay you know how to convert them okay you know how to convert them into n first order equations and write it like this so what will be f of in this particular case what will be the f vector let us go back and write that see in this particular case in this particular case my f vector after the transformation actually my equations are d by dt of x1 x2 x3 xn is equal to x2 x3 xn okay and f x1 x2 xn t this is my f of x this is my f of x okay this is the transform problem this is my f of x and I am given the initial condition okay so I am given initial condition x0 which is whatever this is y0 d by 0 by dt all these are given to me right this is my x0 this is my x0 this is given to me okay this is my f of x this is my f of x the original equation will appear as one scalar non-linear function in your function vector this is my function vector okay and this is the transform problem so I do not have to worry about nth order equations I am not going to do separate methods seen in the first course of differential equation you have second order differential equations one chapter on second order differential equations then you look at you know nth order equations we are not going to separate we are just going to look at n differential equations which are coupled okay if you are trying to solve dynamic trying to do dynamic simulation of a chemical plant there will be thousands of differential equations which are solved simultaneously together okay in fact they might be differential and algebraic equations not just differential equations so we are worried about right now to begin with solving large number of differential equations simultaneously together in one shot that is my aim okay and this form is very very generic applicable to any set of other way of getting these kind of equations we have already seen where do you get these kind of equations in problem discretization where did we find them finite difference method orthogonal collocations of partial differential equations that involve time and space we discretize in space we got differential equation in time we got n differential equations they were first order okay so if those are of second order you can convert them into first you know two first order equations all that is possible that is that is not difficult so converting nth order equation into first order equations is not a problem we are going to look at the generic form this could be arising from any of the sources it could be arising from the one which you have done right now it could be arising from discretization of a PDE okay it might be arising from some other some other context the context is already you know we have we already have studied about in what context this kind of problems will come we will look at only how to solve this abstract form okay of vector differential equation the other thing which you might worry about is that where does this time t come into picture okay most of the times the differential equations that you get as an exercise I have given you to solve differential equations for one particular system and I have given you a program which solve differential equations for a CSTR okay I suppose you remember to submit assignment soon that equation is of this form dx by dt is equal to f of x u there are some free variables x are dependent variables and there are some free variables okay like feed flow coolant flow coolant temperature inlet concentration all these are this u variables so in that particular problem CSTR problem x corresponds to concentration of a and temperature and u corresponds to inlet flow rate cooling water flow rate inlet concentration cooling water cooling water temperature at inlet and so on so these are the free variables but if you go back and look at the problem statement these manipulated variables or input variables have been defined as a function of time say this is sinusoidal this is whatever we have defined these as some functions of time okay once these are given as functions of time we can substitute them here okay as some function of time and then once these are these are specified once these are specified functions of time then only we can solve the initial value problem and for that particular for those specified functions of time this problem is then transformed to dx by dt is equal to f of x t because u will be function of only time some specified function of time okay a RAM function step function sinusoidal function or whatever random function whatever you want to study the dynamic dynamics of the particular system for so you are specified these free inputs and then this becomes a problem which again is this this generic form okay so these parameters or these input variables we assume that we are we already know them okay and then we want to solve the problem for the known inputs how does the dynamics evolves in time that is what we want to solve okay so that is why we are looking at in general dx by dt is equal to f of x t okay how this is specified as a function of time how these are specified how these are specified as a function of time let us not worry about that right now it could be an operator who is giving this values it could be a controller which is finding out these values it could be some environmental conditions which define the cooling water in the temperature okay we do not bother about that right now we want to solve the problem when this is specified how do you how do you actually find out how do you find out x as a function of time so I want to find out given this given these input trajectories in time I want to find out x trajectory that is concentration trajectory starting from time zero to whatever finite final time you want and temperature trajectory as solution of this problem solution of this problem is going to be not one vector see when you are solving non-linear algebraic equations you got one vector as a solution fixed point right now the solution is going to be a trajectory in time okay it is going to be a trajectory in time trajectory in time over the finite time if you are solving over finite time or whatever t goes to infinity if you want to look at now linear equations linear differential equations of this type you probably have already looked at in some other course wherever we need them we will visit them while those of you who have not done the other course on analytical methods in chemical engineering I will just briefly mention those results which we need here but we are going to look at the problem when this f of x on the right hand side is non-linear not when it is linear okay that is very very crucial we will use the results for linear later on to get some insights into the convergence properties under what conditions the methods that you have proposed will converge that is where we will use some linear system results but in general what we are going to look at is methods for solving non-linear ordinary differential equations given initial conditions how do you get trajectories in time or it could be trajectories in space okay we have seen that for example method of lines for you know converting Laplace equation it you discretize only in one spatial direction the other one is treated as a differential equation so you get a set of differential equations in time or space so you want to integrate these differential equations so T here in general need not be time alone T here treat it as independent variable in some context it could be space so maybe I should write a generic form that neta so neta is some independent variable okay could be time on space depending upon the context and initial condition at neta equal to 0 is given okay and you want to integrate this set of differential equations the way we are going to proceed is will briefly peek into the issue of existence of solution very very briefly and then move on to the different methods of doing numerical integration okay again what is going to help us is Taylor series approximation and polynomial approximations okay so we are going to meet our old friends Taylor and Weistras again and use them repeatedly to solve these problems okay what I want to stress here is that the same ideas are used again and again to form the solution methods okay there are few fundamental ideas which if you understand those ideas and if you know how to apply them you can almost do everything from scratch okay same idea is repeatedly used if you get this view point then you know I think you have learnt a lot in this course so next class onwards will begin with how to solve ordinary differential equations and algorithms and then finally we will move on to the convergence properties under what conditions these converge try to get some insights into relative behavior of different methods and so on.


The Clyde Beatty Show. The world's greatest wild animal trainer Clyde Beatty with another exciting story from his brilliant career. This master of the big cats captures ferocious jungle beasts and trains them to perform under the big top in the circus where there are always thrills, action, and danger. Hundreds of dramatic behind-the-scenes adventures are all part of the Clyde Beatty story. Here is the story of death in State Room B. Well, let's see honey, this is number 12 Pickworth Street. The building just ahead must be the Grazner building. Oh, I hope Mr. Dawson has really located something. Clyde, you realize we've been to every theatrical agency in London and we still haven't found so much as one good act to take back with us. I know. Well, we'll soon find out what Dawson called about it. I can't understand it. Before the war, there were all sorts of good European acts to choose from. It takes time to get back to normal, honey. Oh. Well, here we are. I say Harriet, too, let's take the lift, shall we? After that beastly tram ride through all the petrol fumes, all that sort of things. The elevator operator will hear you. Gang up, please. Step right in. Oh, third floor, please. Right, Your Honor. What time do you have, honey? 4.30 on the nose. Good. We're right on time. Which way would the Dawson theatrical agency be? It's the first door on your right, dear. Oh, thanks. Maybe it'll prove a lucky thing running into Mr. Dawson at the courage yesterday. Well, let's keep our fingers crossed. Well, here you are. Nice to see you again, Betty, Mrs. Betty. How are you, Mr. Dawson? First read, thanks. Please sit down, won't you? Oh, thanks, Dawson. That's a nice office you have here. Well, it serves my needs very well. But I know you want to know what I call about, so I'll come right to the point. We're awfully anxious to hear what you've located, Mr. Dawson. And wait until you've seen this act. It's made to all of your circus. Well, just what kind of an act is it, Dawson? I've decided not to tell you anything about the act itself. I'd just rather you saw it. Well, that's fair enough. Now, look here. We have a rehearsal and audition room just through that door. There are the Dumas in there now getting ready to show you their act. The Dumas? Yes, a man and wife team. Jacques and Roselle Dumas. Oh, French, huh? Yes. They've been in England only three or four months. And actually, I haven't represented them while they've been here. As a matter of fact, I'm not representing them now. But I don't understand it. I just happened to hear they were about to return to France and they discharged the chap who was acting as their agent and manager just a couple of days ago. Dylan was his name, rather an unsavory character. I see. This act isn't the sort of thing that I've been handling, so all I'm doing is to try to bring you two together. Well, it's very nice of you to do this for them, Mr. Dawson. Well, they seem like very nice people and I'm forgiving them a hand. I think with the proper opportunity they'd take a new lease on life. They haven't got on too well of late, I'm told. Little quarrels, misunderstandings. Their act will have to be sensational before we'd hire anyone who might give us more headaches. Circus people have to get along with everybody, you know. Oh, yes, of course. But I don't think you need worry. I think Dylan was responsible for a good deal of their differences and he's out of the way now. Do you think they'd want to come to America with us if we liked their act and offered them a contract? Oh, I'm sure they would. That's the goal of practically every performer in Europe. Well, they should be ready by now. Shall we go in? Let's go. You've got my curiosity around. Good. And I'm anxious to see if you agree with me. In my opinion, the Dumas Act has more thrills and danger packed into a short period than any I've ever seen. More danger, you see? Mrs. Batey, in this act you're about to see. The slightest miscalculation could end in death. Now back to Clyde Batey and death in State Room B. In the rehearsal room next to Dawson's office we met Jacques and Roselle Dumont. They were an attractive pair and I guessed their age to be around 35. My curiosity about their act was heightened still more when the only pieces of equipment in the room turned out to be a small table and some 15 feet away a large wooden wheel or disc attached in an upright position to a metal frame. The lower edge of the disc just cleared the floor. Well, Jacques, the Bateys are very anxious to see your act. And we are anxious to show them, Monsieur. It would take but a moment if you are ready now. Yes, quite ready. Oui, Monsieur. Come, Roselle, do not waste time. They seem very nice, Mr. Dawson. I knew you'd like them. Now what? You know, Jacques is strapping Roselle's arms and legs to the big wheel there. Oh, so I see. I wondered what those straps attached to the disc were for. Now she can't move, you see. And now Jacques turns on the switch there on the framework. The wheel is starting to turn. Hey, what is this? A human pinwheel? Exactly.  He bows to us, removes the scarf from the table. Knives? Mr. Dawson, surely this isn't a knife-throwing act. I warned you, you'd be surprised. But boy, you can't possibly keep from hitting Roselle while she's turning that way. Can't he? Let's watch. And here goes the first knife. Great scot! The knife stuck in the disc an inch from her throat. Not bad, eh? I tell him to stop. I can't watch. Steady, honey. They know what they're doing. I hope. He'll have Roselle completely outlined with knives in a matter of seconds. I can't believe it. That's the greatest exhibition of timing and precision I've ever seen. I don't like it. He's done. Oh, come now, Mrs. Batey. Jacques doesn't miss. Or rather, he does miss just in a minute. There. That's the last one. Thank heaven. Brother, I can see him now. We'll feature him in the Wild West show. Jacques is stopping the wheel. He let Roselle down now. Well, Batey, what do you think? I think that's the most sensational 30-second exhibition I've ever seen. If they want to come to the States, they're hired. So, it is signed, Monsieur. Good. And now that the business of contract is out of the way, I feel like celebrating. Why don't you and Roselle have dinner with us? You are most kind, Monsieur, but... We will accept with pleasure, Monsieur Batey. But first, we must hurry home and tell Yvonne the good news. Yvonne? My cousin, Madame. She has lived with us since way before the war. Oh, and will she go to America with you, too? Well, I... But of course! We could not leave Yvonne behind. Could we, Roselle? No. No, of course not. Won't you bring her along to dinner? We'd love meeting her. Merci. Suppose we pick you up at your hotel around 7. Then we'll go over to the Savoy. The chow over there is... Climb. Oops. Pardon, Madame. The cuisine over there is magnifique. Monsieur Batey, it was so nice of you to invite me along. Oh, not at all, Yvonne. We're enjoying it as much as you are. And I'm so thrilled about going to America. Oh, tell me, when will we be sailing? Well, I'm going to book passage for all of us on the Atlantic Star. It sails day after tomorrow. And we promise we will not, how you say, miss the boat, Monsieur. Look, look, coming toward the table. What? Who is it, Roselle? It is Monsieur Dylan, the man who was agent and manager until recently. Well, well, the small world is indeed. How are you, Roselle, Jacques? Mind if I join you? Well, you see it. Thank you, thank you very much. It might be far too. You know you are not welcome here. Why do you not leave? Because there's something I want to say first. But this is hardly the time or the place. Hold it, Jacques. Might as well let Mr. Dylan speak his piece now. As later, it won't bother us any. Well, there's a man with some manners. You must be the bloke, what's I, the act, right? That's right. Batey's my name, and this is Mrs. Batey. Well, you seem to know who I am. Oh, pardon me. Every cigarette? No, thank you. Yvonne? You know I don't smoke, Monsieur. Oh, sorry. Roselle, Jacques? We have our own, thank you. I would not smoke the foul-smelling cigarettes. Oh, yes, I keep forgetting. Nobody else appreciates tack is stamped. Ha! Suppose you come to their point, Monsieur. I'll be very glad to, Jacques. The point is this. I'm on to your game, and I don't like it. Yeah. Two days ago you gave me the sec. Now you've signed a contract for a year in America. Three times what you've been making. Now it's a co-incident, isn't it? What do you mean by that, Monsieur? Just this. I heard about what happened this afternoon, and I know we're not being treated. You knew before you sent me what was in store. Well, wait a minute. Are you suggesting these people knew I was going to hire them and fired you to keep from paying you a commission? Of course they did. I never trusted them from the beginning. Look, mister, you got it all wrong. I never saw or heard of these people before today. And that is the truth. Monde Dieu, we would not do such a dishonorable thing. Don't end me that honor, and you won't get away with it. Please, get out at once. Rose, do not let him upset your soul. I think maybe you'd better go with that, Dylan. Oh, certainly, I'm going. But they needn't think for a minute I'm going to forget what they've done. Lord Mortier, I... Oh, they're adorable men. He has no right to speak that way to Jacques and Rose. Well, there's nothing he can do, so let's not worry about him. In three days, we'll be starting to put lots of distance between Mr. Dylan and ourselves. Once aboard the Atlantic Star, you can forget him. It's about time you got back, crybaby. I've had three turns around the deck, two cups of coffee, and a game of shuffle-ball. I'm sorry, but even on a ship, business seems to keep interfering with pleasure. Business? Yeah, the purser collared me down in the lounge and wanted to know if I'd get the Dumonts to do their act in the dining salon after dinner. Oh, I thought the ship carried its own entertainer. Well, it does, but it seems one of the acts didn't show up by sailing time, and somebody from another act is sick, that leaves them pretty shorthanded for a ship's performance. I see. Well, what's funny? I was just thinking. It's a lucky thing we don't have any lines and tigers along. They'd probably want you to set up the steel arena and give up a performance, too. It wouldn't surprise me in the least. But the Dumonts said they'd be happy to fill in if I agreed, and I sort of hate to have them go on when they're upset the way they are, though. Well, what do you mean? Well, they're all in a storm again. They found out this afternoon that Dylan is aboard. Dylan? What of what? Well, don't ask me, honey, but he's on this ship, and they're worried. Jock and Roselle are both on edge. I see now what Dawson meant when he said they weren't hitting it off too well. Well, they'll come down all right. I think a lot of their personal troubles come from having a relative with them constantly. Oh, I don't see what Roselle's cousin has to do with it. Yvonne's young, charming, beautiful. Three good reasons for a wife to worry, dear. Well, whatever their problems, they've still got a good act. I'm anxious to see the passengers' reaction to the wreck. Oh, look at the time. I'd better start getting dressed. That you had. When that dinner bell rings, I'm going to be plenty ready to put on the feet bang. Oh, that was a beautiful dance team. It's so graceful. Yeah, they were fine. Thank you, ladies and gentlemen. And now it is our privilege to present the only act of its kind in the world, the ultimate in skill and daring. This act requires for the matter of seconds to perform. But during those few seconds, a life is literally at stake. For this reason, when the drum roll starts, it's like a drum roll. It's like a drum roll. For this reason, when the drum roll starts, we request that the audience be as quiet as possible. I give you now from Paris, Jacques and Roselle Dumont. I didn't know they had to have quiet while their act was on. They don't, honey. That's just for effect. It makes it sound more dangerous. When Jacques's throwing those knives, he couldn't hear a bomb go off beside him. Oh, they're ready now. Roselle has trapped on and the disc is turning. Yeah. There's the first knife. Two. Three. Four. Look at the spectators. Nine. They can't believe it. Six. Seven. Eight. Nine. Oh, no. Roselle, that knife hit her. Come on. Now back to Clyde Beatty and death in Stateron B. The moment Roselle was struck by the knife from her husband's hand, the dining salon broke into panic. Harriet and I made our way quickly to the small stage, reaching it just as the curtains were drawn. The ship's doctor arrived at the same time, and together we released Roselle from the straps which held her to the wooden disc. And then, while the doctor bent over her unconscious form, Harriet and I tried to comfort Jacques. Roselle, mon chéri, whatever I done. Now take it easy, Jacques. Getting hysterical won't help. Please, Jacques, try not to think about it. I cannot help it. I do not understand how it could have happened. Jacques, you're very darling. What an awful thing. I have just learned what is it. If I'm, how can I? You do not talk now, Jacques darling. It was not your fault. Clyde, the doctor's motioning for you. Oh, well, I'll see what he wants.  Will she ever forgive me? Of course she will. With an act such as yours, it's a wonder it hasn't happened before. The doctor's going to have Roselle move to her stateroom. Do you want to show them the wave on? Oh, but of course. It is day to room B on the deck below. All right.  Very well. And Jacques, please do not worry. Everything will be all right. I hope so. Of course it will. Come along, Jacques. It's no good staying here on the stage. Here, Jacques. I had the steward bring some brandy. Merci, madame. You are most kind. Just make yourself comfortable. If only I could know how... how Roselle is. Well, the doctor's doing everything he can. We'll know as soon as he can tell us anything. I still cannot understand how this terrible thing could happen. Perhaps you were nervous. The ship might have rolled a bit. No, no. It was neither of those things. I was perfectly sure of myself. But after the knife left my hand, I realized it would hit Roselle.  But we have done the act so many times. My timing has always been perfect. But tonight it was almost as if... But what is the use? Excuses are not any help. What do you mean, Jacques? What were you going to say? Well, it will sound foolish, but it seemed to me as if the wheel slowed down just a little bit just as I threw the ninth knife. Wheel slowed down. I know it sounds ridiculous, but it seemed that way to me. Of course it is impossible. Wait a minute. Is it impossible? Well... After Roselle was hit, did you turn off the switch to stop the wheel when you rushed over to it? I do not remember. I was so confused, Monsieur. Tell me something, Jacques. In your act, isn't it absolutely imperative that you maintain a regular tempo or rhythm when you throw those knives? But of course, Monsieur. And if that wheel did slow down just a bit, the knives would strike two or three inches off their intended mark. Oui, that is true. God, you mean you think someone might have tampered with the motor or something? I don't know, but it's a thought. The knife that struck Roselle was number nine. The one which should strike just under her left arm, Pete. Yeah, only this time it struck her heart, or mighty close to it, anyway. I'll see who it is. You are Mrs. Bady? Why, yes. I'm Forsey, ma'am, ship security. Oh, come in, Mr. Forsey. Thank you. This is Mr. Bady and Mr. Dumont. Mr. Forsey. Sorry to barge in this way. It's all right, Forsey. Did I understand you to say you were the ship's security detail? That's right. I was just making a routine investigation of the accident. Tell me, how is Roselle? She's dead. Oh, no. No. It cannot be true. Easy, Jacques. Oh, I'm so sorry, Jacques. Harriet, you stay here with Jacques. I'll go with Mr. Forsey and give him whatever information he needs. That's a good idea, Bady. Come along. Well, what you've told me may throw a different light on things, Bady. That is, if Dumont's imagination isn't playing tricks on him... Well, I thought it might be worth checking on it, anyway. Apparently, nothing's been moved or touched here on the stage. Is this where Dumont was standing? Yeah, by the table there. I see. Let's have a look at this wheel contraption over here. Let's see now. Oh, what is it? Nothing. The switch to the motor is turned off. Jacques didn't remember whether he turned it off or not. Bady, has it occurred to you that this Jacques might have deliberately thrown that knife so it would hit his wife? Yeah, I'm afraid it has. Would he have any reason to want to kill her? No, I don't think so. They didn't get along too well, I guess, but I don't think their troubles were the kind that would prompt him to murder his wife. It would be simple, though, wouldn't it? Who could prove it wasn't an accident? That's right. There's no way in the world to prove it. Somehow, I think Jacques was telling me the truth about the wheel slowing down and throwing his timing off. I am. Do you see anything unusual about this apparatus here? No, I don't. Yes, it was... Hmm. Oh, what's the matter? This electric cord running from the motor. Oh, what about it? Well, let's find the socket where it's plugged in. All right. The cord runs across the floor and under that piece of scenery at the end of the stage. Here, we can get around this side. All right. There's not much light back here. I've got some matches. Don't bother, baby. I've got a small flashlight. Here we are. Yeah, it's better. Over here, Forsey. Look, this plug's pulled halfway out of the socket. Well, I'll be there.  It looks that way all right. Of course, it could have been done by accident. Someone might have tripped over it. Maybe. Where does that door lead? To the promenade deck. And somebody could have come in here behind this scenery and gotten back out without being seen. Oh, yes. Wouldn't have been any trick at all. This is beginning to get intriguing, Forsey. Yes, isn't it? Hello. What's this? What? On the floor near the light socket here. Oh, I thought it was a scrap of paper, but it's only a cigarette butt. Oh. Hey, wait a second. Let me see that. Here you are. Hold the light closer. Well, you don't see those very often. Turkestan. What? They're a Turkish brand. Very strong smelling. Don't care for them, myself. Turkestan. It's Dylan. He smokes those. What in heaven's name are you talking about? The Dumas former agent, Dylan. Oh. He's aboard this ship, and the Dumas were afraid he might give them some trouble. I see.  Then today they discovered he was aboard ship, too. Well, I think perhaps we'd better find him and have a bit of a talk. If he smokes Turkestans, he's probably the only person on the whole ship who does. Most people don't care for that strong aroma, but, well, you probably know all that. I know practically nothing about cigarettes, Forsey. I don't smoke, but... Hmm? Nothing. I just thought of something I'll tell you later. Well, come along then. Let's find this Dylan person. See, you're blokes, I've done nothing, and you've no right to barge in and put me to the third degree this way. If you've done nothing, you've no worries, Dylan. No, look, please. I understand you were angry with the Dumas, but you even threatened them. Oh, sir, they never threatened to have someone killed. How do you happen to be on this ship, Dylan? My being on the same ship as the Dumas is quite by accident. I decided to go to Canada, something I've been meaning to do for years. But I didn't even know until this afternoon that Dumas were aboard. And where were you during the entertainment this evening? Well, right here to MacGabbin. Can you prove that? No, no, I was alone. Well, see, I suppose someone did pull the plug from the socket, and you say, what make you think it was me? This. What? We found this on the floor backstage, near the light socket. I understand this is the brand you smoke, Dylan. Well, yes, but that proves nothing. Maybe not. And then again, maybe it does. You'll see. In the meantime, I'm holding you on suspicion of murder. Look, if you can't, I didn't do it, I'll tell you. Now calm down and come along with us. Well, well, you're taking me. I want to see you face to face with Jacques Dumont. All right, just sit down, everybody. This won't take long. I guess you all know Mr. Dylan here. They all know me all right. Come on, let's get on with it. Yes, sit down, Yvonne. Thank you, Malcolm and Betty. Now I'll be very brief and to the point. We have reason to believe that tonight's tragedy was not just an accident. Dumont, you were right about the wheel slowing up. We found the electric plug had been disconnected. Well, you didn't stare at Jacques Argin, did you? It could not have been anyone else. You wanted to get even with us. I apologize. All the evidence points toward Dylan all right. But the evidence is strictly circumstantial. Evidence? Pig, you haven't moved at all, then. Here, Yvonne, wait. What evidence did you find, Monsieur? We found the half-smoked butt of a Turkistan cigarette. It was right near the light socket. Ah, so you weren't careful, is there, Monsieur?  If I were playing something odd, I wouldn't be guilty of making... Oh, no, no, no, no, no, no, no, no, no, no. I admit, such evidence isn't much to go on. But since it's all we have, I'm forced to hold Dylan here. But Mr. Beatty came up with an idea that might be worth living. Well, uh, I'll need a lady's purse to demonstrate. Honey, where's your... Oh, it's in the other room. I'll get it. Well, never mind. Here's one. Is this yours, Yvonne? Oui, it is mine, Monsieur. You mind if I borrow it a moment? A woman's purse is private, Monsieur. What's this all about, anyway? You'll soon see, Dylan. I think there's a good chance I may have the answer to this whole business. You don't understand? What do you mean, Monsieur? I mean to have a look inside here. Oh, but, Monsieur, you cannot... Hold her, Forsey. I've got her. What are you doing here? Are you all right, Betty? A pack of cigarettes. A pack of cigarettes. Yeah, with one missing. Yvonne, what does this mean? Young lady, you've got some explaining to do. Take her away, Forsey. Oh, haven't you turned in yet, Harriet? You want me to sleep, not knowing what's going on? I'm all confused, Clive. Yvonne confessed. Oh, she confessed. But what? Why did she want her own? Hey, yet. When I've got a hunch, it was so she could fall heir to being Jacques's wife. It's pretty plain she's in love with him, and was jealous of Brasile. And she actually planted that cigarette spot, so if her pulled cord were found, it would point toward Dylan? Sure. She knew that in their act, the ninth knife came close to Brasile's heart and that by slowing down the wheel, she shot through it, it would probably be fatal. Oh, awful, Clive. But I still don't see how you knew it was Yvonne instead of Dylan. Well, I didn't, for sure. But remember when we were having dinner at the Savoy the other night, when Dylan sat down with us, he offered his cigarettes around? Yes, I remember. And Yvonne said she didn't smoke? Yes. Well, tonight, when she rushed backstage after the accident and said she just heard what had happened, I noticed something subconsciously that didn't register until later. Go on. Well, not being a smoker myself, I always noticed the smell of tobacco on others and I could definitely smell tobacco when Yvonne came backstage and stood next to me for a moment. You never miss a thing. It sort of made me wonder if when we found that cigarette butt back to the light socket, so I checked up with the cigar counter in the lounge and the man there told me it only sold two packs like that all day and one of them was sold to a young French lady. Well, well, I do believe I'm married to another Sherlock Holmes. Elementary, my dear Harriet. Elementary. Clyde will return in a moment with a preview of our next story. Once again, here is Clyde Beatty with a word about our next show. When the lights under the big top flicker out when the brilliant costumes are hung away and when the last dab of grease paint is removed, those dazzling people who thrilled you with their feats of daring become ordinary folks. When next we meet, you'll hear the tender and exciting story of a circus family, the Fly Politos. All stories are based upon incidents in the career of the world-famous Clyde Beatty and the Clyde Beatty Circus. The Clyde Beatty show is produced by Shirley Thomas. Death in State Room B was written by Robert T. Smith and Frank Hartowski. All names used with fictional and any resemblance to persons living or dead is purely coincidental. This is a Commodore production.


Welcome to Think Tech Hawaii and my second episode of Movement Matters. I'm Christine Linders and I'm a licensed physical therapist. I've been practicing physical therapy for over 23 years in California, New York City, Connecticut and now in Hawaii in a variety of settings including sports, orthopedics, neuro and even on-site corporate wellness platforms. I'm a board certified orthopedic clinical specialist. I'm certified in applied functional science and I have my manual therapy certification. This is my show, Movement Matters, designed to bring you the most cutting edge and effective treatment strategies so you can help your body perform better, decrease pain and get back to doing the things that you love. Today's topic is low back pain. In the first part of the show, I will discuss why low back pain occurs and teach you about the specific group of muscles that needs to be activated in order to end the pain quickly and prevent re-injury. And in the second half, we'll be hearing all about Michael's experiences with low back pain and learn the simple way to reactivate these critical muscles so you too can heal your low back pain quickly and be back to enjoying your life. Welcome Michael, thank you for coming. Thank you for having me. I'm so happy to have you on the show. So we're talking about low back pain and I don't know if you know Mike but 80% of us at some point in our life will have low back pain. Now there's some statistics I'll post a link for from an article that I just had published but that's about 540 million people in the world that will have low back pain at some point in their life. Did you know that? No. I'm not sure if the audience knows that either but it's a big problem right now in our society. We do sit more and we're active but I think the problem is that when we sit, it's very difficult for us to sit without rounding our spine and it's the rounding of our spine that puts a little bit extra stress on their back. So I mean do you sit or are you pretty active? I'm pretty active. Yeah, what do you like to do? I surf, I play volleyball, I hike, I paddle board regularly and I don't know, play a little golf now and then, not very well. That's great. It's great to stay active and it's great to be active but what do you do when you have low back pain? So I think for the audience when you have an episode of low back pain it could be something that you don't realize you know you're doing and that's what I wanted to show. So in image number two, some of the things I tell people are that poor posture that you don't realize you could be doing while you're sitting reading a magazine. In the upper photo, you see this person sitting reading the magazine and their back is only slightly rounded but that slight rounding creates stress and increasing percentages of stress on your low back.  In the picture below, you could see this person sitting with good posture supported reading their magazine so those are the things I think people don't realize add up that create extra stress on their back and then they go and they bend over to pick something up and hurt their back. In image number three, you see posture while you're washing your face, brushing your teeth, shaving, anything like that and in the picture on the left where the spine is rounded that's how people typically do it because you would never know that there's anything wrong with that but on the picture on the right with the flatter back that's what you need to do to keep your low back safe while you're brushing your teeth, while you're washing your face. That's something that we do every day, twice a day for all of our life so that's a lot of repetitive stress that we can have on our low backs. So why do people have recurrent low back pain? So I'm going to explain to you guys right now these two critical muscles that are responsible for keeping your low back free from pain. In image number one, you will see the picture on the left which is our deepest abdominal muscle. It comes in from the sides, kind of like a course that you're wrapping around yourself that is the transversus abdominis. The image on our right is the multifidus muscles. Now these muscles are the deepest segmental spinal stabilizers. They run right up just to either side of our vertebrae, the spiny oaky bones in the back. If you bend forward, you can feel them.  Now there's been research proven, which you'll see in the link that I'll provide, that even after one episode of low back pain you don't regain the function of these multifidus muscles in your back without specific training. And that's what I want to show you today is how to retrain these muscles. So let's meet Mike. Picture number four. We talked a little bit about Mike. He's very active. He serves, he paddles. He goes hiking with his grandkids. I heard this weekend that you had to go serious four by fouring on your trip. What is serious four by fouring? Well, it's a road on the west shore that goes up over a mountain and there's a campsite on the other side. Oh, that's awesome. How much time did it take you and was it bumpy road or was it climbing up and down? It was a good four wheel drive. It's the first time I've had to use four wheel drive in Hawaii and I had to use four wheel drive. So could I get there in my Hyundai hatchback?  So four by fouring is a great thing to do but I have a lot of patients that also tell me that it's jostling on their back if they hit a bump in the road just because of the sudden compression that has on your back or if you're riding a jet ski or you're in a boat on the bumpy wake of somebody else's boat that that can be tough on your back if you're having back problems. So I was wondering how you did, but you were okay. Yeah, that part was okay. Excellent. So let's talk a little bit about your history of low back pain. You know, it's hard to say. At one point I thought I took a pretty good fall when I was about 28 years old doing some construction work and I just flipped over and landed right on my butt and it hurt for a while. It was pretty painful for a while. Did you break anything? No. Oh, good. And so when did you notice you started having back pain? Well, that's probably the start of when I noticed having back pain. Now, earlier, when I got married in 73 which was about five years earlier I looked at the pictures from the wedding and I'm leaning in every single picture. I see. Yeah. So, you know, chances are there's something that happened prior to that fall. Yes, that's right. So Mike and I met on Labor Day and we talked a little bit about his back and he was telling me what he was just telling you about having noticed being crooked in some photos and one of the things that someone could have that notices that is a thing called scoliosis and there's two different kinds of scoliosis that you can have.  Idiopathic meaning no known cause. So what are your symptoms like? You're very active. You surf, you play beach volleyball. I think you mentioned you played this morning. Fantastic. Yeah, you know, I mean, I can last so long and then I have to relax and if I don't relax and I have a bed that conforms a little bit it has a zero gravity position which helps a lot for sleeping at night until then I didn't ever get a good night's sleep. Generally speaking, when I'm sitting at anywhere in a meeting or here even, I'm very fidgety. I have to move all the time because my back is uncomfortable if I leave it in the same place. Okay, yeah, so that's an important thing to know is that you need to keep moving. If you do have an episode of back pain one of the bad things that you can do is just to be still. It's important when you first hurt your back to take some rest, maybe lay down on some heat, lay down on some ice, work on getting rid of the muscle spasm which is the body's protective mechanism for letting you know something's wrong and not allowing further injury. But after that it's very advisable to take short walks, keep moving but remember you've got to pay attention to how you move now.  You want to maintain what is called the neutral curve of the spine which is the path that our normal spine takes back and forth. We met on Labor Day and we looked at your body a little bit and I think if we look at the video number five this is what I saw when we looked at Mike walking and this is where Mike and his wife were mentioning how oh yeah I think he tends to walk a little crooked especially if you've done a lot that day. It's so interesting looking at this because I have no idea I'm bent off to the side until I look at this video. And for the audience what's so important there is every time he steps on his left leg you see his shoulders drift off to the right. Now your vertebrae are stacked on top of each other so when that happens there's a thing called shear which is like a sliding back and forth that can happen. Now if Mike was just walking down the hall a few times like that that's not a big deal but if Mike is walking down the hall all day long down the beach out to surf running around playing volleyball like that that's where you get that wear and tear that leads you to having pain injury, disc herniations all kinds of problems like that. Now I think what Mike mentioned to me is you had an x-ray at some point I forgot when that was. Actually a couple times my daughter worked for a chiropractor in San Diego and he took an x-ray on my back I bought a series of adjustments it was very helpful. And my daughter was in talking with a doctor and she sees this x-ray up on the monitor and she goes oh my gosh who is that and the doctor said that's your dad. Oh So is that the you had another one since then or is that the most recent? No I've had another one since then I had a kidney stone and they had to take some pictures of my lower back area and that showed that everything was tweaked I had a follow up appointment and when I walked in the doctor was astonished that I was still moving around. That's unbelievable and I think for the audience the take home point there is that you may have an episode of back pain and you go to your doctor and you get an x-ray or an MRI and it looks horrific like Mike's daughter saw and said oh my gosh who is that how are they walking the doctor said I don't know how you're moving around like that the x-ray and the MRI what you find don't necessarily predict how bad you'll be because I've seen some patients who had terrible x-rays and they're walking around with a stiff back they don't even have pain but you're worried that they shouldn't even move because it looks so unstable in the in the image so when you get your x-ray and your MRI it's just telling you about what mechanical faults you have going onto your spine narrowing of the canals is it uneven are some of the vertebrae sliding side to side is there discs bulging and pushing on the nerves and that's just information that we as health professionals use to try to match up with some of the symptoms that you have so we know the severity but someone like Mike who is surfing doing exercises he does a lot of heavy construction as a hobby right with your own plates not necessarily as a hobby but as a request from people I'm I can't say no to yes people like me I can't wait to have you help work on my future home someday but Mike stays with a strong core Mike has exercises that he does that keep his really messed up x-ray healthy despite how bad it looks like and that's so important so is there there's some exercises I think you mentioned you learned from massage therapist or Rolfer who we both knew from San Diego I went to a Rolfer in San Diego knowing that I had these issues for a while and he he showed me several exercises and probably if I had kept up with the things he taught me and learned more of what he had to offer I might not be as bad as I am now but you know I still do the exercises that he showed me on a regular basis and you showed me some that are just really good too they fit really well with what I'm already doing and that's great and I think the combination is has some kind of effect I'm not sure what it is yet but well that's good so this is movement matters we're going to be taking a short break I'm here with Michael Brennan and we'll be right back to learn a little bit more about how I've provided Michael with some exercises and exercises he's done in the past that can help you as well get rid of your low back pain fast we'll be right back Hi I'm Rusty Kamori host of Beyond the Lines I was the head coach for the Punahou Boys varsity tennis team for 22 years and we're fortunate to win 22 consecutive state championship this show is based on my book which is also titled Beyond the Lines and it's about leadership creating a superior culture of excellence achieving and sustaining success and finding greatness if you're a student parent, sports or business person and want to improve your life and the lives of people around you tune in and join me on Mondays at 11 a.m. as we go Beyond the Lines on Think Tech Hawaii Aloha Aloha my name is Becky Sampson and I'm the host of It's About Time on the Think Tech Hawaii a digital non-profit organization that's raising public awareness join us on Wednesday at 2 p.m. where we talk about real issues some of the topics will include entrepreneurship health life skills and growing your business so once again this is Becky Sampson on It's About Time on Wednesday at 2 p.m. on Think Tech Hawaii Mahalo Okay we're back we're live this is Movement Matters I'm talking with Mike Brennan and we're going to be learning about these secret muscles that you can activate to help get rid of your low back pain fast now so Mike we were just discussing some of the exercises that a massage therapist had given you and then also I'd given you a few I believe it's one week ago have you tried some of those exercises yeah I try to do them regularly we were camping this weekend so I didn't get to do all of them okay but yeah I've been pretty diligent regularly now do you notice I know it's been quick but do you notice any possible changes or feel anything different in doing the exercises more regularly I don't know how to say this it's hard to yes but it's hard to admit I mean I've gone through so many things with my back and some of them have been helpful but this I mean I'm here and I don't feel like I need to fidget okay good to me that says something positive excellent after I played volleyball this morning I'm usually pretty sore and I'm not that kind of sore right now okay so I'll take that as a positive trend but I think it's important for us to follow up with Mike in a few weeks see if that trend has carried on with being regular with the exercises and trying a few new things right that's great I hope that continues for you another thing is I actively work my core as much as I can and I believe that that's probably the only thing that keeps me going can we say that again I actively work my core I do I actively work my core with different sit ups and stretches and all kinds of stuff to make sure that I've got enough muscle there to keep me playing that is the key right there so for all of you to activate the core those muscles I mentioned in the beginning the transverse abdominis and the multifidus I didn't mention this part yet when they contract together you get an anatomical corset or girdle that wraps around your girdle that helps to stabilize your low back and I'm going to show you in a little bit how to activate those muscles in your own body in five seconds we're going to do it here while we're sitting that you can get control of your low back pain right away but first let's look at Mike so we looked at Mike walking let's look at image number six Mike standing so I apologize for my sloppy finger on my phone but this is what I saw when I watched Mike stand there after we watched him walk I drew the bottom line as just a parallel to the floor and then I angled up where his pelvis is so what that's telling me with the the weavy line that I drew above is that his spine is bending a little bit to the left at the bottom so there's a little bit more compression that's happening on the left side of his spine and a little bit more opening happening on the right side of the spine meaning that the right side back muscles are going to have to work harder because they're getting a stretch reflex telling them to turn on as he walks you saw before his shoulders shifting to the right creating that that shear every step he took so that's one of the some of the exercises I did give Mike were to help level off his pelvis and make his back muscles a little more symmetrical what we also found in the exam image number seven is that on one side Mike's hip doesn't have good rotation and it happens to be on that left side that left side where his spine is getting a little bit more compression now you can have all kinds of symptoms from having a little bit extra compression on the spine it could be that you have pain down your leg or some numbness and tingling but also you can get a muscle spasm in the glute from nerve irritation that might not hurt but might just tighten up so this was one of the exercises that I gave Mike it's one of my favorites that I give to my patients that have had disc low back pain I have them lay on a flat surface you can do it on your bed if you can't get to the floor I don't want you to have more pain getting to the floor but I call this the knee to opposite chest stretch so your knee is lining up with your opposite chest or shoulder and you're pulling it that way and the lower hand is used just to gently pull your shin that way as well you get a deep buttock stretch a lot of people want to pull their knee way over outside of the shoulder but then their back comes off and you're twisting your back and I want your back to be safe while you're helping to stretch out some of the other muscles in your body so now Mike I think you mentioned to me before I checked your hips that you had one when you crossed your leg that you look I can do this but the other side you can't is that something you had noticed before we were looking at it or a long time I'm much more flexible with my right leg hip combo than I am with my left leg hip combo okay yeah and that's something that when you have a little bit of curve in your spine or you do have a scoliosis it changes sometimes the rotation of your pelvis as well and your hip bone the femur sits inside the pelvis so when the pelvis shifts the femur shifts and that can change the angulation with which your deep hip rotators have to hold on to that socket but again we're trying to achieve symmetry because the closer we are to symmetrical the less pain that we're going to have we don't have to be perfect none of us are perfect we want to get you a little bit closer so the muscles on both sides of your body can function optimally so let's talk about neutral spine in image number eight neutral spine is something that we all need to be able to perform in order to engage the transverse abdominis and the multifidus the anatomical girdle so in this image you can see a small space behind that person's back so it doesn't have to be that large neutral spine is the position that when you lay on the floor with your knees bent up and your feet flat on the floor it's the position that your spine is resting uncomfortably you're not pinning your back flat you're not arching your back it's just your most comfortable position the low back has a lordosis which is a backward curve and so you want to have a slight backward curve which is the normal curve when you're learning to engage your anatomical girdle and what I will tell people in image number nine is a cue called suck it in so what you want to imagine pardon me what you want to imagine is that you are pulling your belly button in towards your spine towards your back so you don't want to flatten your back to the floor you want to pull your belly button in just towards your spine and what that's doing is that's going to engage your deep back rotators so that they can form the anatomical girdle so if we go back to the camera Mike do you know how to engage your transverse abdominis in neutral I can't even say it I know so I call it TA they call it t-r-a-t-r-v-t-v-a it's nicknamed a lot of things I nickname it TA because transversus abdominis is a big mouthful so if you're going to activate your TA you're going to take your belly button and pull it in towards your spine your spine you just did it right now and that's great and you do do that anyway because you're working your core so when you pull that in what activates that what we don't know about is those deepest back muscles from that first slide they fire together and give you this anatomical girdle and they run from one vertebrae to the next to the next to the next and they go up a couple levels so that gives you the stability to your low back so with you doing all that excellent core work you're activating your anatomical girdle which is enabling you despite your x-rays to be able to play sports golf which is a lot of rotation on the spine surfing paddle boarding hiking acting like a five-year-old with your grandkids right so let's go to image number 10 image number 10 is specifically for the anatomical girdle so what you're going to do with that is you want to be on a supported surface so if you're laying on a floor or a bed you're going to need a sizable pillow under your stomach so that your spine will be a neutral and if you don't have that pillow your spine could sway into more of a lordosis and we don't want that so I often have people lay over their countertop, their kitchen table their bed with your leg hanging off so that you can have your spine stable and you're going to suck your stomach in pull your belly button into your spine and lift one leg up if you're going in the top photo where you're laying on the table you want to lift your leg up not so it goes beyond your back but just enough so you feel that buttock muscle tighten and then you put it down you pull your belly button in, you lift the other leg up just so you feel your glute muscle tighten and then put it down the reason why I don't want you to have your lift your leg higher is then you're extending your spine and you're putting too much stress on the low back if you go to the picture in the bottom which is laying on your bed, laying on the floor laying on a massage table you want to have a pillow that enables you to keep your spine in neutral and again you can see from the image the leg is just lifting an inch or two up off the table you don't want to go higher and you don't want to have pain that's the key but when you do that you're activating your multifidus muscles and on the bottom picture as well if you try to lift your leg and you experience any discomfort whatsoever in your low back when you do it you need to make sure that you suck your stomach in first before you lift it and if you still have discomfort all they have people do is pull your belly button in and lift your hands off of the table so you're like in a little W position you don't lift your head, you don't lift your chest you just lift your hands up and you put them back down doing that also will activate those multifidus muscles in your back.  Yeah it does, it's actually part of martial arts That's right. I did that for about eight years and part of what I did was you think about your one point which is two inches below your belly button and that is what helps you stabilize your core Okay that's fantastic, so thank you so much for coming on the show Mike this is Movement Matters thank you so much, thank Tech Hawaii for hosting our show, aloha everyone


You guys have been asking me a lot about skincare lately and I recently did an upload on Skincare where I talked about my favorite skincare of 2019 We did hair care and nail care in there, too What this video is going to do today in a second You're gonna see me without any makeup and I don't have my hair done and I just got out of bed And I'm gonna be doing my morning skincare routine I will be doing my evening skincare routine for you as well But this way you get to see them broken apart and what order that I do use them in So I hope that you do enjoy this video. Please give it a thumbs up if you're new to my channel I would love it if you would subscribe. We have a whole big YouTube family here love makeup We love skincare. We love all things beauty and that's exactly what we get into And the mature side of it the things that we need as we age as we get into menopause as our skin changes As the weather changes we talk about so many of those different things and how the different things that can help us So if you're interested in all that make sure you subscribe and hit that bell down there so you can be notified of any future videos Okay, we're gonna get into it like I said no makeup. No hair done. You're gonna get to see the true Undone me set back to factory settings. So let's get into my morning skincare routine so you can see how I do it Here we go. So to start out We're gonna take off any of the residuals from all of the skincare we had on the night before and this is just my Regular cleanser. It's the purity made simple I don't use a lot in the morning because I don't feel like I have a lot on there to you know, really Needing to be cleansed.  This one is from Pixi. This is the vitamin C tonic Love to use vitamin C in the morning because as you know, we have all of the pollution They're free out free radicals and vitamin C is a really really good anti-pollution free radical Almost like a preservative when you're looking at it Have you guys ever made a fruit salad and you put some lemon juice on top of your fruit salad before you? You know served it because what that does is it helps preserve that fruit from turning brown?  I really feel like the past Oh, probably 18 months I've been using a lot of skincare on my hands and my arms because I had such bad dark spots And I really feel like everything has even improved on my hands and my arms as well The next thing I'm going to go in with is my buffet from the ordinary This is just an all-around awesome serum that helps with so many things of anti-aging. It has great ingredients in here I just use one pump for my face and one pump for my arms because this isn't too expensive And I feel like I can do that at this point I'd like to just bring out to you the best order you can do your skincare in When you do a cleanse if you're going to go ahead and you're going to do an exfoliation in the morning Do your exfoliating and then give your skin just a second to rest and then after you do that You go in with all of the rest of your skincare And when you do that you're always going to go from the thinnest of your skincare to the thickest But as you can see this is really thin on the vitamin C formula very very thin And then what I'm going to go in with next is the buffet and as you can see that's a little bit thinner I've got all over my eyes because I feel like that's so good for my eyes, too And it doesn't cause any irritation up those ears all the way around that neck Clear down all the way across the decollete and then I'm going to use one more squeeze of it And I'm going to go in and I'm going to put it on my back of my hands Fingers and clear up my arms So I've kind of gotten away from putting on too many layers in the morning for my skincare Vitamin C in one serum is usually about all I do and then I go in with my hydraters I feel like my hydraters are probably the most important part because not only are they going to lock your skincare in But they're also going to help with a little bit of other things that because of the hydraters that I do choose to use So the first one that I use is the Hada Labo and this one is their brightening serum or their whitening serum This really helps you guys can see that I have you know dark circles Of course, but I have a lot of redness that I fight right through here through my cheeks and Especially on my chest where there has been so much sun damage and this is really helping that right after it I go in with my Sukoyakasuhara Your real ocean on top of it now I feel like these two together are just a one-two punch So I don't use a cotton pad because I feel like it wastes my product And I'm not I'm one that wants to get every bit put a few drops in my hand I'll run that around you can feel this has a little bit more slip to it as well And I'm just going to kind of run that all over every single spot on my face ears again Down the chest and I love this stuff you guys It's not very expensive and it's worth it to get it all over those parts of you that Need that brightening, you know, maybe you have a lot of dark spots or something But immediately after that I go in with the Sukoyakasuhara I'm almost out of this I do have a backup, but this is my moisturizer for the day So when I said earlier that I do this a little bit different It's because I love this for my moisturizer for the day I don't use anything else to put on my face in the daytime because I find that this is so hyper hydrating and I love it it has been one that I just I can't live without this stuff in the morning when I Don't want too many layers. I don't want there to be a cream that might ball up. I use that's my hydrator for the date That's all I do now I do go in with the Sukoyakasuhara cream the eye cream and I will use a very generous amount about the size of a large P and then I will do this all around my eyes and I just pat that all around my eyes Smooth it around I go all the way from eyelid all the way underneath all the way out way far into the eyebrows into the Elevens I go all around this orbital bone and then whatever's left I will go and I'll put on my top of my lip because I feel like it's super hydrating There'll be days when I use a product that does sting around my eyes a little bit It's a medical grade product that I actually haven't mentioned I'm still testing it but on those days that I use that and I get that stinging sensation It's from a probiotic that's in there And I guess I have a little bit sensitive eyes to that specific product But my reason for telling you that is because when I use the the Sukoyakasuhara Urea lotion after that it takes down any of that stinging I don't know whether it's the urea what product it is in here But it sooths so much and the lotion as well.  Okay as a last step I will go in with my sunscreen and I talked a lot in between there because I wanted to wait for this all to Kind of sink in because I usually do wait about 10 to 15 minutes Between my skincare being on and then putting on my sunscreen I like to have all of that locked in before I go in with my sunscreen. This is from Misha. This is their Sunmilk Essence did talk about this having alcohol in it. It does but it is a super super thin product Which I love under makeup and it doesn't seem to dry me out It sinks in so fast and that's the part that I love about it You can't even see it hardly at all and it covers everything and goes around and I just feel like For a sunscreen to not interfere with my makeup at all is so good So I will just go along pat that in I pat it across my eyes Everywhere so everything is very protected and then on my arms and hands. I use the Australian gold mineral This is a tinted SPF and I really like this too.  Okay, that is it for today's video you guys I hope that you did enjoy it Please let me know in the description box if you guys are using a skin care product that you just think is Fantastic that has been so good for your skin and helped your skin so much I would love to hear that because I'm always up for trying different things for a little while I hope that you guys are happy and healthy and I hope you did enjoy this video Please give it a thumbs up if you wouldn't mind on your way out of here. Take care of yourselves everyone I love you so very much. Catch you all in my very next video. Bye. Bye