So let's go on with our study of rule surfaces, we interrupted, it took a bit of time speaking about spaghetti, but now let's go back. Remember, which are the data of such a surface. We have a curve, the directrix, and a one parameter family of lines, so we call it alpha tw of t, and we are looking at the map xt v equal to alpha plus tw, plus v w. Meaning w is working for us just to give a direction, and for such a direction we put on the surface the whole line, the whole straight line in this direction, while something is moving, the point is moving on a given curve alpha. We saw a few examples. Let me add just one example, which will turn out to be important later, which is called the tangent surface to a curve. I mean, this is adds to the list. Last time we made many examples of this type of surfaces. Tangent surface to a curve. That means just, so it's a one line example, in the sense that given any curve alpha, for any alpha, I can take w to be the tangent vector to alpha.  Geometrically is a particularly simple situation. You have a curve in space, and you add for every point of this curve the whole line tangent to it. We are going to give us a classification of surfaces where this type will be one type of such surfaces. Let's keep it in mind, but of course it's a very special example. In any case, given this type of surfaces, so now let's go back to the general case, alpha in w. Let's compute first, second fundamental forms and curvatures. We know how to do it. X, X, XT. In this case, the variables are not UMV, but TMV. The first one would be XT is alpha prime plus vw prime. I drop the dependence on what they depend. It's clear, it's written there. XT is this one. Xv is just w. Xtt is alpha double prime plus vw prime. Xtv is just w prime. And Xvv is equal to zero. So then what is the normal to such a surface? Remember, actually one general comment. We are performing the usual theory without worrying too much if this is the parameterization of a regular surface. Remember, there is the problem of, for example, of injectivity of this map, which is highly false in general. So we perform it formally, and the computations we are going to do will work on the parts of the surfaces for which this is a good map. So we have to be careful where to make interpretations. But let's go on. Now n would be what? Would be Xt wedge cross product. Xv normalized. Well, there is not much more that we can say. I mean, we have to do the cross product of these two, and in general we have no information. Of course cross products simplify if you know that something is parallel to something else or something is orthogonal to something else. In this case we have no information in general about properties, mutual properties of these vectors, alpha prime, w prime and w. They are almost free to be whatever they want. So let me, I leave n in this form and then depending on the situation we can make further analysis. So how much would be e? OK, so I don't write it because I would just write the general formula. OK, so what is the, which are the coefficients of the first fundamental form? Well, I have to take Xt scalar product n. OK, sorry, first fundamental form Xt, Xt. OK, so, well, in this case, in this case I made a mistake here, but that's OK. OK, this is the norm squared and again I cannot really say anything. OK, so it's just the norm squared of this vector. OK, how much is f? F, again, would be the scalar product between this and this and I have no information to use. So it would be alpha prime scalar w plus v w prime scalar w. And what is g? And g is just the norm squared of w. OK, how much are the coefficients of the second fundamental form? Well, let me not even write little e because little e would be the scalar product between this and this, but OK, I have no formula to use. So besides rewriting the definition there is nothing I can say. Let's see what is little f. Little f would be the scalar product between n and Xtv. OK, so let me introduce a notation. If I take the scalar product, so some notation, if I have three vectors, a, b, c, OK, in R3, whatever you want, in Rn, but I mean I call the bracket a, b, c is by definition the scalar product a scalar product b cross c. OK, this is a standard notation in many books that you will find, so better get used to that. I mean it's not a great simplification. It's almost the same number of symbols anyway because now what is f? F, you see, it's the scalar product a vector and a cross product. So I can use immediately that notation there and that becomes alpha prime, alpha prime ww prime divided by the norm of n. OK, so now I can add the formula for the norm of n, which is alpha prime cross w plus vw prime cross w. This is nothing but substituting the norm of this. I'm not cheating you. I mean this is just what I would get here immediately. OK, and then how much is g? Little g in the second fundamental form? Well, this is the best one because of course it's zero scalar, whatever, so it's zero. That's why I don't have to compute little e. Because in all formulae, I mean for example in the Gauss curvature I would get eg. So if g is zero I don't care about e. OK, so having done this how much is the Gauss curvature? Well, it's eg minus f squared. So eg minus f squared divided by eg minus f squared. That becomes, so eg is zero, so minus this object here, alpha prime squared divided by alpha prime cross w plus vw prime cross w squared. This would be just f squared, minus f squared, so times one over eg minus f squared. And I don't even write it. OK, divided by eg minus f squared. You substitute those almost generic functions and that's it. Because this is enough to make some considerations. You see, any ruled surface has Gauss curvature less than or equal to zero because whatever this is, this is a positive function because it's the determinant of the first fundamental form. One over. So this is positive, this is of course positive and this is minus something squared. So this is less than or equal to zero and in fact it's zero if and only if this number is zero. Now I don't go through the examples we made. Now you have this nice formula and you can substitute in the one sheet hyperboloid for example or in the cylinder. For example this is another way to compute the Gauss curvature of the cylinder or general cylinders or of a cone. I mean now you pick all those examples you have the explicit expression for alpha, w and w prime. You substitute and you get the result. OK. But now it's clear that the vanishing of this function is measuring something interesting. So let's see if how many times we can reach this position. Now in general, so no loss of generality you see the role of w we said at the beginning is just to point in the direction. So to give me a direction. So there is no arm, I mean I'm studying the same surfaces if I assume the norm of w to be constant equal to one. OK. If I have one parameter family of lines where w is the norm of w is a function of course I can res k and I get the same surface because it's giving the same direction. So since I'm adding the whole line I don't care. It will change the value of v for which I touch a point but I don't care. The image is the same. So no loss of generality in assuming this. OK. But this is technically convenient but before seeing why definition a ruled surface is called developable developable if this function is zero. If alpha prime w prime is equal to zero. OK. Or equivalently if the gauss curvature is constant equal to zero. OK. We have given a name but we could have been giving a name for how many surfaces are developable. OK. Now let's see. So now suppose we have a developable surfaces. So let's see if we can identify it in some form. OK. Well, first and this has nothing to do with the developability we have put this normalization on the direction. OK. If I have a family of vectors of given norm I take the derivative and I get one equation between the scalar product of the vector and its derivative. OK. Nice. So now let's see in how many ways I can get this. OK. Of course this is the scalar product between this vector and the cross product of these two. So, first case when this is zero cross product of these two is zero. Of course, if this is if the cross product of these two is zero whatever is alpha I get zero. So let's first see if I can understand the geometry of such a surface. So, case one if w cross w prime is equal to zero what can I say? Well, if the cross product of two vectors OK. So, this implies that, let's see in which way I prefer w prime of t, now let me rewrite what they depend on is equal to some number but which in principle depends on t times the other one. So they are proportional but the factor of proportionality depends could depend on t. OK. We have this equation here so let's use it. So on one hand I have zero is equal to w scalar w prime but now I have also this equation so if I substitute to w prime this I get what? I get lambda w w but w w is equal to one OK. So this is equal to lambda. So in fact the second equation is a stopable equation forces this factor of proportionality to be constant equal to zero. OK. But then but then the geometry of the surface is almost trivial because that means w prime is constantly equal to zero but that means that w is a constant vector and we have given a name to such surfaces. So that implies w of t is equal to sum w zero if you want independent of t. So the surface is what we called a cylinder. Is a generalized cylinder of course now drop from your mind the idea that the cylinder is only the rotation invariant straight cylinder. So there is a curve and there is a constant vector in the surface which at each point of this curve takes the line in this parallel direction. So cylinder. Very well. But this was of course the first trivial case. I mean this color product is zero because one of this vector is zero. Case two. Of course suppose now that we are not in that situation. Another way to say it is of course zero. That means these two vectors are linearly independent this time. It's the contrary of the situation before. But the scalar product between alpha and the cross product of these two is zero. That means that the triple alpha w prime are linearly independent. Alpha sorry, linearly dependent. Are linearly dependent. Sorry? Sorry, alpha prime, alpha prime. Yes, of course. That's information coming from here. So now it's a little bit like before but it's slightly more complicated because if they are linearly dependent for example, I mean let's assume that alpha I express alpha prime in terms of a linear combination of the other two. So alpha prime of t would be, so there would be now again two functions because the coefficients are free to depend to move in time. But there are coefficients such that something like this holds for some choice of lambda and me for two functions lambda and me. And alpha tilde of t. Let's define another curve by just alpha alpha of t you see, I want to change the directrix of my surface without changing the surface. Remember that the rule surface has many infinitely many directrix. So in this case it's clear that if I take alpha minus this, I take this on this side, I mean minus mu well, before taking the derivatives minus mu w I'm doing something smart because what is alpha tilde prime of t? Well alpha tilde prime of t is of course alpha prime minus of course I introduce a little mistake because I had to take the derivative also of this function w minus mu w prime ok but then again I use the equation defining lambda me and this is equal to what? This is equal to me me minus me prime w of t because alpha prime minus me w prime lambda sorry and where do I put it here? Alpha prime minus so this is lambda minus me prime ok now so this case too must split now into sub cases so case if you want 2a sub case a suppose that alpha prime is constant alpha tilde prime is constantly equal to 0 ok well what does it mean? It means that alpha tilde is a constant vector let's say some kind of alpha not so it's a given point of ok but then what? then alpha well this is of course the trivial case alpha is where is it here? alpha is equal to alpha tilde plus me w is equal to alpha not plus me of t w of t ok but you see that we have not changed the surface so what is x? x of tv now let's write everything just to be sure that we are not making mistake is alpha so then it's alpha not plus me of t w of t plus v ok now we could go on with analytic interpretation but now the geometry is clear what is the geometry of such a surface? you see w of course the only thing I would like to do let's do it but I mean it's me plus v w ok but that means what? that the rulings of this surface they all pass through one point alpha not so in geometric terms this is a cone ok this was again trivial case 2a now let's do 2b ok so now suppose that alpha prime alpha tilde prime is non-zero alpha tilde prime different from zero so what can I say now? well let's take let's see how x again looks like so this is alpha plus v w and now let's put the alpha tilde into the game we found alpha tilde ok so alpha becomes alpha tilde plus me w ok so this is alpha tilde of t plus me of t w of t plus so again w of t ok but now I know that w if I want I can divide suppose for a moment generically I will be able to take this thing and put it here so w is equal to that one so this becomes alpha tilde of t plus me of t plus v lambda of t minus me prime of t everything times alpha tilde prime ok now finally I have to use the equation otherwise ok but then you see why I gave you the definition at the beginning as an example because now my surface looks like alpha some alpha if you want call it alpha tilde beta or however you want plus this one which you can call if you want v tilde times alpha prime so this is a tangent surface to a curve to the curve alpha tilde ok so this is the tangent surface to alpha tilde ok so this was very simple but what is the moral of this remember that the question of understanding what is the geometric meaning of the gauss curvature and also of the mean curvature in 15 minutes ok so the first thing is curvature should distinguish between flat and curved ok so naive question above the big question in general you have a function you want to understand what is the meaning of this function easy level zero question suppose this function is constant equal to zero what does it mean I mean is the surface special and then of course the answer is well yes depending what you mean by special but certainly is not too special so I hope I convinced you now that there are huge families of surfaces with constant zero gauss curvature because all cylinders or cones and all tangent surfaces to curves they all have gauss curvature equal to zero and in some sense they are the only one ok so if you ask my four years old baby he would say ok now that these are flat but this seems to contradict openly our intuition of curvature because remember not even this one is flat but you can find some strange argument but I mean now a cylinder over any curve is flat a cone over any curve is flat so the tangent surface to any curve is flat so now it looks desperate now you say well that means the gauss curvature maybe is not that interesting if it's not able to distinguish ok so we have the only point is that we have to work harder ok now let me enter the problem of mean curvature I would like to play the same game with the mean curvature now in fact let me take a long route and we are going to learn something which is as important as the final point in fact probably even more important than the final point because we have treated the first fundamental form you see for example in the consideration of gauss curvature equal to zero the first fundamental form was not really entering because it was always e g minus f squared divided by e g minus f squared first fundamental form but if you ask is this zero or not you don't care about the denominator so it seems a property completely determined and we also said when we define the first fundamental form we almost said well ok it's just to do something since we called it the other one second let's call something the first because it's not really adding any new information it's the scalar product between tangent vectors but now let's go back and let's throw away this this is not bish attitude because the first fundamental form in fact measures many fundamental things and at the end in a couple of weeks you will see it will measure all fundamental things so for example let's ask the following question remember we start with a regular surface and suppose we have a local chart ok that means we have our usual picture we have some domain we have a map x and that's the image ok now suppose you take a curve on the surface of course inside this chart I'm going to make local computation so I will restrict myself so suppose I have an alpha of t which lies on my surface s so of course since we are doing metric geometry because everything which comes from the scalar product has something to do with the metric ok the simplest question in metric geometry is what is the length of a curve and in fact for when we started curves by themselves that was basically the only thing we could do so how much is the length of this curve well this is a curve in space so the answer does not depend doesn't seem to depend at first look from the surface s and you say well I have alpha of t very well the length of the curve alpha let's say between two points of the interval doesn't matter is the integral of the norm of alpha prime so suppose I have it parameterized by some t in dt between some t0 and t1 you see that this doesn't seem to depend on the surface in fact does not depend on the surface ok so it doesn't seem to be treated just as a curve in space but then if the curve lies in a local chart of course there is the usual correspondence that we use everyday there is some curve here beta of t which is mapped to alpha ok so you might ask now a more refined question and this depends on the length of this curve in the plane and the length of this curve on the surface so somehow how much is distorting lengths the map X so the map X was fantastic from the study of surfaces but it will change lengths of course ok there is no reason why in general lengths of curves will correspond ok if I use this trick of course how much is alpha prime so because alpha becomes X composed beta ok as usual so what is alpha prime well alpha prime is u prime X u plus v prime X v ok where of course I am assuming as usual all the usual notation in two functions u of t v of t ok but then how much is the length of alpha well in fact before doing that how much is the norm of alpha prime of t so then I substitute into the integral ok how much is that well it is the scalar product between this and itself well divided by I mean square root so this becomes what it is for the square root this times this so u prime squared E plus and you can imagine twice ok u prime v prime F plus v prime squared G everything to the power one half ok well this is interesting because ok now put it I don't even write it but I mean now I love alpha becomes what is the difference of course is that if I write it in this way I am thinking of beta I have the two functions u and v of t no and in some sense I am comparing an integral here so I am comparing I mean this means that the philosophy the geometric interpretation of the first fundamental form is exactly what I said before because what would be the length of beta as a curve in the plane it would be the integral of beta prime but what is beta prime beta is equal to u and v is just the integral of u prime squared plus v prime squared one half in the t so you see that the second fundamental in the first fundamental form is exactly the distortion of lengths of curves measured on the domain of a chart and on the image of the chart the same curve but once it's here and the other one is transported on the surface ok and here they come E, F and G very important ok we can play the same game as of domain after all so now let me draw the parallel picture but it's always the same I have my ux and now suppose I take some region here ok now now suppose I have some region that I imagine again inside the chart so it corresponds to some region but it doesn't look very different here so let me say this is v and this is x of v ok now again let's ask the same question but now with the area so in some sense I know how to measure the area of v this is not a v prime this was just the area of v compared to the area of x of v ok so let me give you a definition but the definition is motivated by the following classical if you are in R3 now let's say this is a kind of a side note to justify what I'm going to say if you take two vectors which might not be orthogonal of unit norm nothing of course linearly independent in some sense it's a pathological case let's call it xi and eta in R3 now of course two vectors in R3 give me a parallelogram ok I can imagine this parallelogram here how much is the area of this picture ok exactly so the area of this region is given by the norm of this in fact it contains the pathological case if they are parallel you get zero as an area ok this justifies the following definition in fact let me write it here so I save a little bit of space so the area of x of v because I have to define it actually it's not ok of x of v is equal to the integral of ok why I'm going to write this formula because I want to apply this simple observation which are the two vectors which play a role here xu and xv so you see at this point of the surface I have two canonical vectors tangent to the surface and in some sense in this decimal area well generated by these two vectors the norm of their wedge product of the cross product ok but how much is the norm of this strange enough but not strange at all is again the determinant of the first fundamental form so this justifies to define the area of this region the integral over v now as an integral here eg minus f squared one half now as an integral in R2 so I know what to put there as a measure so I put the standard Riemann measure ok now here that now there is a problem because you see I'm defining the area of a set on a surface of a subset of the surface and here I use to define it a chart so now there is the usual problem if I have two charts covering the same region does the area depend on the chart or not because if I look at the formula it does ok but this one it's an exercise in calculus and I leave it to you suppose you have another chart that you want capital Y which now comes from another which covers this the same set but now for example corresponding to a domain let's say capital W ok so this will have another other coefficients of the first fundamental form because the coefficients of the first fundamental form because E measured with X and E measured with Y are different F are different G are different but this integral is well defined of course everything boils down to the change of parameters formula ok for integrals in Rn ok and it works so this is a well defined integral maybe I'll put it in a in an exercise sheet so in case some of you have some problems we can do it together ok but now again you see this simple interpretation definition now tells me that the first fundamental form is exactly measuring the distortion now this time is by definition but I hope I convinced you that it's the right definition the distortion of areas here and here ok so the first fundamental form which seems to be just a name is in fact taking care of the two major geometric things that you can do measuring lengths of curves measuring areas of domain ok ok we will freeze this for later use but now let me go back to in the same spirit we did the ruled surfaces thinking to the Gauss curvature zero we studied ruled surfaces now thinking to surfaces with zero mean curvature we do something else and that's what we are going to do now ok in fact it's better to put it right from the beginning the key definition so a surface a regular surface is called minimal this is kind of the corresponding name to developable if h is constant equal to zero but now I want to convince you that this name has some sense minimal in which sense ok and this is nice so first, so the idea is the following you have a surface and again this is just the piece of a surface covered by some parameterization by some chart I'll try to make a big picture even though I will need so this will be x of u and that means here I have some big domain u and the map x ok now, suppose in fact this u in principle could be unbound and non-compact anything it's just an open subset of R2 so now suppose here I pick a bounded domain d ok so now d is a bounded domain so x from u to s is a local chart d contained in u is a bounded domain actually there I drew it closed but I don't want it to be necessarily closed in fact usually a domain is open in fact to make it proper you have to remove the boundary and now suppose you take a differentiable function h from the closure of d to r differentiable from the closure of d ok I will call the normal variation by h or in the direction by h so this is the definition of this sentence the normal variation of x of d closed determined by h or in the direction of h if you want is the map let me call it phi phi from where small interval so for some epsilon positive which doesn't matter into r3 which takes so d is inside u so in u we have the two coordinates u and v so it takes u v and let me call t the parameter in this small interval and gives me what gives me x of u v bonding point of the surface plus t h is again a function of u and v ok so n at the point u v so what's going on nothing worrying in some sense I should draw the graph of h over d and this is kind of difficult now in the picture but you can imagine most domain d bar so imagine a graph here and now I'm taking phi is what is the corresponding point on the surface so here I have a normal vector at this point h has some value for example it's positive otherwise I need to draw it the other way and here I take the point phi of u v plus t u v t so that means geometrically it's simple if you think of what's going on you have a graph here on the flat r2 and basically you are drawing the graph here in the normal direction the graph of h transported here so this is the idea of course the graph at height t it's a family of graphs but of course the role of t is just to push it to translate it to translate it in the normal direction it's not a translation in r3 because n depends on the point so you see for t equal to 0 this of course is not there and phi takes the value x so for t equal to 0 I'm just taking this disk for example in my case and I'm putting it here somewhere so this will be x of d closed and then as t moves I start moving this it's not even a disk anymore but the image of this disk using h as a kind of a graph function ok I'm writing something nearby not for t it will be very close to that one by using h ok is it clear geometrically what's going on so for example if h is equal to 1 constant equal to 1 what am I doing I'm taking really x plus tn ok so this is the subject this piece of the surface in the normal direction ok every point at the same speed so h is a parameter which gives you the freedom that one point is moving quick and one point is moving slow or maybe one point is going up and the other point is going down because if h changes sign of course we have an imagination of what's going on ok now notation and you might say why it's not the initial notation so for fixed so I call this a normal variation ok it's clear why it's a variation because for t equal to 0 it's the old one and it's normal because every point is pushed in the normal direction ok so for t in minus epsilon epsilon I want to think of this of a variation as a family of regular surfaces and of course I can do it because I define x and now I put the t here of uv to be phi you see if I fix t I'm taking the slice ok it's a surface which is very close to the old one it has moved with the function h ok so to each of this surface so if I keep t fixed I can study the geometry that we studied up to now of the surface xt let's do it ok so basically I want to know how much is xt u so now I have a parameter which is t and two parameters umv well how much is xu xu will be xu without the t now plus something xu plus t hu n plus t h nu ok how much is xv something changing ok so how much are the in principle I could compute the coefficients of the first fundamental form for each t ok let's do it so what is and I call it et et will be the first coefficients of the first fundamental form of the surface xt well that means it's this squared ok imagine you are ok so now let me save three seconds and we are taking the scalar product of this times this plus t h nu ok what can I do and how do I want to do it because of course the only thing I could say is to expand everything and I stop but the thing I'm interested in is this will be a function now I want to see it as a function of t ok so for t equal to zero I want to make in some sense the Taylor expansion in t ok so I want to know what is the t equal to zero or what would be the constant in the Taylor expansion and then what is the linear part in t I want to split it in degree with respect to t ok well for t equal to zero it's easy because of course the only known constant in t now my speaking is referred to the variable t the only known constant object is x u x u so not strange it starts with the old one after all for t equal to zero the surface was the old one so what is there in t plus t what well of course the only thing I don't have to do is a scalar product of something multiplied by t with something multiplied by t because I would get a t squared so for example so this one times what this one times this but x u and n are orthogonal by definition so this one is not there so this is the next one so x u scalar this ok let's write it so plus t times h x u and u ok it will be longer than this and then I am done with this one now what about this well this one goes against x u and then goes against anything because there would be t squared so this one goes away in the linear part and this one gives me t h so t is there h is here and u x u so I get twice this ok plus so let me write it in this form of t squared so the point I want to make is that now I don't care t squared coefficient would be much more complicated but I'm interested in the linear part let's play the same game with f f t is what now imagine we are taking this scalar this so of course it starts with f and then what is t what so this times this zero this times this h x u n v and I am done with this one then this one but this is zero and then everything else is t squared and this is what this is n u x v plus of t squared n g g t is g again so this times this is zero this times this gives me of course by symmetry if you want you know it so it's t 2h x v n v ok plus of t squared ok now I can write in fact I should have left one line because I know what are these these have a name in our language because you see these ones are computed at t equal to zero here there is not a t here and there is no t here so these are the original objects ok so I know what they are for example this one is what is minus e ok so what I found so in fact let me rewrite it, it's a pity so e minus 2h t e ok plus of t squared ok when I write in fact let's invent a symbol ok this means up to second order ok so we save time how much is f, f t would be f ok and now here it seems slightly more complicated but it's not because you both of them are f ok they look different but then you think coming from n scalar x u equal to zero and then scalar x v equal to zero whatever derivative you take you get that these are equal to minus f ok so this would be minus 2 t hf so and this would be again up to second order and g t is up to second order g minus e g ok very well we are halfway through but then remember it's not there anymore but the formula for the area the area of a region the area for the region was given by the integral of e g minus f squared one half d u d v ok so it's quite clear now we are going to speculate on that but for the moment it's quite clear that we want to perform this computation so we want to know how this subject depends on t again with the same spirit first order, first order and higher order ok so e t times g t minus f t squared well the one half bit later now a little imagination and you get this you get of course the zero order is the old one so this is e g minus f squared plus t times what in fact I should have put minus because there is a minus in front of everything minus this object here minus in fact even a 2h is in front of everything no surprise 2h times e g minus 2 f f minus plus e g and here this is again up to second order this is immediate but now you remember I mean you don't remember and you are allowed not to remember even I don't remember but the formula for h in terms of all these functions h of the original surface h of the surface at t equal to zero and if you use it which in fact was actually just telling you that h is this divided by e g minus f squared capital ok so imagine what we are doing we divide by e g minus f squared but that means I can take out e g minus f squared from everything so I can say that this is equal to e g minus f squared times something which starts with one minus 2h 2th and now the only problem is that this is so this divided by e g minus f squared is not exactly h but it's twice h ok so this becomes 4 capital h ok and now let's compute the variation of the area so in you see the spirit is these surfaces are moving but somehow they are all image of the same thing so if I since I define the area of a surface has the area of the coming domain original domain but measured with a strange measure so measured with a measure coming from the first fundamental form because here the integras are always with respect to du dv but there is the function in front square root of e g minus f squared so if I have two surfaces which are image of the same thing I can transport problem of computing the area becomes a problem here on the fixed domain but with measures which are changing ok and that's what I want to do so the area of what of how did we call it of x t of d bar ok so for each t look at this strange graph normal graph and I measure its area but of the same domain so this is equal by definition of the area over the integral over d bar so it's an integral in R2 so I should put here the square root of this so e g minus f squared 1 half times square order because I dropped the higher order terms times 1 minus for th h 1 half but then everything is integrated with respect to du dv let's compute d in dt so how is it changing so the derivative at t equal to 0 of this function of t for each time I have a surface I measure the area I have a number so I have a function I take the derivative of t equal to 0 simple so you see this does not depend on t so you have just to take the derivative of this evaluated at t equal to 0 and you get this this is minus the integral over d bar of 2 h h times minus f squared 1 half du dv ok, this is take the derivative and evaluate at t equal to 0 ok, there is nothing there oh, but now we are this is a very interesting formula because you see the key point is that here you have isolated so this is kind of a positive function and here you have in front these two objects every time you see something like this this is easy to interpret because the point is if you have a surface if you have played this game starting from a surface which was minimal so h equal to 0 ok so for example there is a stupid simple error here so if the initial surface is minimal then this object, the derivative of the area in any direction well in the normal direction but using any function would be 0 it's a critical point ok well if s is minimal but here the analysis takes care only of what's going on inside the chart x the important thing is that h is equal to 0 on this part ok, so if you want if s is minimal but I mean, what you care is that the mean curvature is 0 on x of d bar ok but now this formula it's not difficult to see that implies also the other way around so, because if this object is equal to 0 for any h you see that this is a huge condition and you can imagine I mean, I'm not going to give you a complete proof because I hope you see it ok if h was not 0 at some point it's easy to build a little h for which this is non-zero you see why I mean, the idea is in fact, this is even the proof in some sense suppose for a moment that we are drawing our surfaces flat otherwise the pictures become impossible for me so you have a surface and at this point h is equal to 1 sorry, this is the normal vector and at this point p h of p is equal to 1 for example so I argue that this integral cannot be 0 for any h that's enough because I build a little so if h is equal to 1 at one point by continuity it will be so if I draw the graph ok, here it's 1 and here it will be whatever it is I don't care I build h as long as it's non-zero I build for example a function which is 0, 0, 0, 0 and then and this is little h ok and I choose this small interval in a way that on this small interval even capital h for example is positive if it's 1 at one point there will be a small disk where it's positive ok, so you build a little h h is a positive function whatever capital h was ok, over the whole d so this is positive this is positive the integral cannot be 0 ok that's in fact the proof so if the variation if the first variation of the area is equal to 0 for any variation then h is equal to 0 ok, now this is a beautiful theorem it's one of the building blocks of the modern differential geometry ok, in some sense but now in the spirit of what we did for ruled surfaces so we wanted to say k is equal to 0 was meaningful I mean, for example was it the plane or maybe just the plane and the cylinder and we came up with huge family of examples so whatever the truth will be it will be more complicated than that now h is equal to 0 now we know a variational characterization of surfaces with h is equal to 0 but in principle it could be the empty set well not really empty because you know the plane the plane is a minimal surface ok but are there other minimal surfaces well first just a language problem the name minimal is very misleading in some sense because it's a critical point of the area functional nowhere is written there that it's a minimum minimum would mean compute the second derivative and prove that this is non-negative and this is highly false so in fact the search of minimal surfaces in R3 it's the search for unstable critical points of a functional which is more complicated of course much more complicated than if your functional has minima you minimize and you try to do something ok here all these objects except for the plane the plane is the only minimum ok so the word minimal but unfortunately it comes from Lagrange and when something comes from such a big guy you shut up but it's kind of remember minimal in this theory it means critical point and not minimum ok so the problem is are there other minimal surfaces or not because maybe we are speaking about the empty set we did a beautiful exercise in differential geometry but well you know another one in fact you know two ok I knew one and you do another so we know two so the first one or I don't know if it was the first or the second but this is one catenoid ok the catenoid is a minimal surface ok but then why this was a kind of a huge problem in mathematics because this is a very natural interpretation in term of if I just find ok so this was the way the theory of minimal surfaces was born kids playing with soap in fact physicists playing with membranes ok take a wire put it in a in soap and pull it up ok now believe that there is a correspondence between the area of the surface that you get and somehow the amount of energy that the soap has to waste to cover to span this wire the soap will make an effort ok you have fixed the boundary and the soap will try to minimize the effort to span the boundary so the soap is going to produce a minimal surface it is going to produce the minimal surface when you have fixed the boundary somehow you are telling the soap something I want you to span this piece of wire usually kids play with in fact it is not true anymore I have been playing with round soap wires and then the solution was quite boring because you put it there you peel it up and you get a disk ok but we are mathematicians and we think why if we play to soap bubbles with something like this this is a wire you put it in soap and you pull it up so what is the soap going to do well this is already an interesting example because this shows you that the soap gets crazy there are two minimal surfaces ok in fact one is bigger area so in fact the soap doesn't get crazy this time because it will choose this one ok but you see now you have formulated the problem given a boundary a closed curving space is there a minimal surface with this boundary in R3 this was a classical problem in physics ok it's formulated with soap because it's easy because everybody have played with soap bubbles but it's clearly a very general family of problems ok give me a rule for which area corresponds to energy or effort or whatever you want and you will end up studying soap bubbles ok so that's why you will find many books in mathematics library speaking about soap bubbles because it's the prototype and it's already very difficult ok and in fact you can go in fact the helicoid I didn't print a copy of the helicoid I have a picture of the helicoid but the helicoid is another minimal surface so you start seeing that they look very different but you cannot even imagine how different ok because for example this is another minimal surface it's periodic in space it's contained in a cube and then you start duplicating the cube with this surface inside and you invade the whole of R3 the gray areas are the holes ok so with something like this for example you construct a minimal network of plumbing ok these are plums ok minimizing the area means you are using the least possible steel ok since nature does these things in fact this surface exists in nature it's called the Schwarz surface ok but then you start getting more and more complicated objects so this looks more or less like a catenoid you see for example if you take a cat the stupid thing would be stupid but I mean you take a catenoid and you take a plane it's orthogonal to the axis of the catenoid for example in fact the union of these two things is a minimal surface the only problem is not really a regular surface because they intersect in a nasty way it's not a differentiable surface so the question we have been for many many years is it possible to smooth a configuration like this keeping h equal to 0 because in some sense formally it's either the catenoid or the plane it's ok and the answer turns out to be yes but this is an extremely beautiful and difficult problem which was solved actually in the beginning of the 90s so we are speaking about very recent stuff and there are still people asking in how many ways ok because actually these objects do exist so it's one of those points in which mathematics and nature work together so you really like to have a classification of minimal surfaces because these things exist so we know enough to say that there are plenty of them but we are very very way behind the classification in some sense it's a beautiful research area because of course you can imagine that behind this picture there is also a huge 3D graphic problem because if you think that you are going to prove that this object is a minimal surface by telling me which is X forget it ok you will never be able to write down the explicit solution unless it's the catenoid or the helicoid you are lost either you have some method to prove that something exists so it's proves are very mathematical and then you wonder well there are the pure problems and the applied problems and say now I want a picture ok very well I think but just since we have told you a little bit of history let's go back to the boundary problem because I'm drawing you pictures I showed you pictures of complete surfaces things which go to infinity and so on somehow the kids problem was when you take a wire you put it in soap and now you have forced the soap to span this wire and you want the minimal surface with this boundary ok so does it always exist well this is a very nice I mean some one of the crucial problem basic problems in the calculus of variations in 1930 for any curve and fixing some kind of topological type of the surface that you are looking at by two mathematicians an American and Hungarian Douglas and Rado for some reason which has never been completely clear to me the first fields medal you can dream I'm one and a half two-hold I stopped dreaming one and a half ago so I joined a big group of people who didn't win the fields medal ok I like to be in the majority but so the mathematical community so we don't have the Nobel prize so we invented another prize at the beginning of the 20th century which is called the fields medal which has to be given to one mathematician below 40 so the first fields medal in the history of mathematics has to be given to Douglas for solving this problem which is called the plateau problem plateau was the physicist who has set up some kind of basic physics to speak about this problem in the late 19th century so somehow it's also the beginning of this I don't know why they chose Douglas instead of Rado I mean now we consider the two proofs correct so probably at that time there was a more controversial issue they really came up at the same time with the proof so ok, that's it and we'll see on Thursday


What's up everyone? It's your girl Kenzie here from Leighton Sports Cards and I am ripping last obsidian basketball of the night here We've got 21 22 Panini obsidian basketball. I'll be 12 box full case break number six tiered random teams again Here is the list of the random results. I'll leave this up here for a second. You see what team you got again You can always find these a leighton sports cards calm underneath the results tab and the link is also in chat for you All right, best of luck everybody Let's see what we can get again if you're not a member definitely recommend you get in that membership only $1.99 a month Tons of benefits you get funny emotes. You get a first look at some members only content that we post every now and then You'll get coupons we usually do About a coupon a month that will definitely pay for itself, you know with the first use Usually 10% off or some different perks that we can give you for being a member's only Again, if you have any questions about that, just let me know be sure to help you out But you only a dollar 99 a month You can become a member Get first looks behind the scenes kind of stuff Definitely worth it All right, so here we go Bobby porous Herbert Jones rookie at Pelicans and Mike Connelly jazz We've got to 50 lightning strike Jay Sean Tate auto for the Rockets and 235 we've got Pelicans rookie Jersey auto of Herbert Jones Pelicans Sub-zero I got orange to 50 clay Thompson and Pitch black Jalen sucks for the magic 32 of 99 You see who won the last time We've got Elijah on Kelly who brave junior Hornets and a Jeremiah Robinson Earl Thunder rookie Out of 99 spurs lightning strike of Doug McDermott. No Liberty. Oh, wow Out of 50 Aurora Dave Bing for the Pistons Dave Bing man there's seven One I believe yeah one or two Malcolm Brockton to 199 Pacers and we've got magnitude green clay Thompson for the Warriors at a 25 Bears and Seahawks play tomorrow Yeah Jason Tatum Zyra Williams rookie Grizzlies and DeMar DeRose and bulls Drewlock really did not look bad first week Didn't look horrible out of 50 Rockets galaxy Kenyon Martin Jr. I mean, I don't think he's gonna do absolutely fantastic things for the Franchise, but he looked better than what I think a lot of people are giving a credit for out of 75 RPA Pacers Christu Arte You know Pacers nice hit there Danny see with the Pacers You got COVID again. Yeah Got James Harden on purple to 99 sixers and pitch black green LeBron nice one there for the Lakers out of 25 Hey Like I said, he's not like MVP worthy at all, but I definitely think he looked better than what people were so rightly all over Instagram comments And Facebook comments saying how much he hated the Russell Wilson trade said that they gave up way too much Julius Randall Nick's Terry Rozier Hornets and David Robinson for the spurs Out of 149 volcanic little Wendell Carter Jr. You go Orlando Jasper T and going back to Jasper at a 99 rookie Jersey auto Jalen Suggs magic Oh my down nice little pinch straight there congrats magic another solid team in Obsidian Nice Purple to 99 CJ McCollum Pelicans and orange to 50 Damien Lillard Blazers Who I remember who was I'm not gonna like But there was someone who's big in sports cars on Instagram the other day We got Kevin Durant nets Corey Kispert rookie wizards and Lamarcus Aldridge net saying I'm investing a hundred thousand dollars and drew That I think that was a joke That dude is probably is known for saying stuff like that out of 149 volcanic Chuma. Oh Magic Jasper T on that one I have to Jasper magic are heating up here to 75 purple rookie eruption Jalen Suggs boom Wow Nice one there magic purple shingoon rookie Rockets to 99 and pitch black Devon Booker to 99 Suns No, Brad Bradley hated the Russell Wilson trade hated it Think you may have cried Said why would we ever give up drew lock? Ray Allen Carly Anthony Towns T Wolves and Brandon Boston jr. For the Clippers Ricky Um Chad just put a note in your order that you would I believe that judge should be fine Put a note in your order, and I would just shoot us an email and I can definitely Forward that to the shipping department that you want to add additional insurance on it Should not be a problem again put it in your order note And then shoot us an email late in sports cards and gmail.com and I will forward that on to the shipping department out of 75 Aurora Joe Harris for the Nets Will you type our email into the chat room so you can just copy and paste it? I've got Pacers to 50 lightning strike of TJ Warren Yeah, no problem. There is our email right there for you green it to 25 20 to 25 Anthony Edwards and Volcanic short print rookie Zyre Williams Grizzlies Nice SP there on the case hit Volcanics All right second not here again last city and basketball tonight got a city and team all football up next love those guys such beautiful products Yeah, Brandon Boston jr. Rookie Clippers Alex Caruso for the bowls and didn't we need for the maths? We've got nicks out of 75 Galaxy Auto Evan for me Looks like him And Cavs out of 25 Rajon Rondo on the Galaxy Auto It looked like Mowgli for a second from where you're sitting probably look like him, too Got orange to 50 Jeremiah Robinson Earl Thunder and purple in 99 Jean-Té Murray Spurs Parker, right?  We've got the blue flood electric edge rookie Josh Christopher on the SSP Rockets Heist one there back to back big rockets hits here Grats again Gary like that one and Magnitude Josh giddy for the thunder that is a goal to 10 Nice giddy. What a box Benjamin M It's like that. I can't tell if it was gold or green the light hitting it It's got a little two colors on that one almost nice giddy. We'll mag that one too. Wow. What a box What's a box Gray Brown the third rookie Blazers Jared Jackson jr. Grizzlies and a Brandon Ingram Pelicans Out of 35 we got Hornets rookie Jersey Otto JT fourth and To 10 we've got gold rookie Jersey Otto Zion Williams on the three-colored patch. Here you go Grizzlies Beautiful looking card there Chris G with the Gris This is a really cool photo It's really cool out of 99 the Andre 8 the Andre 8 and for the Suns and magnitude Joshua Cremo Spurs That is 33 of 75 And I'll switch in the WBA playoffs We had on the Liberty in the sky. I can't believe the Liberty beat him Isaiah Jackson rookie Pacers That was a close game Fred Van Bleet Raptors and Harrison Barnes for the Kings. What's up, dog? You got to 99 Nick's rookie Jersey Otto Quentin Grimes Nick's Out of 149 sixers a Derek Coleman Sixers have quite the throwback names Purple rookie to 99 James book night for the Hornets and a pitch black Lamella ball Hornets 11 of 99 on that one. Most of the golds are very noticeable Just that one giddy for some reason looks like it was like possibly green to me. It might just be my eyes But the golds are definitely noticeable It's just the way the light is coming off of it Doing good dog hanging there man. Oh, yeah, how about yourself? Got last box here Joe Weiss camp Spurs rookie Clint Capella Hawks and James book night rookie for the Hornets We've got Grizzly's RPA another nice pad Grizzly's have nice patches in this for sure. Santiago Dama You go Grizz on the three-color patch To 99 Nick's how would a Jamal Crawford nice J cross over on the Volcanic Otto Last couple here, and then we'll do the recap And we've got purple to 99 rookie Keon Johnson and magnitude orange Jason Tatum to 35 Alright recap time very nice case there, too Obsidian has been solid all night long very nice product this year for sure We had Jamal Crawford nicks a Santiago Dama RPA Grizzly's Derek Coleman sixers. We had the rookie Jersey Otto of Quentin Grimes for the Knicks a Gold rookie Jersey Otto Grizzly Zaire Williams Ricky Jersey Otto for the Hornets JT Thor Daniel Gibson calves Duncan Robinson Heats Jason Preston Clippers Rajon Rondo calves. We had Evan Forney nicks TJ Warren Pacers Joe Harris for the Nets Chuma Oki magic Wendell Carter jr. Magic a RPA Chris Duarte Pacers Rockets Kenyan Martin jr. Dave Bing for the Pistons Doug McDermott Spurs Jay Chante Rockets and the rookie Jersey Otto Hedbert or Herbert Jones Well said Hedbert Perez Don't ask me where that came from. They haven't ripped baseball in a minute We had these gold Josh giddy for the thunder on the magnitude nice one there. Okay, see again. That was nine out of ten the Blue electric flood etch Josh Christopher for the Rockets on the SSP rookie Otto on the rookie eruption purple she does a mouthful Jalen Suggs for the magic the 75 We had the rookie Jersey Otto Jalen Suggs to 99 magic and the rookie eruption Jalen Green rookie Otto 80 of 99 Great way to end obsidian tonight. Thank you again everyone so much for the film. We've got some football coming right up


YouTube will disable your monetization on all your videos on your channel on April 1, 2016, if you do not accept the YouTube Partner Program. How do you accept it? Well, if you go to your channel, sign in to YouTube and then click, let's say the YouTube logo anywhere, then you will get this message telling you you need to accept the YouTube Partner Program. Simply check all the boxes and then choose whether you want to be notified with updates and then click accept. That's it. Once you do that step, you will not see that box again and you will not get your monetization disabled on April 1, 2016. If you don't know whether you've accepted the YouTube Partner Program, simply sign in to your YouTube channel, click the YouTube logo and do you see the YouTube Partner Program big message? If not, that means you've already accepted the program and you don't need to accept it again. This is very important, guys, because it's only three weeks before April 1st and we don't want you to lose your monetization on your channel. So accept the YouTube Partner Program. It's a new YouTube policy. It's a requirement. It's not optional and it's not going to unlink you from freedom. It's not going to disable your monetization in any other way. It's not going to change anything. What will happen is you'll gain access to new revenue streams such as cover song licensing and other revenue streams as YouTube rolls them out. So there is no negative to joining the YouTube Partner Program. It is required if you want to keep monetization. So I hope this helps you understand why you need to do it. Tell me in the comments below if you have any feedback or questions about this process and who am I? I am George, CEO and founder of Freedom and I have a bit of a red sunburn if you didn't notice. Where did that come from? Well, click that eye up there to see how I got that sunburn a few days ago and you've been watching and we will grow together as a family because this is the freedom family. You are part of it. We are all part of it and we're all growing together. So grow with us and partner with freedom. That play now button right there on the video bar is all you need to click to get started.


How can we support children who are struggling with separation anxiety from a parent or carer? That's the topic of today's episode of Pookey Ponders. Let's dive straight in. Okay, so this question was posed to me by one of my lovely patrons who said, how could they support an autistic child who was struggling a lot with separation anxiety from parents? And actually, the ideas that I'm going to share today could equally be applied to autistic or non-autistic children, and hopefully they'll get us thinking a little bit more widely about how we can support any child who's experienced kind of significant distress when it comes to that goodbye normally in a school or nursery setting, but any setting where they might be needing to let go of and be separated from a parent or carer. First of all though, I think it's really, really important that we acknowledge that sometimes children and young people experience anxiety, not about being separated from the parent or carer, but about going to the new environment. And when they're experiencing that, because in some way that environment doesn't feel or isn't safe for them, isn't meeting their needs, and this is relatively common amongst our neurodivergent population, then that's not traditional separation anxiety. This is about a child going, hey, I can't go there, that's not going to work, I'm going to feel horrible all day, this is really terrible. And this is something that I feel strongly about as a mum, both a parent and a carer, have two daughters, one biological one adopted, both of whom experienced really significant school anxiety, both of whom are autistic, and both of whom weren't having their needs met well at school. And I did all the traditional things about really encouraging that attendance and working hard on that goodbye. And I had got to the point where I could successfully get them to attend school and get them to say goodbye and leave them behind. And then they did not have a good time. And the challenge with this, and we just need to acknowledge this, I think it's really important, if we are going to continuously say goodbye to a child and leave them in a situation that doesn't feel safe to them or isn't safe for them, then we're going to erode that trust, erode that relationship. And the anxiety isn't ultimately going to go away, it's going to kind of come out in other ways. But for many children, separation anxiety is about finding it very, very difficult to say goodbye to a parent or a caregiver. And there are things that we can do to help. I just had to say that from the beginning because if the problem is not that they're finding it's hard to say goodbye, but they're finding it hard to go to the environment, we just need to get curious about what's going on for this child. Why don't they feel safe here? Are their needs actually being met? Does anything need to change in this environment with the faces, the places, the people, the activity in order to allow the child to thrive here? Is it the right place for them? Assuming that we're confident that this is the right place for them, their needs are being met, the fear, the anxiety, the worries are about the saying goodbye, rather than they're going to, then here are some things that we can do to help. Okay, so first of all, I think it's really important when we're thinking about separation anxiety that we don't think of it as just about the child and fixing the child. This is something where we need to think about all the different people involved. So yes, the child is right at the centre of it and we need to think about them first and primarily, and they are the key person who we want to put at the heart of this and to help, but we also need to think about how to support their parent or carer or supporting adult who's saying goodbye. It's really, really challenging having a child who's really distressed when it comes to that separation and this can provoke anxiety for both child and for parent or carer and so we need to think about how to support both and to make this feel okay. If we don't do that then we can end up just feeding into this cycle of anxiety because as a parent or carer, if what happens each morning, say it's a school drop-off thing, is that you have your anxiety peaked and you are taken to the ends of your emotional limits just trying to say goodbye to your child and then you go away and the last thing you saw was a distressed child. Your assumption is they're going to stay in that state all day unless someone tells you otherwise and that really doesn't sit well. You feel horrible, it's really distressing and you are likely to struggle with your own mental health, with your own emotional well-being, it really takes its toll and it's kind of expensive in terms of our emotional energy. What that means then is by the time you know fast forward to the end of the day, picking the child back up, having time at home, whatever, we're not necessarily in our best happy brilliant places to be positive and happy and to do this all again tomorrow. It wears us down over time and that makes it hard and that is not in any way trying to blame parents or carers or say it's their fault or say they're provoking their child's anxiety, it's just acknowledging that when your child is struggling as a parent or carer you love them and it means that you probably are struggling too.  It's really hard. I've been there, I've found myself flailing and failing and I teach this stuff for a living. It's super hard so we need to think about how to support both the child and the adult so if you're working in a school or an education setting for example don't just be thinking how can we get the child in here, be thinking alongside parent and carer how can we support both to have a positive experience of the school to reduce the separation anxiety so both are feeling happier. We also need to think about the adult who is doing the meeting, the greeting and the supporting within school as well because we need to try and find the person who can have a good relationship with that child and who feels confident building up this trust and who believes in this process that's really really important. If there is concern from an adult that they don't feel they can connect with the child or that they think that things need to change we need to look at some of those other sort of bigger issues. So involve all the people. This isn't just about the child, the child sits at the heart of it but we need to involve all the various different people. We need to work as a team, we need to be united, we need to be confident that we're doing the right thing. The next thing is that making sure that anything that we're doing here to try to break that cycle of separation anxiety is not going to jeopardize the relationship that the child has with any of those important significant adults whether they're the adults from home or the adults from the environment in which they're trying to enter each day. Those relationships really matter they're absolutely crucial to the child's well-being those attachments as they form ideally we want them to be secure we want them to be positive and those secure attachments are the things that are going to enable the child actually to separate to go off to go and spread their wings and so what we don't want to do is say hey you've got a really great trusting relationship with your parent or carer here we're going to try and break that and bring you away against your will and make you both unhappy that's not the way to go here. The way to go here is to recognize that that relationship has strengths and that leaning into that and really making sure that that relationship is positive is strong is secure is the very best base from which this child is going to have to go off and explore the world so we need to lean into the strengths of that relationship we need to enable and facilitate trust rather than do anything that might jeopardize it. Things that might jeopardize it would be things like tricking the child so doing something like getting their attention to look elsewhere for a moment and then parents suddenly slips off what message does this send to the child it just says I can't trust that adult and I can't trust that adult and tomorrow this is going to be harder you can play those tricks once twice not many times and you will be left with a child who feels really alone with nobody they can trust and the likelihood of getting to a point where they feel really confident and happy saying goodbye gets very low you might get to a point of learned helplessness where the child will be separated but they won't be happy and they are much less likely to engage with the day so this is where we just get to the point where we give up as a child we give up no longer need to be with that adult I don't trust them why would I want to be there with them anyway I don't really want to be here either but I can't do anything about it I'm just going to go with it but you end up with a kind of a shell of a kid to be quite honest and this actually was what happened particularly with one of my own children and this was the thing that made me think I'm not going to send them to school anymore and they weren't little children and teenagers at this point and it was that recognition that they had given up they no longer trusted us when we were we were dropping them off and so there was no longer the tears the difficulties the challenges with that drop off there was just this resignedness and they did go and they went through the motions and they were really really unhappy because the problem wasn't with separation anxiety the problem was with an environment that didn't meet their needs so in order to maintain that trust which is crucial we're going to be thinking about making sure that the people involved feel safe to the child that these are people that they have chosen or that we've observed that they have got positive relationships we're going to be doing things like making sure that this happens in a place that feels safe to the child we'll think about where the actual kind of goodbye might happen we'll be thinking about doing things gradually so the child isn't pushed beyond what feels possible to them where it always feels just about like something they might manage to do so this might be a real kind of gentle transition over time this is about gradually gradually gradually building that child up never losing that trust listening to them and giving them channels to communicate with us how this is going and what we can do to support and scaffold and working with them this should never feel like something that's being done to the child but rather something that we are doing it together because if we love this child we want to support them we want to care for them then we shouldn't be forcing them to do things if they're distressed already that feel really really impossible and yes sometimes this will mean working always to reduce that distress thinking with the child about what might motivate them to want to be able to go into for example school there might be great reasons why they want to be there but this is going to be about taking at a pace that feels comfortable where they feel safe and where that trust with those important adults is not eroded because that trust is key to making this work in the medium and the long term that takes us kind of nicely really onto the the next idea which is about using the child's motivations when it comes to that goodbye so actually thinking with the child or observing the child depending on their age and how much we kind of know about them and how much they're able to share why do they want to be able to say goodbye because many children where it's kind of genuine separation anxiety and the issue is about saying goodbye then often they do want to be able to go and enjoy all of the things that your setting might have to offer maybe they want to be with their friends maybe there's a particular activity they enjoy maybe there's a particular adult or place that they really really like here and they can be really really conflicted between wanting to stay and wanting to go but exploring with the child what are the things that they love who are the people that they feel safe with which are the places that feel good here can help us to think about what might most motivate the child to actually want to be here being told it's drop-off time and it has to happen now and that's it isn't really going to build that that trust that confidence that joy within a child about the idea of being in the new setting so we need to think really carefully about what might make them want to go there and use those things as our kind of our pools if you like and this can mean thinking about exactly how does the day start for that child what activity what place what person might make this feel like something that they're more excited more keen more comfortable with engaging with as they say goodbye to their parent or carer what will really motivate them to want to to come in to do this to do this brave thing and we must never let go of how hard it is for a child anyone who's experiencing anxiety to kind of feel the fear and do it anyway it's so so hard and our children when they're doing this they need our our respect it's a real privilege if we can build that trust with them and enable them to begin to to let go and to enter our settings it takes real bravery we need to respect that we need to understand quite how hard it is and create something good for them to come to the next idea is around routine and ritual if you listen in regularly or watch a lot of my videos you'll be like oh yeah she's on routine and ritual again it comes up all the time this is something I talk about a lot in terms of our neurodivergent kids it's really really helpful with anxiety routine and ritual just finding ways to do things in the same way again and again and again and again it's kind of what can help us cope sometimes the first time we do something it can feel impossible the second time slightly slightly easier by the time we're doing it for the tenth time we're getting there and it feels a bit more manageable by the hundredth time it's okay it might not feel easy but we know we can do it we've done it 99 times before we're pretty confident we can do it 99 times again so finding routines and rituals ways of doing things in the same places with the same faces in the same order doing the same activities just finding a bit of routine and rhythm to that can really really help and in particular over how that goodbye happens so we know how long it is so we know what's going to happen so but both child and both adults the one they're leaving and the one that they're going to knows what their role is here knows what the expectations are and kind of understands when and how that goodbye is going to happen and there are all sorts of things that we can build into this so we in our household love a special handshake and we can use a special handshake as part of our goodbye or indeed part of our hello but it's about finding what works for the child what works for the adults around them and just finding a ritualistic way of doing this that feels kind of positive fun connecting and so that the child knows there's a there is a goodbye moment but they know that they're going to be reconnected again later on I just can't emphasize enough the don't sneak off thing it it might feel like the right thing to do but actually getting to the point where the child can confidently say goodbye is going to be really really crucial you don't want to break down that trust are the little things that can make a difference here are things like transitional objects so allowing the child to have with them something that can remind them of or connect them with the parent or carer that they are finding it hard to sort of say goodbye to and let go of some children might wear piece of jewelry that is in common with their parent and carer they might carry something in their pocket that they can hold or touch that their parent or carer has held or touched or kissed as well and sometimes we do a laminated kiss for children where we kiss a piece of paper maybe write a message on it laminate it and they can carry it with them having a kind of a touch point something we can return to in the day if we need to feel that connection with that parent or carer can really really help just knowing that they're still out there and they still love us and parents and carers can be encouraged to remind their child that you know well i'm going to go off and i've got to go to work or i'm going to go home um i'm still going to be thinking of you i'll be thinking of you all day and loving you all day and i'll be here at the end of the day and i can't wait to see you again but i hope you're going to have really good fun and actually being able as a parent or carer to get to the point where we feel confident leaving our child we know that there are people that we're handing our precious cargo over to who love them to who have professional love for them in a safe environment that's going to meet their needs and where they're going to have a good day that makes it much much possible for us to be the adult that that child needs being able to say you're going to have a great day and to really really believe it that makes a huge difference but those transitional objects something so the child knows that parents still out there i can kind of connect with them i can imagine being with them i know that they're thinking of me because that relationship is able to really help the child to move away if they know that the parent is always there to return to that's absolutely key another seemingly small thing that can make a massive difference is just contacting back home and reassuring them that the child is actually having a good day um so i talked before about the emotional load if we leave and we feel that the child is distressed and we're not completely confident that they're going to be okay and how we leave the child is kind of in our minds how they will have remained all the day long and so actually if you as the person within the school or other environment are able to drop us a text an email a quick call just to say hey your child's doing great actually they're really really happy and they're playing with lego or they went into maths and they were quite happy with that they sat with their friend whatever it might be just reassuring us that actually things are okay so we don't have to hold all day in mind that idea that our child is distressed all day and this is really crucial because as i said before we need to think about all of the people involved the child's at the centre but everyone around them really matters too and if the parent necara believes the child is distressed they will spend their day feeling distressed and that will make it much harder for them to support facilitate scaffold um their child as they come home and then when it comes to the next day as well so relieving them of that burden letting them know actually child's doing okay um it's really important but don't lie again so in just the same way that we need to build up trust between the child and the parent and the caregiver and we don't want to destroy that trust by doing things like sneaking off when they're not looking actually it's really important that we don't look to break that trust between the different adults involved here we need to know that if you call us and say the child is okay that the child is okay otherwise if the child comes home and tells us a completely different story we're then not going to trust you we're going to feel less confident dropping our child off the next day and so on and so forth trust relationships communication is so important for all of this for all of us okay and then finally um we can think about just celebrating small successes and taking really really tiny steps so if we've got a child who is really significantly struggling with this don't think that you're going to fix it right away but rather have a plan to do this gradually over time to build up the confidence of the child and the adults around them um and ensure that they have a positive or at least a neutral experience each day what we're trying to do is help them to realize that this is safe I am safe this is going to be okay they also need to feel confident that the adult that they are going to be separated from will still be there will still return and they need to keep experiencing that and so sometimes at the beginning those separations might need to be short or not complete so it might be that at the beginning the parent or carer is still within eye or ear shot but that the child just practices being with those new other safe adults and safe places within the parent or carer's presence but without being directly attached to you and we might gradually increase that until they feel able to have the parent or carer out of the room and then gradually for short periods as they stay and it feels like it might be a lot of work but what we're looking to do here is to long term build children who are strong who are capable who are able and who are not going to have that relationship put to the test by trying to trick or cheat them into kind of short-term wins that might seem like wins for us they're attending but that actually may yeah jeopardize that relationship and make it harder than to really engage and thrive with our setting long term so taking it step by step by step if we can celebrating those small successes and throughout all of this making sure that we keep open those lines of communication with child with the various different adults and we keep working on this as a team it's not going to be a quick win but we're looking at growing little people here into brilliant humans and sometimes it's worth really persevering okay I hope that there were some ideas in here that have helped you and got you thinking and I do find this to be quite an emotive one as you'll have picked up and there are certainly other approaches out there and some people would advocate a much much harder harsher approach but I've been thinking a lot lately about gentler ways of approaching some of these anxiety issues and actually really respecting the voice of the child and beginning to really think about what is the genesis of this anxiety because sometimes we're not listening and I've got that wrong as a parent as a carer despite the fact that this is my whole world my whole life and I suppose that I always talk about gifting mistakes and that's a mistake of mine that I guess I would like to gift to you just to stop and just think is this actually about saying goodbye or is this about fear of what I'm going to and therefore what appropriate actions should we be taking I really hope that you found this helpful if you have please consider supporting my work you can do that by subscribing and sharing what I'm doing that's the number one thing I would ask you to do please I love it when my work can reach further and personal recommendation is the very best way for that to happen you can also support me by heading over to patreon and becoming part of my little community there who get their questions answered and access to things a little bit sooner or you can support my work by going the nuclear option and inviting me to speak with you at your next setting or event or via a webinar all things which I absolutely love to do and do get lovely feedback on so that's also an option to however you choose to support whether it's big or small thank you ever so much thank you too for the support that you are giving to the children and young people in your care and in your lives it really does matter and I'm grateful to each and every one of you for all that you're doing every day until next time over


In my son's case it will probably never be solved. It's always going to be an open investigation. Deb Nothi says she didn't know the depth of her son's addiction until his first time of inpatient treatment in 2011. He told me and it was like I was shocked. Just shocked. In just a short time he was out of treatment and went straight back to using drugs. 22 days is nothing.  Garrett cycled through rehab and criminal convictions for drug possession and paraphernalia until he had a fatal dose of heroin. When the people he was with called the police it was too late and Narcan, an overdose reversing agent, did not work. He was 28 years old. No family needs to go through this if it's at all possible, not to. Family and friends gather on the anniversary of his death to reflect using the day to ensure that others know that there are people out there who care about them. Grand Rapids is one of the most generous communities I've ever lived in but a lot of people don't know where to go to get help. They're leaving blessing bags in areas where people experiencing homelessness may be with a list of area resources, food and water and other necessities while sharing their story. Melissa Lane has experienced homelessness and said these bags would have meant a lot to her. I would have been surprised that somebody actually cared you know I would have been thankful. Doing something good for a good man's legacy. This is a cool person but I don't know say the good die young I guess I don't know. The family says the best way to end the opiate crisis is for parents to get educated and talk to their children. In Grand Rapids, I'm Mal Meyer, Lakeland News. If you've enjoyed this segment of Lakeland News, please consider making a tax-deductible contribution to Lakeland Public Television.


I get so angry sometimes.  Who's that? I haven't done the good work. It's down. Oh, not the scissors. Oh, I'm sorry, Sam. I'm sorry. Two patrolmen out here. FX Clancy, third precinct station. From Samuel's fate, license number 137-596, subject to one hour, Capers. Dare Clancy. In case you were aware of it, and I think you were, Clancy, the staff that owned in San Francisco was warm, quiet, and so. I was sitting in my underwear in front of the open window of my office, reading the news of the flippers back east. I was not employed by the Chamber of Congress to do this. My secretary had really taken my shoes downstairs to get a press. It began as the clock on the Church of Dismas to leave around the corner was hammering out the hour of four. I'll drop in tomorrow. I'll be glad to talk to you. Tomorrow may be too late. I'd rather not be seeing you. Orton, Harry. Well, I don't. I'm telling you what it meant that he seemed very worried about it. And to him, I should claim his belongings right away. So it was nothing terribly serious, but he'd get these little spells, and he'd have to stay on for more than a week or 10 days. But Mr. Suley, the informant at the Prince Shop, it'll get to things you know. I even thought he was getting there. He fell down. But at 45 seconds, trying to figure out what time it was, but a calendar watch my secretary gave me for Christmas, and a black eye. I took the case out of my pocket and opened it. I took one out and read it. With Clancy, I saw something green behind the cigarette. It looked better. It looked like money. When I examined it more closely, I wasn't sure the funding on it was Dutch, and the amount was $100,000 on Montgomery. The gold glattering on the plate glass window showed Van Pelt and Meisler. Amsterdam, New York, San Francisco, MacArthur, and Curacao. Mr. Meisler, that's only van Pelt. And I'm Hendrik van Pelt, I'm so sorry. Oh, don't feel like that. Maybe you can help me. What can I do with you? Well, somebody paid me off for a job in Dutch money. I want to know how much it's worth. Oh, it's better than Meisler. I know the value of money. Show me, please. Give me a cigarette, please. My brand, Schumacher Queen, thank you. No, the money, $100,000. High under the light low, OK? General number, here is M, order eight, clear is. Seals with color is. Paper, paper, excellent. Give it a chance. What's the word? La, look, latest quotations, flooring against the dollar. Aha, $53.34, that's what these games we take out. And you like $10 or not? I love them. You mean that money's real money? Who knows better than I should. My brother was engraved to the Royal Dutch Treasury. I myself in the manufacturing was, until the occupation coming was. Pardon me, would you mind saying that again, please? In the manufacturing from all kinds of money, including already currency from the Indies, East and West, Java, Tel Aviv, Borneo, and Holland, Netherlands. Also six months in very long Bali, where I'm English learning. Oh, you learned English? Several foreign languages. Well, I'll take it in 10. Go, shit, right, right. 30, 40, 5 cents and 20 cents. Did I say that? OK. I spoke a little about it. Yes, eventually you had lived in San Francisco for how long? Oh, eventually quite some time. I'm Hendrik van Thel. How are you? I'm good, though. I know this on the cigarette case. You have the same initials. H-P, who your name is, please? Paul House, Herman Paul House. You know, Paul House, I like that cigarette case. With the coincidence, you tell me you're Dutch money. Maybe also sell me the cigarette case with the Dutch cigarette. You like the cigarette? Oh, I love that tomato and cream. You're going to have fans for nothing. No, no, such a pity to remove them from the beautiful case. They go together, cigarettes and the case. How much? What would you say it's worth? Well, that goes $500. What do you pay? Nothing. I take it out of that body. Get out of the grave, Robert. Help, please! Stop the... OK, OK, Mr. Grandfather, go and get that. I was just crossing when I saw it the second time. First, if this was the case, it was shining about half hour. The night, I'm sure, with... Why, he invited me to have this drink with him. I guess I could one. He wants to buy Hank's cigarette case. And, of course, much as I say, an Indian brother and Mrs. Bates. Yes. Well, you know, Hank didn't need much. Only a little insurance. And the printing shop is a partnership. You know, Hank and Mrs. Sallay. The cigarette case is just a windfall. I told him you wouldn't stand in the way of a widow. I wouldn't think of it. I knew you'd do the decent things with Bates. I was going to ask him. Uh, please, please, Mrs. Bates. My booze is besides. I haven't got it. You haven't got it? Oh, no, look, look, Mrs. Bates. Blanche. OK, now Blanche, try and think. Did you ever hear your husband mention Van Peltz? No. No, they... No, they never even met Mr. Van Peltz said so. Well, Van Peltz find out about the cigarette case. He said you showed it to him. Oh, well, do you know your husband's partner, Mrs. Sallay? Oh, we're total strangers now. I told my husband everything. He forgave me. Uh-huh. And what of Mr. Sallay, don't you? Having much, nothing. He's led a very dull life almost as soon as he got out of reform school. He took a forgery rest. Oh, very dull. But that's where he became a master critter. And sir, if you say so. Oh, yes, I do. But I'm not much help. Oh, contraire, Blanche. Oh, contraire. I was sitting in a desk in his shirt place, checking off figures and a ledger.  He shook hands, and then he waved me to a chair across the desk. This is awful, Sade. What was one thing and another where he was over head and work, and got pulled with these books? And I don't know a thing about it, Mike. Hey. Well, we're handling it at this stand, but we'll be very busy for a while. What point are you dropping by tonight? I guess, like, the news of the boss's death would make some difference to those customers, but no. He'd take that full of the desire and so on. What makes you think that car deliberately ran down the boys? Did I say so?  He got me tagged. Anybody have anything against him? Why, as you know? Well, he fired two printers last night. Why? Well, they couldn't spell in English. You, uh, sleep at the pace, asap, or not? Yeah, he came in for about 10 minutes, Sade. He'd be back on the job tomorrow morning.  How do you work? Oh, good Lord knows. Boss wanted his resume there, so he took it with him. This man felt no about Page's heart.  You didn't know Page was killed with that soft car. Uh, that's a long shot, sir. Thanks. There's another one. You're lying straight down the line. Huh? You didn't see Page today.  He didn't take that Dutch money for a souvenir. If he had you to mention the cigarette case. You said enough. Cutting down to the knife? You'll find out. Put your hands on top of the set. What I told him, sir. The press room's door was directly behind him, and I knew his body would screen my guns in the view of the response of the signal he sent. It didn't have long to wait. I could sound just one up before the other three were on me. I knew it, and they knew it. Back here, would you do that? Sure. I threw myself sideways, but I wasn't quick enough.  I flung it high, and let go. Brass cut to it into a plate glass window into a rush hour crowd. He broadly bought his way out of the country with the same kind of money he insulated printing here. Genuine Dutch blanche printed from the original place. Metalworker. He designed a gold cigarette case in which those plates were fit with uncanny anchors. He filled the case with an odious brand of Dutch cigarette, which only fools or criminals could possibly smoke. It was the safest hiding place in the world. So clever was it. Now get this fancy boy into the base metal until it stopped that slug and fell through with me in the alley.  Thank you, Wenzel. All right, just in case you'll promise to me. Yes, that'd be. But what hurts even more than these wounds is the thought that you doubted my words. Oh, I didn't say that, Sam. I only inferred that you had no sense of time. Yeah? Well, I guess you've changed your mind about that, eh? No, I haven't, Sam. May I ask why? Well, I'd rather not discuss it during working hours, Sam, but as soon as I type this report, I'll tell you exactly what I mean. I said... Get the stamp, that's what else? Oh, harness it. It won't be meeting me anymore. I'll... It's down. It's just nearly 5.30, Sam. Not told the photographer not to come. That's a fine thing. After all I've been through. You still say I have no sense of time when you told the photographer not to come. Why?  Thanks a lot. Oh, Sam. That's why I must have been so cranky. I must have had a premonition. I didn't listen to that with me. I'm so well-tempered. I think I'm training into an old maid or something. Oh, calm down. I must be. I must be. Oh, by the way, honey, eh, remind me to write the, uh, 5G at Washington, and have them to send me that cigarette case for the souvenir if they take it off van Pelt.  Your life? Okay, I've been nagging and saying all these terrible...


Seed is a living material. Crop seed is a living material. To keep this living material in good condition up to the next rice season, there are a series of activities called seed preservation. More than 95% of rice seed in Bangladesh is retained by farmers for sowing the following year. Storing seed is the responsibility of women here. That's why women members of the family must know the reasons why seed quality deteriorates during storage and how this can be overcome. The quality of seed during storage, many depends on moisture content. If the seed is not properly dried before storage, it becomes soft, hot and moist and this will affect the level of disease and insect infestation. This in turn will cause seed quality to deteriorate. Using poor quality seed will give poor yields in the following crop season. Based on hard real-life experience, the women of Maria Village in the Majira-Apazilla of Boghra district have devised some very effective techniques for seed preservation. Let us learn from some of these innovative women. We select good panicles in the field and harvest them separately. We then thresh them and then dry them on the table. We then store the seed in a container placed on a table. We test the dryness of the seeds by biting them with our teeth. We wash, clean and dry the seed container before using it for storage. So the container is free from insects and their eggs. The women in our village use plastic pots, tin containers and drums as seed containers. In addition, painted earthen pots are widely used for seed storage. Keeping seed in good quality helps us in reducing the seed rate needed for the next crop. Some women in Maria Village paint their earthen pots with varnish or tar on both the inside and outside to prevent the passage of air through the container. Others use used cooking oil instead of paint. They also know that air can pass not only through the lid but also through the capillary holes in an earthen pot and destroy the seed quality. If there's still some empty space in the seed container, some women of Maria Village use an amazing technique from the past. They put a burning candle inside the container and then they cover the container with the lid. They know the candle will burn until there's no oxygen in the air left. So in this way any insects inside the container will die. After proper drying and resting, the seeds are placed in a container which is filled right up to the top. If there's not enough rice, some puffed rice or dried sand can be used to fill up the airspace. It's also common in our village before pouring seeds in the container, we put dry Bishkatali leaves at the bottom and at the top of the seed. As well as Bishkatali leaves we use dry neem leaves, tobacco leaves or naphthalene inside the container and then tightly close the lid. These help a lot in controlling insects. If you keep the seed container on the ground, very quickly the seed absorbs moisture from the soil and it becomes mouldy and discoloured. That's why we keep seed containers somewhere off the ground. It prevents mouldiness and minimises insects and the seed also looks bright and clean. So what have we learnt from the innovative women from Muria Village? We've learnt that we need to select healthy panicles. Only clean and sorted seeds should be stored. Before storing seed we need to clean, wash and dry the container. We should put seed right up to the top of the container. If there's insufficient seed then properly dried sand or puffed rice should be used to fill up the airspace. The container lid must be completely sealed to make it fully airtight. Additives like Bishkatali, dried neem leaf, tobacco leaf or naphthalene will help in controlling storage insects. If the container is an earthen pot, tar, paint or use cooking oil can be used to paint both the outside and inside of the pot to make it fully airtight. We should always keep the seed container above the ground. We must remember, properly preserved seed is good seed and good seed means a good crop.