Ok, so, last time, we learned something crucial for the rest of the course, namely, gauss curvature, principal curvature, curvature, principal directions, and principal curvature, of course, I mean curvature, ok. So, the point was improving what kind of differentiable functions if they are differentiable and where they are differentiable, we learned also the way to compute them in terms of a local chart. So, if around the point you have a parameterization of your regular surface, now you have very simple formulae, well, some are very simple and some are less simple, but I mean, in principle, looking at your notes, you are able to compute k, h in terms of the components of the first and the second fundamental form, ok. So, if you have a local chart, if you have x, so that's kind of the strategy you have to keep in mind. You have a local parameterization of your regular surface, so around the point covered by this chart, you can compute capital E, capital F, and capital G immediately, and now I'll show you in two examples.  So that's the strategy, if you need to know the curvatures, all these curvatures for a surface, the only point is to be able to parameterize locally around some point with a given chart. Let's see this procedure in practice, how it works, and you will see it's very simple, ok. So, for example, so example one, let's take one of our standards, I mean, we did some examples already, the sphere, the plane, and so on, but let's, for example, suppose we take the elliptic paraboloid, one elliptic paraboloid. So, for example, the set of x, y, z in R3, such that it satisfies the quadratic equation, it came up already, we used already this example somewhere, ok. So, this is called the elliptic paraboloid. Well, maybe it's a good idea to try to imagine first how it looks like, usually the way to do it with, so actually, now I remember where we used, we proved that this was a regular surface by checking that zero is a regular value of the function x squared plus y squared minus 2z, because this is the inverse image of zero, ok. So, we know it's a regular surface, in fact it's also a graph, if you want you can put this two down here, and this shows immediately it's a graph of something, ok. So, that's also another way, and now we will use it to parameterize it, what I was saying. How does it look like? Well, the standard way, I mean the simplest way for this kind of locus to understand is to cut with planes, ok, and see what kind of curve you get.  So, it depends, if you are able to detect which curve is it when the plane is moving, you understand how the surface looks like, ok. Well, of course, in this case it's simple, because if I cut with planes, so I have R3, and if I pick the planes, the horizontal planes, z equal to a constant. Well, of course, for example, if z is equal to zero, the point, the origin, ok. So, the intersection with the coordinate plane z equal to zero is just the origin. Then if z moves down, becomes negative, the intersection with the plane becomes empty. So, there is nothing below the horizontal plane at height zero. What is above? Well, if z becomes one, it becomes what? It becomes a circle. If z becomes two, it's another circle. And the radius of this, I mean, the center of the circles lie on the z-axis. And the radii, of course, are proportional, I mean, besides these two are proportional to the point z. Ok, so it's a growing family of circles. Ok, so this tells us how it looks when I cut with horizontal planes. If I cut with another plane, for example, x equal to a constant, or y equal to a constant. Of course, this is symmetric in x and y.  And at the end, of course, you know, you realize that the surface looks like this. Ok? Like this, if you want a better picture. Ok? This is called the elliptic paraboloid. Ok? Now, well, this is one elliptic paraboloid. So let's see our machine at work. Well, are we able to find around the point? So I fix now a point, and I would like to know, Gauss curvature, mean curvature, and principal curvatures. Well, I need to parameterize locally. I should find a local chart parameterizing a neighbor of this point. But actually, in this case, it's simple. Because this is globally. So there is one chart, which covers everything. Which is just the graph. The graph chart, in some sense, ok? X of uv, where u and v are free to move over the whole r2. So our domain u is equal to r2.  Ok? And this works for every point. Very well. So, let's do everything. Let's compute first the big letters. The big letters are what? Are the coefficients of the first fundamental form? Now, remember the coefficients of the first form, the entries, in some sense, they are usually called the coefficients, but it's kind of misleading in this moment, why? So the entries of the first fundamental form. What is xu? Is xu scalar product xu. And so on. So now you realize immediately, if you imagine what kind of computations that you have to do, that it's better first to compute xu, xv, xu u, xu v, xv v. Before even thinking.  Xu is 1, 0, u. V is 0, 1, v. Ok, then you write.  So, xu u, xu u is 0, 0, 1. Xu v is 0, 0, 0. Xv v is 0, 0, 1. And then what? We know. I already imagine what kind of computations I have to do, I need to compute a unitary normal vector field. So I pick the standard one. Remember there are two. Ok, so this is a moment where there is no one answer. I pick n to be xu, wedge, xv divided by its norm. Ok, so let's compute it. These are the vectors 1, 0, u and 0, 1, v. So what kind of vector is this? Ok, so this is the wedge product and then I have to divide by its norm. So this is minus u, minus v, 1 divided by its norm, which is square root of 1 plus u squared plus v squared. Ok, now we can switch on the brain. We have on the blackboard every ingredient. How do we mix them? Well, capital E is xu, scalar product xu by definition. So it's 1 plus u squared. F is xu, scalar xv. So it's uv. And g is 1 plus v squared. Ok, now how much is little e? Little e is the entry of the second fundamental form. Ok, so in this case let me rewrite what it was, the definition. The definition was it's minus dn xu, scalar product xu. Ok, but now instead of this remember we proved a little, we have a little observation that this is equal to n xu, and now you see why this formula looks a little bit better not to have to take derivatives of this. Ok, this is just a technical thing, of course they are the same, but here it's less likely to make mistakes. Ok, so how much is it? Well, I have to take n, scalar product xu, so it becomes 1 over the square root. Ok, so what is little f? Little f I use the same trick and I find out that this is nxuv. Ok, so n is that one, xuv is here 0. And what is little g? Little g is n scalar xvv. So it's the same as before, it's 1 over the square root. Ok, so now you have everything and you pick the formula and you find what? That k, now if you want just to be careful, k at which point? Well, k at the point x of uv, so if I fix uv on the domain I'm looking at the point x of uv on the surface and I'm computing the Gauss curvature at this point. Ok, so k of xuv is given by this formula. Ok, so it's eg, so that becomes 1 over this without the square root. Ok. Minus f squared, but this is 0. So this becomes the numerator actually now becomes a denominator, but it's divided times 1 over eg minus f squared. Ok. Eg minus f squared is this one. It's 1, so let's compute here eg minus f squared which is something which appears everywhere. Ok, so let's compute it on a side.  And this is what?  So that's it. So that's again the same thing. So I write it just for once, but then of course this becomes 1 over 1 plus u squared plus v squared. Everything squared. Ok. And that's for the Gauss curvature. Well, now you pick h, ok, let's not waste 10 minutes. You have everything here. You substitute and you get something. I can tell you, I mean h, h at the point of x, u, v at the end of the substitution will be just one half.  Ok. Is there anything we can draw out of this computation? Any conclusion we can draw out of this computation? Well, for example, these functions are known constants, but, for example, k, and in fact even h, both are strictly positive, for example, if you want just to observe something. So you can say in the language we are building that the elliptic paraboloid is all made of elliptic points. Fortunately, otherwise the name was kind of very unlucky. Ok. If the elliptic paraboloid had hyperbolic points it would have been a language crisis. Ok. Every point is elliptic, for example. Just one conclusion. Ok. Let's make another example. Much more difficult is to draw any conclusion out of the sign of the mean curvature. In fact, the sign of the mean curvature is not a geometric thing, because we have already observed that if I had picked the opposite n, this would have changed sign. So the fact that this is positive is just an accident of my choice. Ok. If you had picked minus n, you would have gotten minus h, so it would have been negative. So maybe the only thing which is geometrically interesting is that this is never zero. Because that's the only thing which is invariant by the freedom of choosing n. Ok. Let's build another important example. Well, I mean, this is not particularly important, it's just one example. But now the next one is a famous surface and sooner or later exposed to that. In any way, the point of this computer is to show you how easy they are. So now do ten exercises and then everything will become quick and automatic. Ok. Example two. Well, this is a famous surface which is called, it has a name, it's called the helicoid. What kind of surface is it? Well, the idea, actually you have seen it many times. Some of you might even live with one of these in their house. So this is R3. The idea is to take at a given point here a line orthogonal to the z-axis. Ok. I take an orthogonal vector for example of norm one, just to fix. I mean, it's irrelevant, which norm it is. It's a line in this direction. So in particular it has to be a surface covered by lines. And then I take the object. So then as I move this point I would like this vector to rotate at uniform speed. Ok. So if I take as a parameter the height of the point I would like at the same time, so if I move this height vector to change at the uniform speed always in the orthogonal direction to the z-axis. Ok. How can I get an explicit formula for an explicit parameterization for something like this? Well, that's not too difficult. Look at this. So you take X of UV let me remember now. So the height so it will have third one is kind of the parameter of z. Ok. So it's what I would like to call the speed of this special point. Ok. So let me call it AU for some given number A non-zero. Ok. Pick a non-zero number and U is exactly the movement of this point there. As here cos U sin U. So what is this point? Which point is it? So the point given by, let's say suppose A is equal to 1. I take the point at height U and then I take the point the vector cos U sin U. Ok. Which is exactly a vector orthogonal to the z-axis of norm 1. Why do I like this expression? Because when U moves this is actually describing a rotation at uniform speed in the circle. Ok. So this becomes the angular momentum of this vector. Ok. But now this would be a curve. In fact this is the helix, one helix. Ok. Now basically for every point of this helix I want to take the whole line in the second parameter, which is V. I mean this is not a surface of course it depends only on U. It's a curve. But I want to which line do I want to take? The line passing through this point with this velocity. Let's see. Ok. So for V equal to 1 is exactly this point here. For V equal to 0 is exactly this point here. And then it's linear. I'm picking all points on this line. So why do I say that you should be very familiar with this surface? So now if you want to draw the surface this becomes a bit more complicated. So I've printed well, this is a helicoid. No.  I mean here there is no problem in going down. Ok. So this keeps on rotating or if you want a nicer picture but it's less clear it's something like this. Ok. So this is, you should be familiar with this I mean certainly you have seen it many times because for example stairs, I mean circular sphere, stairs are built out of this. You discretize this and you make stairs going up. Ok. So now, so I don't draw it keep those pictures in mind. Well, the first problem is is it a regular surface or not? Because now I've given it as the image of this map but now it's not clear in principle that this map satisfies all the properties required by the definition of a regular surface. Ok. So you do it. Ok. So it is a regular surface. It is automatically, so actually where U and V lie. Well V is the parameter on the line. So V lies in R. The whole R. U lies in R too. Ok. Because you want Z to be free to go up and down over the whole Z axis. Now. So, you can check in 5 minutes that this is a regular surface. It has the nice feature of being covered by one chart. Ok. By construction. So, let's see the machine. What is the machine telling us now? Again, now you switch off the brain.   X U V X U V, for example. Let's take this one.  Which is ok, because it's linear in V. If I take two derivatives in V. Ok. Then what else? N, normal vector. N, I pick again the standard choice. X U cross X V divided by its norm. So instead of writing N let me write immediately minus, let me make the wedge product.  So how much is this wedge product? Ok, so this becomes 0 minus A sin U.  And on the other line minus V sin squared minus V cos squared so minus V. So then I can write N. The normal with my choice is exactly this one. I don't repeat it, but it's divided by its norm. And the norm is what? D squared plus D squared it's A squared. A squared plus V squared Ok? And now let's start. Ok, now we are back in business E. E is the scalar product of this with itself. So it's V squared sin squared plus V squared cos squared so it's V squared plus A squared. V squared plus A squared. F, F is the scalar product of these two. So it's minus V sin cos plus V sin cos 0 plus 0. F is 0. G is what? Is this times itself, so it's 1. Little letters N scalar X U U. So besides the square root we will remember about this at the end sin U so it's plus A V sin cos Ok? Minus so plus, sorry, plus A V sin cos minus A V sin cos. So they disappear plus 0. So E is 0. Very good. Little f is this times this.  So it's A, but now I don't have to forget the denominator. So it's A divided by the square root of A squared plus V squared.  Little g is, well, it's 0 times something. Very nice. So Modulo mistakes let's see. K at the point X of U V is how much?  OK? So this becomes just minus. Choose the way you like it most. OK? For example. H I can tell you, you substitute all these numbers. In this case it's easy because there are plenty of 0s. Of course when you get the 0 you are very happy because formula becomes much easier. Then in this case XU little miracle is constant equal to 0. OK? We will talk about these kind of surfaces at some point. This is a first well, it's not the first example because the plane is another surface like this. But it's the first meaningful example of what is called the minimal surface. In this case well, the only thing, the only conclusion again we can draw is what? Now, this surface all the points of the surface have negative gauss curvature. So the helicoid is completely covered by hyperbolic points. OK? Now, the point of this lecture is to be able to say this kind of conclusion just looking at the picture. OK? So you could have told you that the helicoid was made of hyperbolic points just by looking at the picture and also for the elliptic paraboloid we did before that there were elliptic points. There is a simple way just by so what is the difference between these two surfaces? Well, they have nothing in common of course. But now this is this one. On this surface imagine that you pick a point and you draw the tangent plane at this point. Where is the surface? On one side. OK? It's all on one side. So there is only one point where the two things touch which is of course the point where you are taking the tangent plane and the whole surface lies on one side. OK? Well, the key properties of course it's locally but I mean it is true. Now what if you do the same game here? Pick a point and draw the tangent plane at this point. Where does the surface lie? On both sides it crosses. OK, that's exactly the geometric property that characterizes elliptic points well, almost characterizes we will see. There is one delicacy that we have to be careful about but I mean this is the key property which tells us just by looking at the picture which are possible usually elliptic or hyperbolic points. OK? How do we do that? Step further in our kind of analytic machinery. Given a function on a surface we learn how to compute the differential of this function and we spoke about critical points and regular values. This was all about first derivatives of a function at a given point. What about second derivatives? After all the function on R2 and you ask me to look for minimum and maximum I need second derivatives to distinguish between a minimum and a maximum point. I would like to do the same thing on a surface. Well that's now S is a regular surface now we pick a function with real values OK? A differentiable function I don't even say because otherwise it would be impossible to do derivatives. Now I pick a critical point on S. So suppose that there is a critical point and I call it P so P critical point for F. OK? Now I would like to define the derivative of the function but of course the surface is two dimensional so there is not anything like one second derivative even if the surface was the plane you don't have one second derivative you have a second derivative in one direction second derivative in another direction mixed second derivative in fact you have a second derivative in some sense in any direction exactly that's exactly what I want to define this symbol so this squared at the point P it has to take one direction and give me a number so the second derivative in that direction OK? So it should be a map from the tangent space to R doing what? Well it has to take a tangent vector hence I have at least one alpha such that alpha of T alpha of zero is equal to P and alpha prime of zero is equal to V but then it's obvious what I I take F compose alpha now it's a function of one variable T and I take the second derivative now this object whatever it is at the moment we don't even know it's well defined so in any case it's called definition of F at P so now immediately we have to prove that this is a good definition and actually the philosophy is always the same so what is the problem is that in this formula I have alpha exactly as for the differential now it's slightly more complicated but it's exactly the same problem and we are going to solve it exactly in the same way so we produce a formula for this object quite complicated but we don't care in which alpha is not there ok how do we do it so remember that was exactly the philosophy for the differential when we proved we picked a local chart and we said the differential of a map is the linear map which is associated with respect to the standard basis to the Jacobian so we play the same game it's just that we have to take two derivatives instead of one so proposition and in fact let me list few remember that we have this condition here we are not taking any point this is a definition given at a critical point so but let me repeat let me stress it just for if p is critical some sense proposition zero is that the action d squared f at p is well defined and then we have also some properties it is also a quadratic form on tps this is like an ambiguous statement but it's clear what it means second if p because after all it was a critical point so now let's distinguish if p is a local maximum and in brackets the other case respectively minimum then the action is in the maximum case is semi definite negative in the maximum case and positive in the minimum case and then exactly as we did in calculus for functions of more than one variable is the converse so true not exactly so three you have to put a stronger assumption to so if d2 squared is strictly negative is negative definite negative and of course in the other case positive definite then the converse holds then p is a local maximum or in the other case minimum this is exactly the same statement and in fact you will see it's exactly the same proof as for functions of R2 so how do we prove it basically we prove it at the same time everything we pick a chart so take a chart around p and give it a name for example to the point there will be one point on the domain u which is sent to p via x suppose that this q for example q will be a pair of numbers corresponds to some pair of numbers ab now we have taken a curve alpha on the surface s as usual by picking its domain sufficiently small we can assume that the image of alpha lies in the image of x ok so let me just say all these things by saying I pick alphas and epsilon such that the image of alpha is contained in the image of x ok and then exactly as we did for the differential define beta to be the curve on u by pulling back alpha ok but then beta is a curve on a domain of r2 so beta will be given by a pair of functions u of t, v of t ok and the fantastic idea we used already for the differential was to write f composed alpha which is the one we have to differentiate twice of t as f composed x inverse composed alpha ok that's exactly what we did last time, few times ago ok so I want to see it as a composition of these two maps and now I start taking derivatives I mean let me just under this notation of course this becomes f composed alpha f composed x composed beta but beta I've given names ok so this is f composed x at the point u of t v of t ok I've given names to this to the components of this so I use that ok so in fact we should pick exactly the first part after all we have to compute the second derivative to compute the second derivative and then to take another one so pick your notes if you so how much is f composed alpha d in dt but now I don't have to evaluate it at t equal to 0 because I'm going to take the second derivative how much is this well once I write it in that form of course that becomes u prime since I'm not evaluating at 0 so this will be functions of t now ok but I don't write it otherwise I need 3 blackboards u prime times f composed x differentiated with respect to u plus v prime at the corresponding point by chain rule plus v prime f composed x differentiated with respect to v at every point now I take the second derivative and now if I want I can already evaluate at t equal to 0 this is what the definition of the action is asking so I need to take another derivative of this with respect to t let's do it this is a function of t so this becomes u double prime now if you want you can put 0 just to I'm hoping to stay in one line so it's u double prime f composed x with respect to u plus u prime times times the derivative with respect to t of this but then I have to apply again the chain rule because this will be a function of u of t v of t so this is u prime times u prime of f composed x u u plus v prime f composed x in fact there was no hope to stay in one line u v and this that's it for this piece then I have to play the same game here so plus plus v double prime f composed x v plus v prime times what in this same game u prime f composed x u v plus v prime f composed x v v there is something I can group not much but you see well actually in fact here which is exactly this one but that's it, that's the only thing that I can group you see it's u prime v prime times f composed x u v so this will become if you want drop one and put here two but it's not a great improvement now if you understand the problem you would say that the proof is over what is the problem we wanted to find a formula for the action where alpha did not appear where is alpha here well everywhere but the point is what of alpha is here now remember everything here now is evaluated as zero ok when you evaluate this as zero u prime at zero but that's ok because this does not really depend on alpha it's contained here in this bit of information so when I see first derivative of the functions u and v I'm ok these objects here do they depend on alpha no these are kind of intrinsic of the chart and the function ok so the partial derivatives of f composed x are ok in my spirit so where is the problem the problem so this this observation tells me look this piece and this piece are ok this piece would be ok and this piece would be ok but here there is u prime u double prime and here there is v double prime now two different alpha passing through the same point and with the same velocity do they have to do they need to have the same second derivatives no the velocity at the given point cannot determine its acceleration at that point the second derivative is independent of the first at the given point ok so this is the problem I can easily construct two different alpha with the same p and the same v and completely different these so why the theorem is true well notice that up to now we haven't used anything that's exactly the moment where this comes in critical and the two given curves of the same thing so the only way the theorem is true is that it's true because this is 0 ok so I need to put but that's exactly I know that the critical point f composed x at u is equal f composed x v is equal to 0 at p at the point corresponding to p at everything here is evaluated I've given names so at the point a b because I'm translating everything back to the domain of R2 ok so if p is critical this first derivative is vanish so then these subjects are not there and then that's it now so now what is the output of this output it's good to so we found also an interesting formula so the action at the point p at v if p is a critical point we have a local expression in terms of a chart which is just u prime squared it's also quite easy to remember u prime squared f composed x u u at the point a b it's interesting about where I'm computing what ok and this is this plus 2 u prime 0 v prime 0 f composed x at a b but this is differentiated with respect to u v plus sorry u prime squared u prime squared times this plus twice this plus v prime 0 squared v v at the point a b so you see it's easy to remember no? because basically it's a quadrat now in fact this formula proves essentially everything why? is it a quadratic form well yes it is a quadratic form ok in the entry v well it proves it's well defined for everything we said up to now it is a quadratic form if I write this quadratic form in terms of the standard basis I've already given you a matrix representation of this it's the matrix given by f composed x u u f so on the first line f composed x u v then on the second line f composed x u v again symmetric v v but now this is the matrix representation with respect to the standard basis and I'm writing in terms of the second derivative so the standard dash this is the old dash of the function f composed x do you agree? so f was on a surface but f composed x is on the domain of R2 so you can apply all the theorems you know the quadratic forms are the same basically I'm telling you the quadratic form associated for the function on the surface is the same thing as the old dash for the function f composed x on the domain of R2 but then you know all these theorems and these theorems are exactly the old ones of course you need to translate p to q but then of course maximum minimum who cares something is a maximum on s if and only if q is the maximum of f composed x and so on so everything is done in one shot so this formula tells you immediately everything now is this making us closer to understand the geometric interpretation of all the of all the curvatures that we introduced well it's not clear yet it will be clear after we make a couple of examples of computations of the action for a couple of interesting functions height function remember this was one of the examples listed also for the differential what are we doing we have our surface we pick a point p0 and we pick one plane actually for making my picture a bit ok and we take a plane passing through p0 ok sorry a plane this is capital p ok a plane now what was the height function as a function on s with respect to this plane well the idea was simple just take a unit a unit normal to this plane call it a ok so p the plane capital p is nothing but the orthogonal complement to a vector and now I assume that I already picked a vector of norm 1 ok there is no so the definition of the height function was h h we call it h of p was nothing but p minus p0 scalar product a ok now what was which were critical points for these functions we gave a geometric characterization of those so p is critical if essentially the normal remember we wrote it in lang in english ok saying if the normal line through the point p passes through p0 but that's the same thing as saying that if n of p since now I've picked the unitary normal to the plane n of p has to be a or minus a suppose it's a we don't care ok if it was minus a change a with minus a ok minus a is again a unit normal to the plane p so up to change of sign this is ok ok do you remember this? now compute the action so in fact geometrically in my picture it's a guessing name it's true it's somewhere here ok so the normal at this point should be exactly a ok so this would be my point p now this is the new the new ingredient so how much is this well it's the second derivative at t equal to 0 of h composed alpha I have an explicit expression for h so alpha of t minus p0 scalar product with a ok so how much is the first derivative constant vector p is a fixed point the only thing which is moving is here so the first derivative would be alpha prime at t scalar a so the second derivative is alpha double prime now I evaluated 0 scalar a ok but a is the normal vector because p is a critical I mean this would be true at every point but now I know that a is n of p so this is alpha double prime scalar product n of p why this reminds me of something this is exactly what we found in Euler's theorem ok this is exactly what this is by Euler's theorem the second fundamental form at the point p evaluated twice in the direction v ok so the action at the critical point of the height function gives you a quadratic form exactly the second fundamental form now in particular there is always so this is true for any plane p0 and so on but in particular there is always one special plane to look at if I now I reverse the process I was fixing here p0 and p and I was looking for a special point on p now suppose I take a point here and I want to construct all this picture based on this point basically what I'm saying this is my p now and suppose I take p0 equal to p and I take as a plane in R3 the tangent plane to this point of course I can construct a height function on this why this is interesting well because of course p is automatically p is always a critical point for this height functional so I have automatically a nice critical point for the height function with p0 and a equal n of p if you want you can rely on by construction p is critical if in fact this was an if and only if if I take the tangent plane of course the unitary normal will be the normal by definition of the normal vector ok so in this case this is obvious and so it p is a critical point ok today I see you a bit maybe I'm not particularly clear today but I see you a bit confused is it hard I mean is there some point you want to go through again ok so if I reverse somehow the process I first fix the point on the surface and I construct the plane that the point the original point is a critical point for this height function but this property was true for every height function ok so in particular it is true for this one ok but then this implies without proof that's what I hope you agree the following theorem one if the Gauss curvature at some point is positive meaning IE the same thing IEP is elliptic sorry IE is a Latin form to say the same ok you know Latin more than your Italian colleagues very good so if you take an elliptic point then there exist it's better to write it in English there exists a neighbor a neighbor of P in S which lies on the same side of D and let me say now in quotation marks affine tangent plane to S at P P is the only point of intersection I stop here but I mean of intersection between these affine tangent plane and the surface itself so just one comment before commenting why this is true what is the affine tangent plane remember when we define the tangent plane I mean the tangent plane to a surface at any point so really if we want to make some precise pictures all these pictures are wrong ok so the vector space it's really this one parallel transport to the origin I mean a vector space something a two plane which does not pass through the origin cannot be a vector space ok on the other hand it's very convenient at P ok that's exactly the affine tangent plane so it's the translation of the tangent plane to the point P ok ok so let me write also part two and then we make comments part two the opposite if K of P is negative then P is hyperbolic then in every neighbor of P in S there are points which lie on both sides of the affine tangent plane well why this is true and in fact it's true now without a proof you have already the proof because pick this point construct this special height function the action of this height function is the second fundamental form so what does it mean that the Gauss curvature is positive is telling you exactly that the second fundamental form is strictly definite now here there is the obvious indeterminacy between geometry and analysis because depending on the choice of N whether it is positive or negative well no no it doesn't matter the point is you fall exactly in the cases of the theorem we stated before about the action of a function where you know so it was the last one so the action of this function is strictly negative or positive depending on N and that means the height function has a maximum or a minimum depending on orientation but that's exactly the analytic way to say that the surface lies of course how much is the height function at the point P if P is equal to P naught is zero so at this point is zero and suppose it has a maximum that means it's always negative it's negative in a neighbor I mean of course you cannot draw global conclusions this is a local analysis but there is a neighbor where the function has to be negative but being the height function negative means exactly that if you draw the graph it lies below but that's exactly what's written here it lies on one side and it's impossible that there exist of course if I go sufficiently away at this point I don't know but certainly there is a small neighbor where there is no other point where the height function is zero because it's a strict maximum ok you see why there is nothing to prove and the same thing here and now exactly as in the case exactly as in the case of functions of two variables so remember there is this kind of annoying between the two theorems about the Hessian of a function if it's strictly positive then you have a local minimum if it's strictly negative you have a local maximum if you have a maximum or a minimum it's not true that they are strictly something they are semi-definite so in all cases where there is semi-definite you cannot draw analytic conclusions and in fact you cannot draw now I'm going to show you examples that in fact you cannot even draw geometric conclusions so the point is is there a line 3 in this theorem starting if k of p is equal to zero then the surface does something the answer is no you see k of p is equal to zero is exactly for all this for all we said today the case where the determinant of fundamental form is zero so it's semi-definite examples to convince you there is no no such theorem with k with zero Gauss curvature so all the following examples are interesting because they are examples but the point of this observation is there is no such theorem for k equal to p k of p is equal to zero ok how do I convince you that there is no such theorem explicit examples well, remember that there are two types of points with zero Gauss curvature because essentially the second fundamental form could be identically zero in every direction or it could be one eigenvalue equal to zero and the other non-zero ok, so planar and parabolic so planar so I want to show you that in fact you cannot even if you distinguish these two cases there is no theorem well planar of course there is example one which is the plane so give me a surface with the planar point well the plane any point what happens to this theorem well in some sense it's through the ok, this is kind of delicate I mean if I take the plane the height function is of course identically zero the height function based at the point of the plane is identically zero ok but there is also an example like this take x of u v equal to u v in fact it's a graph u v and then you pick as a function u cube minus 3 v squared u I should have practiced before coming to this lecture to draw you a picture and so I didn't so now let me try it's something like this ok not too bad and so on, so here you have curves going now this is a famous name it's called the monkey saddle what is the point that we are really interested in x of p is equal x of zero zero which actually goes to zero zero zero it's exactly this central point of the saddle ok let's do the exercise there is no time to move on to other things let's make these examples complete let's compute the Gauss curvature of this surface well first I should prove I should argue that this is a surface it's ok, it's a graph of a differentiable function it's a polynomial so there is no problem it's covered by one chart so whatever I'm going to say will hold at every point at the corresponding point I hit every point of the surface so x u 3 u squared minus 3 v squared x v 0 1 minus 6 v u x u u 0 0 6 u x u v 0 0 x u x u v minus 6 v x v v 0 respect to v 6 u normal vector so in fact you start learning tricks about being quick so write x u and x v one on top of the other order in a reasonable way so that you can make the wedge product automatically so what is n n will be 0 so minus this thing so 3 v squared minus 3 u squared then so this is 0 so I'm glad this is 6 v u and then 1 divided by its norm that I don't care it will be some function I will always indicate it by square root and then ok then e capital e capital e is this times this well actually and again now for you it's the second one so it's a bit too early but for a graph how much is e g minus f squared we know it let's see again and check the previous example so e is equal to 1 plus 3 u squared minus 3 v squared squared this is e f f is equal to this mess ok there is really no point e f and g whatever functions they are ok little e little e will be n scalar product this ok so it's 0 0 6 u over the square root f is minus 6 v over the square root and g is minus 6 u over the square root that's actually the only thing I care I mean I'm not trying to get the complete expression so how much is k well k is e g minus f squared so it's minus 36 it's not really important the function because we have not computed the other one so in any case e g minus f squared minus 36 u squared plus 36 v squared divided by e g minus f squared that I don't know so what was the only point I was interested in was x so I was looking for a planar point did I find it? well for u equal to v equal to 0 of course the corresponding point as curvature 0 so it's either planar or parabolic and then I go back to the single coefficients for u equal to v equal to 0 they are all 0 so p is a planar point that actually is the only conclusion I wanted to draw otherwise you write down the complete expressions ok it's a planar point but if I draw the in this case actually it's since the point is the origin the affine tangent space and the tangent space are the same thing ok if I draw the affine tangent space where does the surface lie on both sides in every neighbor whatever small neighbor I pick there will be points where it's positive and points where it's negative you can actually check it immediately here ok whatever small disk in uv you pick this function is somewhere positive and somewhere negative so this seems to tell you look the theorem could be if you have a planar point the surface cuts the tangent plane but you have the plane and the plane does not do that it's not true that the plane stays on both sides so for planar points end of story well what about parabolic points well what is the prototype you have of a parabolic point give me give me a hint so here we had the plane and then we had to construct something strange what is kind of the model of a parabolic point that we have seen well on one hand we have cylinder every point of the cylinder is parabolic because the Gauss curvature is zero but it's not true that both principal curvatures are zero ok so if you look at this example you might say so the theorem could be the same as for elliptic points because what happens here on the cylinder is that you throw the tangent plane at a given point well the surface actually lies on one side now it's not true anymore that p is the only point of intersection there is a whole line but maybe that's the only part of the theorem which fails no here is an example in this case as an exercise so take the yz plane in the yz plane take the curve z equal so it's a graph of a function z is equal to y cube ok where z so the part of this which lies for example z in minus one one ok how does it look like well of course the graph of the function y cube is something like this ok and I'm taking exactly this part the part I drew is the one I am interested in now take this curve and rotate so of course for y z equal to one corresponds also to y equal to one so this is the point one one so draw this line here so this is the z equal to one line in this plane and rotate this curve around this axis and now I made a mistake in my blackboard because I have no space to rotate it so I need to cancel this so that means if I pick a point on this curve I have to add the whole circle just with this center now if I pick this point this is this would be the center and this would be the radius so more or less something like this and so on how do I draw it in one shot well I draw the symmetric curve if I if I am able to do that which looks more or less like this and then every point is rotated every point I join now the interesting part is this I don't like drawing with colors but let's so points on this curve here are more interesting than the others so first observation this surface actually my picture is lightly misleading because here there is a problem because actually the curve itself looks a bit more like this I mean so when I rotate I really get a casp a nondifferentiable point so in fact that's why I'm taking the open part of this curve so this point is not on my surface every other point since it doesn't touch the axis of rotation produce a regular surface so it is a regular surface and now the point is that all these points so yellow points are parabolic this is something you need to check you need to find the parameterization of this of this subject here exercise but that's the only way you know how to compute if something is parabolic or not you pick a point find a chart around this point and compute e f g e f g e v metal form so you can detect whether the gauss curvature is zero and you will find zero but you have also to argue that it is zero not because everything vanishes but only one of the two curvature principle curvature vanishes do that and these two objects cannot go together in a theorem in this case the surface was on one side and in this case clearly the surface cuts the tangent plane sorry after one hour and a half you start getting a bit myself but it's okay imagine the tangent plane of the picture it will cut the surface in every neighborhood no surprise I mean it's easy to construct examples of functions where the action is semi definite and the function the original function had not a minimum and a maximum so it would have been very strange that the height function so the only thing that we are doing is that we are taking instead of any function the height function of a surface on the tangent plane so it's not particularly surprising but in any case the moral of this lecture is clear show me a picture and I can tell you with minor possibilities of mistake okay minor there is some but I'm almost able to tell you immediately which are the elliptic points hyperbolic points I must be very unlucky to get I can make mistakes only at flat points okay, that's it


Yeah, so hi guys Thanks for sticking around until the last talk of the day. I know everybody's probably pretty tired This is a little bit less less high-tech than Olivier So hopefully Hopefully easy to follow along with and if we go through it a little bit quicker than half an hour Don't hold me to accountable This is a subject that I've been thinking about quite a bit since more kind of with the army About two years ago taking a look at domain names in the public infrastructure I've been quite curious about what's possible with a low sort of a low-hanging fruit style of attack Probably a lot of you especially if you're more technical familiar with the internet and the way it works DNS not a mystery full of vulnerabilities In the army we talk about something called an asymmetrical attack So the cost of putting a roadside bomb for example is a fraction of the cost of defending against that attack I think this is a similar a Similar thing in cyberspace that's worth considering. That's something that's been around for a long time and we'll dig into that a little bit Yeah, so introduction Currently sock analyst at commissioners to Quebec big thanks to JP and the rest of the team for encouraging me to Come up here today and give my first talk at a security conference It's a hell of a place to do it too, so it's exciting Also cyber protection team lead with 34 signal regiment in Westmount were a reserve unit in the army Previously spent many years working as a home automation integrator smart home guy a lot of custom AV home theaters Lighting control HVAC that kind of thing Also software product manager for a mobile app Where I spent a lot of time dealing specifically with user behavior and user issues? So there's a lot of overlap there, too. I think this talk is a bit of a culmination of a lot of my experience So what's on the agenda I want to talk quickly about human fallacy and the role it plays in some attacks I want to talk about DNS specifically some of the challenges that and email security challenges We're gonna take a closer look at cyber squatting and maybe refresher for some of you Maybe some new angles to think about and then I want to bring you sort of on the inside of a little evil experiment that I conducted Over the last 45 days Then obviously we'll talk about protecting ourselves and our organizations and then some closing thoughts Maybe we can talk about more in the panel so It's a cartoon that resonates with me pretty well on one side We've got data security and firewalls and everything that goes along with that on the other side We have Dave who's the walking? epitome of human error Because the point is that you put a lot of technical controls in place But it's really hard to account for some of the human factors involved Now we see that firsthand a recent report probably I'll saw this in the news It was released by Bellingcat a bunch of US soldiers were using flashcards to study overseas and We're leaking top nuclear secrets Some of the most top-secret information in the United States was revealed 100% human error Ops sec issue there More recently city of Ottawa. We saw humans involved in a scam Social engineering experiment where somebody was pretending to be a Partner on behalf of the Salvation Army requesting the city of Ottawa changed the bank account that they transfer money to They made away with $558,000 in city funds 100% human driven Probably my favorite example if anybody should be secure. It's president of the United States Ops sec fail even surrounded by The best people in the world in cybersecurity arguably Still using a password like mega 2020 was the first example if I remember correctly second one was your fired happened twice his account was guessed by security researchers from I think it was Denmark So what is all of this matter? You know according to agress report 94% of organizations have experienced a data breach In the last year with human error being the leading cause of serious incidents On the other hand only 21% of technology leaders surveyed said that human error was their biggest concern Mid-pandemic this issue is only grown Accidental and improper sharing of data by employees has grown to become Probably one of the most critical threats facing were concerning security leaders and then finally the last Five years and even up to today. We see a growing concern with the introduction of things like GDPR Privacy protection laws CCPA in California More close to home bills bill 64 in Quebec and another such privacy regimes Where the penalties imposed on organizations and the costs incurred are much much higher and it's now a regulated Issue it's no longer. Should we care about security? It's a you better care about security or it's gonna cost a lot of money To the organization and in some cases even criminal liability So let's back up a little bit. Let's talk about what is DNS?  You can do that research on your own but essentially DNS is a system that is designed to point users To IP addresses and software applications to IP addresses because humans have a hard time remembering IP addresses names are a lot easier to remember DNS has a long history started all the way back with our penette in 1972 the concept of a host or an Arpanet host name was was introduced and then Dr. Paul mock Petra's pioneer of DNS Publishes RFC's 8 8 2 8 8 3 with some evolutions by 1986 into RFC 973 and Finally, you know the ISE is founded DNS systems that we still use today like bind or introduced by 1987 with RFC 1034 1035 we basically have the DNS that we still have today That's a 30 year old technology that hasn't really evolved in 30 years Long before a time when people were thinking about cybersecurity and what are the security implications of the way DNS works For those of you who study internet history, you'll know that the internet was not designed to be secure It was designed to be robust. So a decentralized naming system was designed to stop a bomb from destroying our Infrastructure and it was to rope packets and be resilient But not necessarily concerned about data security or privacy so important considerations for Putting on the black hat for a second the important consideration is DNS not built with security in mind We didn't talk about it, but you probably all know that it's very trivial to register a domain name You can do it anonymously go buy a prepaid credit card at Shell Throw it into your your registrar of choice and register a domain name It's very inexpensive and finally humans often make mistakes and the component that DNS Relies on is human input more than anything. It's built for humans. So those are the things we want to keep in mind Step aside for a second look at email. Let's look at email You know again first thing we'll notice is as we look at the diagram is that The very first step in composing an email is choosing a recipient and how do you how do you designate a recipient? you put in a name out of domain and right away email is dependent on the domain name system and if we take Forgetting the details of email if we take This concept that DNS is inherently insecure if email is dependent on it. Well, then email is also insecure History of email very similar timeline I'm not going to go through it completely but roughly the same timeline We're looking at email that by and large hasn't changed and I know some of you are probably thinking Oh, what about DNS sec and what about encryption and what about all of these other? Protocol extensions that have come to pass since then. Well, you know, let's do an exercise show of hands Who sent an encrypted email in the last month? Okay, I see about 5% of the room has done that now show hands who sent a sensitive document by email in last month Okay, so different people different priorities. It's Maybe not the effect I was looking for there, but the point is that email is You know basically unchanged since the last 30 years with with some minor caveats Important considerations, we know emails not built with security or privacy in mind really it's trivial to send or receive emails anonymously It's actually quite easy to stand up and operate an inexpensive Email server or email infrastructure and once again humans make mistakes and they make them quite often Let's move into cyber squatting so I'm sure you all know what cyber squatting is but basically it's a practice of registering a domain name with the intention of infringing upon the intended destination of the user so I Could register self-sec that might be a form of domain squatting trying to leverage the trademark of Norsec just one shitty example Many different types of cyber squatting If you break them down into categories, we can look at typo squatting, which is quite common So in this example, we have something like gnail or Facebook with an extra o simple typo mistake put in by the user We have bit squatting, which is really interesting It's when you know cosmic radiation from outer space comes and interferes with whatever is stored in your memory and Flips a bit in the character and all of a sudden you typed in Microsoft But you ended up with mic post off comm in in memory in the computer and that's what gets processed by in this case DNS Combo squatting so taking a brand name Putting a dash in some other trust word or something like that It's not owned by FedEx, but it's made to look like it is Homograph squatting which this is an interesting example Hot mail to a human looks exactly the same as hot mail to a human, but in this case, it's a different character It's not the same character that's registered in DNS Finally paypal this one is a shout out to the franc phones out there PA IE pal dot CA And then level squatting is another interesting technique that we see a lot especially with mobile given limited screen size Sometimes what happens is the domain name is truncated to the left-hand side and everything on the right-hand side is ignored So, you know the user is not actually paying attention to the real domain or TLD so Cyber squatting not a mystery. We see it all the time. It's very common in fishing campaigns You know as a way to bait people into clicking links that seem trustworthy. It's also a very common technique in spam and marketing One thing that's maybe a little less common But but still out there is you know watering hole attacks man in the middle attacks and finally the thing I want to focus on today Is intelligence collection and using this as a tool to to gather information against a target? So let's get to the fun part time for the experiment I want to put a disclaimer out here So when I started this thing, I didn't know what I was going to find I was kind of hoping like maybe somebody's going to send me an email and it'll be interesting It turns out I actually collected an enormous amount of personal private sensitive information None of it's going to be shared. It's all going to be anonymized. I'm going to give you the meta analysis and Everything that I've captured is basically analyzed and deleted on the spot. I'm not storing I'm not interested in anybody's private data, but I am interested in showing what's possible with this kind of attack So I had a few hypotheses going into this thing My first one was that some small but you know given a large enough scale not insignificant number of users Will mistype the domain name of a popular or you know an email address when they're typing in their recipient I'm going to mistype the domain We should be able to register some of those typosquadded variants. They shouldn't be that expensive and We should you know, we should be able to do that easily and then finally The hypothesis is if I put a catch-all email address on the end of that typosquadded domain I should start receiving emails intended for other people, but they're coming to me So some of the key questions I had is what kind of data can we capture this way? Is it possible to remain passive and anonymous in doing so and Can we employ similar techniques? Or sorry, can we detect other actors employing similar techniques? Can we hypothesize who might be out there doing similar approach and finally how can we defend ourselves against such an attack? So remember the email diagram from before so this is a shitty edit. I've made to it to sort of convey the idea so Alice means to send an email to Bob at be.org, but he mistypes and he types in Bob at C.org well The email server has no problem with that the email application accepts it sends it for an NS lookup and There's an MX record at C.org and it says yeah, sure I'll take an email. Yeah, we've got to catch all email address I'm happy to receive an email for Bob at C.org and The email gets processed and delivered right into in this case the attacker or the malicious listeners email inbox so just a little bit of info to set you up for you know, how the experiment is set up We have 12 different domain names that were registered and I want to give a thank Thanks a shout-out to colleague of mine at questionnaires.  Who'd also done some similar work here and was able to help me get the set up We set them up on a shared webmail inbox with a catch all email address and every single one of those domains and we let it collect data for 45 days Okay, so results 123,000 emails. I was hoping to get like five, okay 2700 emails per day is You know, that's like one a minute or something ridiculous. You can watch them come in in real time. It's really interesting You might be wondering what kind of emails we received well a lot of this There's a lot of spam, which was the first challenge that I came across was how am I gonna eat all of this? There's like a ton of spam to deal with It's ridiculous So the vast majority of his spam frankly not unexpected when you stop and think about it Maybe maybe for the next iteration of this talk We'll talk about who the top spammers are and there's probably a lot of work that could be done just around analyzing that But that wasn't the focus of the talk, so I didn't spend too much time on it So emails of interest here's where things get really interesting. I received at last count 204 interact e-transfers With a total value of tens of thousands of dollars A lot of package tracking links and not the spam variant that you're all used to receiving. These were legitimate Personal banking notifications.  You've probably been asked if you want to receive an email receipt Well, maybe the girl mistyped your email and I got it Um Account verification this one is dangerous. So a lot of account verification and activation links You know this this we can talk a little bit more about types of attacks We can pull off with this appointment confirmations dental appointments doctors appointments sales meetings all kinds of appointments Personal resumes and employment applications think about the amount of data that's on your personal resumes Sensitive data There's you know, even over 45 days. This is almost Yeah, almost one a day Password reset links loan applications and service contracts tenant notifications personal medical information x-rays All kinds of information, you know people's private health data lease documents I Got a completed security clearance application form complete with passport photo and 10 years of personal history And finally the most impressive one I received was a I don't want to reveal too much, but it's one of the largest labor unions in the country They were in the middle of a draft negotiation with the government and I got an early version and early look at the contract before Before anybody else basically except for the lawyers So some idea of what can be accomplished and collected and again just 45 days just 12 domains So let's talk about the exploit opportunities here. They're basically endless Identity theft comes to mind extortion social engineering I can gather personal information that people may not be you know Aware is out and available to others Doxing I can do a account in credential theft financial theft Impersonation I didn't I didn't claim any e-transfers. I don't know if that would work. I'm not interested in trying Impersonation, you know, once I know enough about a target I can pretend to be them and finally I'm in the army. I've personally seen people send things over a Gmail account I've seen them send things over hotmail accounts. Is there a national security threat here? Maybe So how common is it I? Was curious to look around I only registered 12 domains, but there's a lot of squatted variants of these things So I looked at three examples. It's by you know Not not anywhere near exhaustive But as some examples if you look at gmail.com and you start looking at typosquatted variants You'll find that there's 155 adjacent domains registered and about 70 of them have active MX records on the domain Outlook.com similar stats Yahoo. I guess it's been around longer. We're highly targeted That's a lot of potential for theft right there And we don't know who's behind these domains and I can tell you when I looked at them Many of them share the same IP address So one has to wonder who's listening who's receiving those emails One thing that's interesting about this and this is an example of What you can do against a public email service, but what about against an enterprise, you know What what about against a company domain or an enterprise domain? Can we do the same type of attack and part of what inspired me on this talk was was one that I saw at the RSA conference Where they use this as a red teaming technique and one of the researchers said hey You know what for 12 bucks. Let's register the domain and just see what we get and They got the keys to the kingdom by like day six of the engagement They they received a mistyped password reset for their most sensitive database They were able to change the password and log in with admin rights to the thing that the company most wanted to protect If you want to see the talk, I highly recommend it. I'll share the slides Afterwards, I guess through north sec. I can probably do that This link will take you right to the interesting part of the talk Okay, so how can we defend against this the first thing I want to talk about is I alluded to it earlier as encryption I wouldn't have really learned anything or at least not anything meaningful if these emails were encrypted They were all in clear text and I could read them However, I wanted there was no as my own certificates and use I didn't I didn't detect a single encrypted email and what I gathered Awareness training is a really big one So I was digging for stats because I started to ask myself. Well, what about digital literacy? Do people understand that this is possible? For me it's shocking, but what about for other people? Stats can the the most recent Most recent stats I could pull were from 2018 where they were looking at student digital digital literacy ratings So in this case 15 year old students only Like in Quebec, I think Canada wide 38% in Quebec were actually the lowest in the country 30% of students reported being taught How to identify phishing or spam emails? You know that that's a pretty big problem I think awareness training across the board maybe in school all the way up into the enterprise is really important And then finally DNS solutions I think solve a lot of these problems so Monitor your DNS take a look at what requests are happening There's DNS firewalls DNS filtering out there Passive DNS is another interesting tool to detect things like newly registered domain names There's a lot of research out there that shows if a domain name is new enough It's probably malicious with a high degree of confidence. So don't resolve new domains There's a lot of tools for this far-set security domain tools There was There's a few vendors here today flare systems was there they have a really neat platform that looks at a lot of this kind of thing and Then finally if you talk to dr. Paul Vixie the guy who wrote bind and maintains bind for for many many years or the father of the DNS They'll tell you to run your own DNS resolver. It's really easy to set up I know it's really easy to go and say hey, you know what get get your domains from your IP addresses from <IP_ADDRESS> Or you know what cloud flare wherever you want to resolve your DNS, but take control back It's actually quite easy. You can throw it up on a Raspberry Pi play with it And you can use things that are called DNS response policy zones. So if you detect domains that are targeting Your enterprise domain you can set up a response policy zone that says hey if somebody asks for Gnail comm well send them to gmail comm give them the result for that instead You can set that up and you can do it in response to the things that you're seeing on your network Maybe specific to your domain or the types of traffic that that's happening on your network So do you ask big one? And then finally some food for thought You know the question is this brought up for me is it was quite shocking the first one was is email an essential service or a critical infrastructure You know for the importance that it has in our lives Is it considered critical infrastructure? And if so who's responsible for protecting it? Does this fall on the public email service providers? Does it fall on us as individuals or maybe the root? Root certificate root authorities and in the domain name system The second question I had is can I automate the analysis? You know I started getting a little bit uncomfortable looking at some of this data and Tried to remove myself from it as much as possible So can we can we do this kind of research in a more automated way that that generates meaningful threat intelligence? But also respects privacy And finally can we use similar approaches exploiting typo squatting against other services is really interesting research done Regarding NTP bit flips some of you may have seen that research the researcher that Basically realized that every I think it's every day every 24 hours a Windows machine checks in with time.microsoft.com And he knew that bit flips were a thing and they're theorized but he wanted to measure the rate that they happened at and He registered time dot some bit flipped version of Microsoft comm and he got millions of NTP requests to his server And in a very short amount of time And then finally man in the middle month excuse me monster in the middle We'll make that a standard Or you know reverse proxying against web pages and popular web portals, you know that could be another abuse That's interesting to look at Thank you


I am not usually hired as a guide, Mr. Poulin. But since you are unfamiliar with our Western frontier, this is one trip I wouldn't want to miss. Have gone. Will travel. Starring Mr. John Daner as Paladin. San Francisco, 1875. The Carlton Hotel. Headquarters of the man called Paladin. Yes. Good morning, Ms. Wong. Come in. It's not condor. You're not in. What did you want, Ms. Wong? Oh, I did not want anything. It's him. Him? Who? This man talks funny. He come look for you one, two, three times. He want to see you very bad. He talk funny, not plain like you and me. Oh. He's here now, right outside door. Well, then show him in. Merci infiniment. Merci. Good evening, Mr. Paladin. Permit me to make your acquaintance. Je suis Jean-Jacques Poulard de la Cité Française. Mr. Poulard. Yes, yes, yes, Ms. Wong. Yes, thank you, Ms. Wong. So, Mr. Poulard, you're with the French Secret Service. Oui, monsieur. What can I do for you? Ah, Mr. Paladin, I need help. What kind of help? I need a guide. A guide? Well, that's not exactly my line. Ah, monsieur, I search for a dangerous man. He has escaped from La Santé prison in Paris and I have searched and searched and now at last I know where he is. Where? In your Sierra mountains. Well, why don't you go get him? No, no, no, monsieur. You do not understand. I am a city policeman. Paris, Marseille, London, New York. I trail my men in these places and I am formidable. But, yes, west, monsieur, the mountains, the wild animals, the Indians. I don't know about these things, monsieur. Do you want a guide or a bodyguard? Perhaps both. My services aren't cheap, monsieur Poulard. Ah, non. I am prepared to pay. Do you certainly understand these matters involved expense? Ah, very well then. Let's say a thousand dollars in advance. Oh, lala. That is, as you say, not cheap, but... D'accord. Very well. Now, how long will it take to assemble the expedition? Expedition. Bien sûr. What will need pack animals, I imagine, a native bearer? We leave in the morning. In the morning? Yes, monsieur Poulard and Don, just the two of us. About, uh, monsieur, I thought... Monsieur Poulard, what does your name mean in English? Poulard? Chicken? That's what I thought. See you in the morning. Again, for the 11th straight year, camel outsold every other cigarette. Filter, king size and regular. The best tobacco makes the best smoke. So if you're smoking more now, but enjoying it less, change to camels. Get more real satisfaction every time. Start to really enjoy smoking again. Have a real cigarette. A real cigarette. A real cigarette. Have a camel. Ah, monsieur Poulard. Oui, monsieur. This man you're looking for, what's his name? Ah, Marius. Ah, who is he, this Marius? How many men is he murdered? Oh, we are not positive that he has murdered any, but we suspect him of many. But what has he done? He escaped from the Sainte prison in Paris, and no one escapes from La Santé. But he did, no? And therefore, monsieur, he must be returned. It is a matter of the honor of the Siereté. Seems like a lot of trouble to go to. Oh, the honor of the Siereté must be upheld at all costs. Oh, yes, I see. What makes you so sure this Marius is up there in the Sierra? Oh, monsieur Thaladin. We French criminologists follow a very useful and profound principle. Oh, what's that? Ah, chercher la femme. Ah, I find the woman. Ah, oui. And I found her in Paris. She had come home after making a small but adequate fortune out here in your west, uh, entertaining. Monique had seen my man. Monique? Monique La Fleur? Ah, bien sûr. Did you know her? Oh, very well. Oh, la, la, la, la. Ah, charmante, m'est-ce pas? Charming? Yes, I guess that would cover it. It was a crazy kind of a job. But I was never a man to argue with a thousand dollars, particularly when it involved no apparent risks. So we spent a pleasant afternoon riding along, reminiscing about our mutual friend Monique La Fleur. That night, we stopped in Hang Town, where I introduced my employer to the dish for which the town is so famous. Now, what do you call this? That's Hang Town Fried. You like it? Oh, Monsieur Paladine, I am tongue between honesty and politeness. Oh, then, Monsieur Poulard, by all means, be honest. Ah, bien, c'est d'execrable et froidable et dégoutant. You don't like it, then. Oh, non. Bah, bah, bah. There he is, Red. A little one with a point of mustaches. He looks like a Frenchie. He talks like a Frenchie. How about it? Are you a Frenchman? Will you excuse it? I am. And who are you? They called me Big Red, and I'm here to tell you Frenchies ain't welcome in Hang Town. We're just passing through. I'm thinking you're going to pass through real fast. How about it, boys? We don't take kindly to having our women stolen. Last Frenchie come through here made off with my girlfriend. I've been laying for him ever since. Yeah, but this, uh, this is a different Frenchman. Makes no difference. He's a Frenchman. Very well. If this is an example of Hang Town's hospitality, we'll be on our way. I'll see you, Will. Scatting? Yeah, Red. Go heat up that kettle of tarp on the Liberty State. Sure thing. And a couple more of you rip up some feather beds.  Oh, wait a minute. Uh, this Frenchman who ran off with your girl, what was his name? I don't know. Everybody just called him Frenchie. Well, was it Marius Moneychance? Well, yeah. Come to think of it, I did hear Lola call him that. Marius, honey, she said a filthy Frenchman. Well, then we have no cause for argument. My companion is from the French Secret Service. He's on his way to arrest this Marius. Yeah, well, he's got to find him first. Oh, he will. Good. When he does find him, he can show him his torn feathers. Yeah. Well, I'm afraid we can't wait for the party. Big Red. Whoa! All right, now, stand back, all of you. With your hands in the air. No, no, wait a minute. You heard me. Stand back. All right, Monsieur Poulin, outside. Get our horses unhitched. Oui, Paladin, tout de suite. The rest of you stay where you are until we're out of town. Now, I'll shoot the first man who comes through this door. Here is your horse, Paladin. Good. Now, as you say in your country, let's fishé le con. Sure. Constipation can be a problem for anyone, even doctors. And when constipation occurs, it's interesting to see just what doctors consider important about a laxative they might use or recommend. Well, a majority of the doctors we heard from had this to say. A laxative should be effective, gentle, as close to natural acting as possible, and a medicine that can be used with complete confidence. Now, x-lax has been popular with many doctors and millions of people over the years because chocolate at x-lax is effective. Overnight, it helps you toward your normal regularity. X-lax is gentle. Next morning, it gives you the closest thing to natural action. And that's why many doctors and millions of people use x-lax with complete confidence. X-lax, the laxative that helps you toward your normal regularity gently. Overnight, is x-lax in your medicine cabinet? On a bright night, you can see your way by the stars in the mother-load country. We took advantage of this ghostly illumination to put a number of prudent miles between ourselves and the vengeful and unreasonable citizens of Hangtown. At last we pulled up on the banks of a little stream and made a cold and cheerless camp. The next morning, the sun was up first. Then quite a bit later, I got up. The little Frenchman was still curled up in a pitiful ball, twitching from time to time with nightmares of tar and feathers. I looked around and saw a bearded, red-shirted miner working his way up the stream, panning the river sand for gold. Eh, Monsieur Poulard, come on, time to get up. Come on. Sleep well? Oh, please, do not mention it. I don't think I can ever move my arms and legs again. Uh, we have company. Oh? Howdy! Hello. Say, we're a little uncertain of our whereabouts, friend. Could you tell us where we are? Well, this here is what used to be called South Fork, but folks have begun calling it Frenchman's Crick. Down stream quite a fur-piece is civilization. Civilization? Yup, settlement called Hangtown. Oh, yes, we're acquainted with Hangtown. But what's upstream? Nothing much. A few abandoned gold camps. Cricks panned out, more or less. But I keep hoping I'll find something the others missed. You said folks have started calling this stream Frenchman's Crick. That's right. Why? Counter-crazy, Frenchman lives up to the head of it. Yeah, Frenchman, what is his name? I don't know, everybody just calls him a Frenchman. What's crazy about him? Well, he's got a claim staked up there. A couple of claims. Only he don't do no mining that anyone can notice. Got a place all fenced in, a big gate with foreign words on the sign. Only two words in English. They say keep out. Well, how far is this place? Well, it's a pesky place to get to. It'll take you the better part of the day, following the stream. Of course, the stream winds a lot. You could go over the mountain that way. Yeah, well, how long would it take that away? The better part of the day. Then it's no shorter. Nope, but you don't get wet. We went over the mountain. And it did take the better part of the day. Towards sundown, we came back once more to Frenchman's Creek and there below us lay a well-built cavern surrounded by clear land. We spurred our tired horses toward it and a few moments later, drew up before a rail gate on which hung the imposing keep out sign that the miner had told us about. Marius must have put up that sign. His guilty conscience is haunting him. Perhaps. Well, where should we go in? Paladin, I am tired and I am angry and I am unafraid of these Marius. Come, we go in. Come on, boy, move. Get in there. Stop! Put your hands in the air. I will relieve you of your weapons, Monsieur. Throw them to the ground. Is it that you cannot read my sign? We read it. You made no effort to hide yourselves. I have watched your approach since you started down the mountain. We have nothing to hide, but it appears that you do.  Give it to me. How did you know me? Your derringer, please. Here. Merci. We have a mutual friend, Monsieur Paladin. Monique Lafleur. Oh, yes, Monique. She pointed you out to me once in San Francisco. And she told you about the derringer. He was in the suite of you, Monsieur, and dangerous to reveal to her the whereabouts of the derringer. Under the circumstances, there was nothing else I could do. But you will know, of course. Eh bien, Monsieur, you will proceed to the house. I will be right behind you with a gun pointed at your backs. Proceed. Paladin? Huh? This is Marius. Black hair, black moustache, scar over the right eye. And a pile of Monique's. That's Marius, all right. I shall ask him. Hey, Monsieur. Oui. Appelez-vous Marius? Marius? Qui est Marius? Ha, ha, ha, ha, ha, ha! Yeah, you are French, Monsieur, huh? I'm French. Then you must be Jean-Jacques Poulard I am, but how did you know? Monique wrote me you were looking for me. Monique is a friend. Everybody's friend, it seems. Dandruff bothers most men, most women, too, so listen. Today, you can get rid of embarrassing dandruff in just three minutes. Yes, with Fitch, dandruff remover shampoo, unsightly dandruff's gone in three minutes. It's the quickest, easiest of all leading shampoos. That's not all. Using Fitch regularly is guaranteed to keep embarrassing dandruff away. Simply apply in the unique Fitch manner. Before you wet hair, rub in one minute. This way, Fitch shampoo penetrates right down to the scalp. Next, add water. Lather one minute to wash every trace of dandruff out of your hair. Then rinse one minute. All that loosened dandruff goes down the drain. In three minutes with Fitch, one rubbing, one lathering, one rinsing, dandruff's gone. And never forget, gentle Fitch can also leave your hair up to 35% brighter. To get rid of dandruff problems forever, brighten hair, too. Use Fitch regularly. Get Fitch dandruff remover shampoo today. Only 59 cents. All right. We are here. This is Shea Marius. You may be smart, gentlemen. You will stay to dinner, no? Yes. Oh, we'd be delighted as we seem to have no choice. Come, it is settled. I shall tell my wife. Come with me. Enter, please, gentlemen. Thank you. I'll see you, sir. Chérie, viens-toi ici. Ah, ma petite choux, we have guests for dinner. Monsieur, this is Madame Marius. Oh, howdy, boys. How you do. Enchantée, Madame. Your first name wouldn't be Lola, would it? Yeah, that's right. We met an old friend of yours in Hangtown last night. Big red. Oh, is he still sore? I ran out on him. He is. Poor slob. He never had what Marius has. What's more, Marius married me. And taught you to cook. Come, dépêche-toi. I guess I hungry. Oui, chérie. Too sweet. A jewel of a woman, Monsieur. A prize beyond price. But she speaks lousy French. Some more coffee, Monsour? No, no, thank you. It was an excellent dinner, superb. Oh, not since I left Paris have I eaten so well. Thank you, Monsour. Imagine to find Escargot in this wilderness. They are not quite as fine as our snail de Bougogne, but they fatten well on my vine leaves. You have a vineyard? Oh, yes. I shall make my own wine in another year. And this fromage de chèvre? From my own herd of goats. Oh, yes, my friend, we live well here. I regret that I must take you away from all this. But you are not going to. I must. I have a warrant for your arrest. And you have no guns. But don't you see, I cannot return to Paris without you. It would be a disgrace to me, to the certé... Quelle dommage. You see the position you place me in? Regrettable. But consider the position of France. In what way? If you insist on being obstinate, you can ruin France. Impossible. No, no, listen a moment. You insist on arresting me. I refuse.  I refuse. The American authorities recognize my rights to freedom from persecution. You return to Paris and report this to the certé. The certé brings it before the National Assembly. The Paris newspapers make it a cause celebre. France has been insulted. The headlines scream. War is declared. The French fleet sails to invade the United States. It is her pulse that great loss at Washington, New York and Boston. The flower of French manhood falls upon the Atlantic beaches. France is defeated and forced to pay huge indemnities, which weaken her economically until she sinks to the position of a second-class nation, thus uninging the balance of power in the Grand Alliance and exposing Europe to such dangers of general war that the mind reels at the prospect and at last refuses to look upon the chaos it portends. Oh, no, no. Are you willing, Monsieur Poulard, to be responsible for such a course of events? No. No, I cannot accept such a responsibility. Of course you cannot. You must think of France. Yes, I am. I shall never see her again. I shall become an exile in this savage country where they fry oysters and eggs together. No, no, no, no, no. That far I cannot go. I will not condemn a Frenchman to that fate. What else is there for me? Stay here with Madame and me. Help me tend the vines and herd the goats and haze the chickens and the snails. Can you think of a nobler activity than growing good things for the table? Only sitting down at the table and eating them, Monsieur. So that's how I left in the next morning, hey boy. Feeding vine leaves to snails. Snails? Oh, no likey.  Well, you know what the old lady said when she kissed the pig.  Hey, what's she say? Chacun à son goût. You're talking funny. What that mean? That means, hey boy, everyone to his own taste. Oh, nearly forgetting.  Oh, a lady? Lady talk funny too, like you just now. Oh, no. Hey boy, was her name Monique? Yes, Monique Flower. She come back this afternoon. I won't be here. Just tell her to... Hey boy, tell her to follow Frenchman's Creek east of Hangtown over the mountains for the better part of the day. For your who? She'll find what she's looking for. Oh, this miserable cold. And my sinuses. Haven't you heard about Dresstan? Dresstan decongestant tablets for real relief from cold misery and sinus congestion. Dresstan is the revolutionary three layer tablet which for the first time makes it possible to unite certain medically proven ingredients into one fast acting uncoated tablet. Dresstan not only helps drain all eight sinus cavities, critical areas of colds infection, but circulating through the blood, Dresstan's decongestant reaches all congested areas, shrinks all swollen membranes, relieves pressure and pain. An exclusive anti-allergent helps keep breathing passages dry and clear. Pain relievers reduce body aches, fever, vitamin C helps build body resistance. This is Dresstan. Today, Dresstan is widely imitated, but the exclusive Dresstan formula cannot be duplicated. There is nothing, nothing like Dresstan decongestant tablets. Of Gunn, Will Travel Created by Herb Meadow and Sam Rolfe, is produced and directed in Hollywood by Frank Parris and stars John Daener as Paladin with Ben Wright as Hayboy and Virginia Gregg as Miss Wong. Tonight's story was specially written for Have Gunn, Will Travel by William N. Robeson. Featured in the cast were Shirley Mitchell, Harry Bartel, Barney Phillips and Marvin Miller. This is Hugh Douglas inviting you to join us again next week when CBS Radio presents Have Gunn, Will Travel.


Now, the waves that we've considered so far are disturbances that occur in a medium. The medium is the substance or material. Water is the medium for an ocean wave. The medium of a stadium wave is people. In a wave, the individual particles or components of the medium are only temporarily displaced. One particle's temporary displacement causes a nearby particle to temporarily displace them, so forth. The particles are not carried by the wave away from their natural position as it returns to their original resting position over and over again. So this wave does not carry particle or medium away, or we say the wave does not transport the particles or the medium from one location to another. The wave only temporarily displaces the medium. One might argue that this is true based upon the following evidence. Despite all the waves in the ocean, there is still water in the middle of the ocean. The wave transports energy. Let's consider only one crest of a wave or a pulse. The pulse is initiated by doing work or putting energy on one side of the medium. This energy is released to their neighboring particles to do work to displace them. This transfer of energy occurs over many particles, over a distance. If the transfer of energy between particles is efficient, the distance that the pulse travels can be very large. If the transfer of energy is not very efficient, say some of the energy is lost in friction, then the pulse does not propagate very far. The pulse transports energy from one side of the medium to the other. You can think of a wave as a consecutive series of pulses, each carrying energy. A wave carries or transports energy. A wave that transfers energy through the displacement of the medium that it passes through is called a mechanical wave. The idealized waves that we've explored are transverse mechanical waves. Transverse because the medium is displaced perpendicular or transverse to the direction of the wave, and mechanical because the wave is a mechanical disturbance of the medium. Now, a striking example of water waves carrying energy is the 2004 Boxing Day tsunami that killed between 240 and 280,000 people living near the coasts of the Indian Ocean. The initial disturbance was an undersea earthquake with a significant vertical displacement of the seabed. This earthquake released an immense amount of energy. The equivalent of roughly 26 megatons of TMT, and this energy was transported by the waves over far distances. Now, in deep ocean water, the tsunami waves form only a small bump, barely noticeable, but those waves travel at very high speeds of between 500 and 1,000 kilometers per hour. Radar satellites recorded heights of these small waves in deep ocean of only about 60 centimeters or two feet, some two hours after the initial earthquake. But in shallow water, near coastlines, that wave slows down to only tens of kilometers per hour, but in doing so it forms large destructive waves. Scientists investigating the damage in Ache found evidence that the wave reached a height of 24 to 30 meters, that's in excess of 100 feet. A lot of the energy released by the earthquake was dissipated or used in traveling across the Indian Ocean, but a whopping 20% of the earthquake's energy, the equivalent of roughly 5 megatons of TMT, was unleashed by the tsunami waves crashing onto the coast surrounding the Indian Ocean. Hence, the undersea earthquake's destructive energy was transported to the coastlines of the Indian Ocean by waves.