 So, suppose f is differentiable. So, consider, now consider what directional derivative is what? It is nothing but the rate of change. So, it is x naught plus T u 1 comma y naught plus T u 2. So, let us consider this function. So, it is a function of T goes to x T y T. The chain rule is coming into picture now. T goes to x. So, this is x T and this is y T. f takes x T y 2 to some value in R. So, if f is differentiable, the function x T that is differentiable. So, composite function is differentiable. Chain rule applies. So, conditions of chain rule are met if f is differentiable, implies what is the derivative of this? What is the derivative of this function? Let us call it as w T. T goes to one variable. So, what is the derivative of this function? Chain rule f of x at x 0 y 0. What is the derivative of this with respect to T x T? So, that is u 1 plus u 2 f y x 0 y 0. That is the derivative of this function. So, that is d w by T T. And what is this? We are looking at the derivative of this function. So, what is this equal to? One end, if I look at the composite function and… So, this is the rate of change of this composite function. What is the rate of change of that function we looked at? That was precisely… What was that? That is the directional derivative. So, it says if I look at the apply the chain rule, this is nothing but the directional derivative of f x 0 y 0. So, if the function is differentiable, then the directional derivative exists and is given by this. I do not have to go to the limit or anything, because limit is taken care by the chain rule now. Is that comfortable? Everybody or not? So, I am looking at this function. What is the derivative of this? This minus divided by T. That is precisely the directional derivative, is not it? If I want to calculate the w T, if I want to calculate its derivative, what I have to do? This minus f at x 0 y 0 divided by T limit of that. That is by definition. So, that is by definition it is this. By chain rule it is this. So, both are equal. That is all. So, it says if… So, theorem says, if f is differentiable at x 0 y 0, then the directional derivative of f in the direction of x 0 y 0 is precisely equal to u 1 f x plus u 2 f of x 0 y 0. From here, let us, a bit of vector, let me write this right hand side in terms of vectors. I want to write. So, let us write. So, I am saying this is equal to u 1 comma u 2. There are components of the given vector u dot product with f x x 0 y 0 and second component f y x 0 y 0. I am creating a new vector. Given the function f, if it has partial derivatives, then this gives me a vector f x partial derivative comma f y partial derivative. So, this can be thought of as a dot product of these two vectors now. This can be thought of as a dot product of these two vectors. This is a very important quantity associated with a function. So, for f, which is on D in R 2 to R, this vector f x at the point x 0 y 0, f y at the point x 0 y 0 is a vector. It is denoted by inverted triangle of f at the point x 0 y 0. It is called the gradient of f at x 0 y 0. For three variables, it will be a vector of the components 3. So, f z will come. But the interesting thing is, so if I write that way, gradient of f dot u at that point is equal to the directional derivative in that notation. Now, you see how things get interpreted. So, directional derivative intuitively is a rate of change of the function in that direction. And that is equal to the gradient times u. Now, here is a bit of vector algebra. A dot b, you can write as norm of this, norm of u into, what is a dot b? Norm a, norm b into cos of the angle between them. So, cos theta. This norm is 1. So, we can forget that. Now, what is a theta? This is a vector, gradient is a vector, u is a vector. So, this is theta is the angle between the two. So, and it is cos of theta. When is cos theta minimum, angle is pi by 2 and then it is 0. So, it says the rate of change of the function is 0 at that point in the direction of the gradient. When u is angle is 0, so direction of u is same as direction of gradient. So, it says the rate. So, interpreting this in terms of rate of change. So, rate of change of the function f is 0 when direction is that of the gradient. Are you following what I am saying? No? So, let me interpret it again. It says gradient of f at point x 0, y 0 is a vector dot product with u is equal to directional derivative f at the point x 0, y 0. And this we said it is equal to gradient of f, norm of gradient of f into cos theta because that is 1 is equal to d u. So, theta equal to 0 implies what? Theta equal to pi by 2 implies what? Cos is 0, implies d u f at x 0, y 0 is equal to 0. So, what is the meaning of saying this is 0? If I interpret it as the rate of change, so rate of change of the function f in the direction of u is 0 when u is same as, direction is same as that of gradient because then only the angle is 0. Angle between the gradient and u is 0, sorry pi by 2, when perpendicular sorry, when it is perpendicular. So, rate of change of the function is 0 in the direction perpendicular to that of gradient. And when it is maximum, when it is equal to 1, pi is angle is 0. So, rate of change of the function is maximum in the direction of the gradient. Is it okay? Yes. So, this is the physical interpretation. You can imagine a plate, surface is some kind of a plate and there is a vector given and the gradient vector is there at any point. So, if at a point I want to know and you are measuring the temperature, function is the temperature at a point of the plate and as you move on that plate in which direction the rate of change of temperature is maximum or minimum, you would like to know. You are walking on a hot plate and you do not want your feet to be burnt. So, you would like to seek the path direction in which temperature rate of change direction should be minimum. So, that is perpendicular to the, at that point find out the gradient that is the minimum, maximum when it is parallel to it. So, these are physical applications, this kind of a thing. Now, it is purely mathematical. Something more I should say about this. Here is another interpretation of this. So, let me just write rate of change of f is minimum u is perpendicular to gradient of f. It is maximum parallel to gradient of f. Is that okay? That we have seen. See how these things lead to physical applications. So, here is another one. So, let us take f f function of two variables again. So, we saw its graph in a surface. So, this is a surface. So, f of f, x, y, f of x, y. Now, in this, what is happening is whether I should introduce the notion of general surface or not. So, here is where z is equal to f of x, y. The z coordinate is taken. It is going a bit away from what we... So, this is kind of a surface, but the point is not every surface looks like z is equal to f of x, y. For example, if you look at the sphere, what is the equation of sphere? It is all points x, y and z, say that x square plus y square plus z square equal to some constant. That is the radius square. So, here there is a relation between x, y and z. So, a sphere, which normally we write as x square plus y square plus z square equal to, say, something, say, 9. What do you mean by that? That is the equation of a sphere. That means, we are looking at the points x, y and z, say that there is a relation. So, as an object in R3, it is given by this equation. Now, here z is not... You cannot calculate z in terms of x and y because there is only a relation given. So, if you want to calculate, it will be z square equal to 9 square 9 minus x square minus y square. So, z will be plus minus square root. If you take plus, it will give you the upper part of the sphere. If you take negative, it will give you the... So, this is what we call as an implicit equation of surfaces. z is not explicitly known. It is only implicitly given. So, there are many surfaces which are of this type. For example, if you take shape of an egg, that is called a lipsoid. That is also implicit. Some are explicit, some are implicit. What I want to do is, I want to look at a curve on this... I want to look at a curve on this surface. So, let us take... So, this is in R3. Surface is in R3. I want a curve on this surface. So, what is a curve? What is a curve? Is it like a path of a particle? Imagine a fly moving around in the room. At some, you start observing at t equal to 0. It is at somewhere. t equal to 1, it is somewhere. t equal to 2, somewhere. It moves around. To locate its position, we have to know what is the coordinates x t, y t and z t. So, a curve is a function from R to R3. t goes to x t, y t, z t. So, let us call it as a function R. Any curve is a path of a particle. You can imagine, at some time, you are observing what is the position of something. You can think it as way. So, at t, it gives a point which is in... So, this is the point p. I want this curve to be on the surface. Then, what should happen? If it is to be on the surface, then what should happen? It would satisfy the equation of the surface. So, 1 s means z is equal to z of t is equal to f of x t, y t. Is that okay? Because z was equal to f of x y. So, z of t must be equal to that. That I can write as f of x t, y t, z t. I am sorry, f of x t, y t minus z t equal to 0. Brought everything on the one side for every t to be on the surface. This is a function of t. Now, mathematics enters into picture. This function is 0 everywhere. So, what happens to the derivative of this? Also, it should be 0. What is the derivative of left hand side? It implies, what is the derivative of left hand side? f x, right? Into x dash t, sorry, into x dash of t plus f y z dash of t plus minus z dash of t equal to 0. I want to rewrite this now as a vector equation. So, it is f x f y minus 1 dot x dash t, y dash t, z dash t equal to 0. This equation, I am just differentiating this with respect to t. This is chain rule and that is z t only. So, z dash. Oh, this one. Oh, sorry. Yeah, yeah. I meant y dash. I wrote as z dash. Sorry, sorry. That mistake. So, that is y dash. Sorry, right? It is okay now. Everybody happy? Yeah. So, this quantity multiplied by this quantity is 0. What is this quantity? Geometrically, x t, y t, z t was the point on the surface. This is the derivative. So, this is the tangent vector to the curve at that point and says this vector is perpendicular to the tangent vector. So, what should be this vector? It should be normal to the surface. This vector is perpendicular to whatever curve I take. So, in the picture, at this point, there is a vector which is not changing because the vector is f x f y. It depends only on f. There is a vector and of course, z component is minus 1. So, whichever direction. But at this point, if I take the tangent, then this vector is perpendicular. If I take some other curve, take the tangent till it is perpendicular. So, this is a vector which is perpendicular to all tangents to all the curves. So, what this vector should be? We should call it as the normal to the surface. We should call this vector as normal to the surface. All the tangents will lie in a plane. That will be the tangent plane to the surface. So, we are going back to saying derivative helps us to define tangent. So, what we are doing is we are saying that derivative in the following sense, the partial derivatives for a function of two variable, we assume they are differentiable. So, partial derivatives exist and are continuous and so on. Then this vector is perpendicular to this. That means this vector f x f y minus 1 is perpendicular to every tangent vector. So, we call this as this is called this meaning f x f y minus 1 is called tangent vector to the surface at this point 0 y 0 z 0. Some confusion is called, sorry, not normal, sorry, is called the normal vector to the surface at plane perpendicular to this is called the tangent plane is called the tangent plane to the surface s at x 0 y 0 z 0. So, I am just trying to bring out the similarity between one variable and several variables. In one variable, we had continuity, we had differentiability. Continuity meant that there is no break in the graph of the function and differentiability was something stronger which implied continuity and said that at every point you can draw a tangent. So, same way we are saying in three variables, differentiability you can define and differentiability implies continuity and differentiability means that at every point on the surface you can have a on the graph of the function that is a surface you can have a tangent plane and that we are coming via normal. We are saying there is a normal, normal is the gradient vector. Gradient vector is a very crucial one. It tells you how the things are changing on the surface name gradient right itself should indicate English word gradient should indicate. So, rate of change is maximum if it is it is maximum when it is parallel to the gradient right and minimum when it is perpendicular to the gradient. Gradient is also the direction of the normal geometrically to the surface at that point. You may wondering why this minus 1 is coming because our surface is explicit z is equal to f x y. If you take a sphere, if you take the sphere x square plus y square plus z square equal to 0. So, let me just let me just indicate if you take the sphere is x square plus minus 9 equal to 0 say that is my f x y z that is a surface is given by a function of three variables which is implicit z is equal to f x y I could write it as f x y minus z is equal to 0 I could write this as f x y. There z was explicitly known in terms of here z is not explicitly known, but still even if I take a curve on this surface it will still have that thing same property x square y square z square. So, what is f of x t y t z t that will be that will be a curve on this surface that is equal to 0 on the surface. So, that is equal to 0. So, what is the derivative of this f x x dash f y y dash f z y z z z dash t equal to 0 that is same as saying f x f y f z dot x dash t y dash t z dash t equal to 0. Still you can what is this now that is a gradient for this what is the gradient f z is minus 1 that is why this minus 1 is coming here there is nothing special happening for explicit when z is explicit that value will be 1 or minus 1 depending on whether you write z minus or this minus in general this will be the gradient vector which is normal to the surface. So, this is very useful in doing what is called differential geometry later on that normal and so on. So, we will not be doing into that my idea of bringing the directional derivative was to indicate that there is something called rate of change in any direction possible and how it interprets geometrically. So, gradient vector is the one which is crucial. So, let us stop here.