Gradient Finally Makes Sense! | Hands-On Visual Explanation (Part 1)

[00:00:00]Hello everyone. If I show you this scalar fields and ask you what is the gradient, let us say at this point or at this particular point or at this point or this point. If you are unable to answer it, this video is perfectly for you. At the end of the video, you can tell the gradients of these figures and even much more complex scalar fields as well. Before going into 3D, let us see how the gradient works in 2D. So here I have a function f of x= x² are parabola.

[00:00:24]Suppose you are at x equal to minus1 and if I ask you in which direction the y is changing more and intuitively you'll tell that direction is in the x. The computer doesn't know intuition it follows a systematic procedure. Let us say you are at x= minus1. Now you take a step from x= -1 to x=0 and you tell that y value is getting decreased. So the gradient should be in the negative x direction in order to make it increase.

[00:00:46]But some other person may say my step is larger than yours. If the person moves from minus1 to directly to two then he may say that this positive x direction I'm having the largest increase in the y. So the gradient should point in this direction. Now which is correct. So in order to estimate the change in the y we need to consider the step size as well by using only the change in y with respect to two points we cannot tell because the step size even matters. So when we consider the step size so it becomes d y by d x. Now we will get the change in the y per unit x. Now when we try to decrease the x less than that unit when we tend the x to zero that's when we'll get at that particular point in which direction if you move you'll get the maximum y value change when we tend the delta x to zero we'll get the slope or gradient at that point computer doesn't know the limits how does it compute it computer takes h to be very less value like in the order of 10^ - 6 or 10^ 8 and then mathematically substitutes the value in the function and then it does subtraction and then division by h so that's how the computer computes the gradient and if you look back at the figure if you calculate the gradient at x equal to 1 you'll get two.

[00:01:47]So when you at x= 1 if you move in this positive x direction it means uh the y value is changing and similarly when x= to negative 1 the gradient is -2 here because it's having that negative sign now the computer knows the direction should be in the negativex in order to get the maximum y value change. Now keeping this in mind now let's go to the 3D figures. Now if you look at the figure this is the xaxis I plotted f of x y = x². You can see that the function is not dependent on the y. If you take a slice of this, you can see a two-dimension parabola like this. When you stack multiple parabas one after other, one after other, one after other, that's when you get this kind of shape.

[00:02:22]For every value of y, there is a parabola. Now, if I ask you what is a gradient distribution in this shape, yeah, just apply the thing whatever we have learned previously and then try to tell me what the gradient is at this particular point. Now, you have two directions to move. One is in the y direction and one is in the x direction.

[00:02:39]If you move in the y direction, it's constant. If you fix the x any given y you basically have the same value of the function. So when you move in this direction your height is not changing it's constant. Now if you move in the positive x direction then you can see the y- value is getting increased. And similarly if you are at this point now if you move in the positive x direction you can see the y value is getting decreased and if you move in the negative x direction then the y value is getting increased. That's how intuitively you can say at this point the gradient will be pointing in this direction for all negative values of x and for the positive values of x the gradient will be pointing in this direction as it is not having any y component the gradient will only pointing in the x direction.

[00:03:16]Mathematically if you see the gradient distribution it's nothing but 2x 0 at x= to.5 and y=.5 the gradient vector will be having f_sub_x is one and f_sub_y is zero there is no f_sub_y component so the resultant vector is only in the x direction. Now if I overplot this in this above field then we'll get this interesting plot. If you see clearly for all the positive x-axis you can see the gradient is pointing in this direction and for the negative x values it's pointing in this direction.

[00:03:51]I'll just rotate it so you can have better view. So this is the bottom view and this is the top view. You can see all the positive axis are pointing in this and negative xs are pointing in this. Now let's try to see for a function f of x y= x cube. So we all know x cube looks in the 2D it's something like this one. When you stack multiple x cubes for every y then that's when you get this 3D structure. Now same you take any given point if you fix the x then the y value is constant because the function is not dependent on the y.

[00:04:29]Now you try to move in the positive x direction if the y value is changing then well and good. So the gradient will be pointing in that direction. But when you are at negative x and if you try to move in the positive x direction and the y value is getting increased in the previous case it was getting decreased.

[00:04:42]In this case if you move in the positive x direction the y value is getting increased. So the gradient is is also pointing in the positive y direction.

[00:04:48]Similarly at even these points and all these negative x values the gradient is pointing towards only right direction and mathematically if you want to compute then if you calculate the gradient the fx component is 3x² always the fx is positive for any value of x.

[00:05:01]So that's how you get it mathematically but individually you can look at the figure and try to go in the positive x if value is increasing then yeah the gradient points in that direction and if value is decreasing the gradient is pointing in the opposite direction. Now keeping this in mind let's test our understanding for this. I have f of x y= y square. Here you can see this axis is y. Previous one was x² and this is y square. For every x value there is y square parabola. And now now you stack multiple parabas behind each other.

[00:05:29]First intuitively if you see at the figure let us say at this point if you fix the value of y and now you try to move in the direction of x the value of the function is constant. The function is not changing the x direction. If you move in the positive y then your y value is getting increased. That's when you realize the gradient should be pointing in this direction. Yeah, exactly correct. And coming to the negative y, if you move in the positive y, then you can see the y value is getting decreased. Hence the gradient should be pointing in the opposite direction.

[00:05:54]Mathematically, when you compute it, you get 0, 2 y. Here you can see the x fx component is zero and f_y component is 2 y. Gradient field is only due to the f_y component. And you can see here there is no fx component. If you overplot these both things, then here you can see this is a y-axis and this is the x-axis. And it's similar to the previous one. For the positive y value, it's pointing in the positive y direction. And for the negative y, pointing in the negative way. So, so far we have understood uh we got to know a little bit idea of the gradient. Now, let's make the figures little more more complicated.