Now, it's time to speak about more central scene here. For single variate case, we've actually discussed that we're going to speak about the change of function and its speed. Thus, appear the derivative, but what for a multi-variate case? What for our two-variable function? Does it have the same meaning and to what does have the same meaning. Because from now on, we do not have our single and one to rule them all. Derivative, and we have only partial derivative here and partial, well, actually means not the whole thing. So first of all, we are going to speak about some basic multi-dimensional geometry, which you are already familiar with by deriving all tangent plane equation, but it's nice to have it covered once and for all. So we have actually spoke about scalar products and what it is. It is just [inaudible] and can be computed by summation of multiplication of coordinates. There is some tricky thing here because we need to start with an idea of what is in a vector, because we have spoke that scalar product is a mapping form of a parallel vector. So what is a vector? That is where it becomes a bit complicated, because vectors are on the elements of vector space. I'm going to use annotation capital V here just in order to stress that that's a vector space. So what is a vector space? It's a set of objects which can follow certain rules. There are basically two rules, and well, as our exam next semester which can be extracted from any search with two faint operations. So what are we going to have in a vector space? We are going to have two operations: the multiplication by a number, in our case, my real number, and the ability to add two vectors and to get another vector as a result. So basically, what we are going to say here is that if we multiply any objects in this space in the set, because there's a number, we are going to get some answer up or maybe the same object in this space. If we add up two objects in the vector space, two vectors in the vector space, then we are going to get in the results another vector. That is fine, that is quite an understandable thing for all the vectors that were already unknown. But here is, what's different? The idea here is that we normally think about vectors as some or ordered sets of numbers, real numbers, rational, and whatsoever. But that is on the coordinates of the vector, and that's true only for the case when we have set the basis of our given vector space. What the basis is, basically, the set of vector which defines the space. The idea is that we do not have to draw all the possible dots on the real plane when we draw some graph. We just draw the axis. Basically, the axis is all the possible multiplication of two basic vectors multiplied by any number, a real one. That's what is the difference between the coordinates and the vector space, the vectors actually. Because vector is something, it's just inulin soup, a polynomial function, whatsoever. But coordinates appears only after you've set some ground rules, some basis, and then you have defined how to actually get all the vectors from this basic set of vectors. It is always important to understand the difference between one and the another. For more comprehensive coverage of this topic, you can consider either our industrial materials or as a part of the course about any algebra. So since we know what is a vector space and vectors, and we actually don't understand all about our scalar products. So our scalar product definition, we have vectors, we have mapping, all the things that we've actually covered all along, and we have our geometrical rule. Our geometrical rule results basically into two things: the criterion of being orthogonal and the definition of vector's length, which can be calculated by taking a square root of vector scalar and multiply it by itself. It is quite easy to understand why because if you can see our geometrical meaning of scalar product and substitute the second vector with the first one, just assume that we have vector b equals to vector a. Then, the angle in between them is put, obviously, is zero, one plus the cosine function equals to one. That is actually we haven't set the scalar product of a and a is square root of which ones. So that's our two [inaudible] from our definition of scalar product. So now, we know pretty much everything to define a very important thing of our course, the gradient of the function. So let us state it in the clearest fashion possible. There is no the derivative for the multivariate case. There is no then speed of change for the function if it has more than one variable. There are only partial derivatives and if we stack them together as a vector, we get a gradient of a function. A vector of partial derivatives. Sometimes, it is said that a vector of first derivatives or first partial derivatives. There are several notations for it. We have stated only the grad of f or this reverse triangle which is Greek character, Nabla. As usual, we just write it down. That's the partial derivative towards x and partial derivatives towards y stack to vector here. You need to remember one key point here is that gradient has a derivative for a single variant function as both partial derivatives are pointwise concepts. So basically, what one should expect that the gradient here exists at every point of the real plane. At every point, it could be different. So basically now, we're looking not only at real plane, we're looking at our real plane as a skin of hedgehog or something like that. Or, we're looking at a plane with a vector at every single point of it. So let us consider some basic example. For example, let us consider just our x squared plus y squared function. We've seen it a lot while we were talking about the function of multivariate case, and let us just try to compute the gradient here. Mostly, what we need to do, we need to find both partial derivatives. So the derivative targets x and the derivative targets y. It's quite easy because in those case, one of the terms here either x squared or y squared doesn't depend on the other variables as we need to just differentiate the other term. So it results, well, everybody know it and turn too slow in two xs and two ys as a gradient here. Oh, I forgot its mean for our picture. Let us draw some very sick x, y old plane and how does this vector look like? For example, assume that we have 0.11, and then we are looking at the vector 2, 2, or we are looking at a point, for example, 2, 1 and then we are looking at the vector. Let us carefully compute it. If it is 2, 1, then it's 2 times 2 and 2 times 1 which is 4, 2. It looks like this. Well, in other words, cells are the vectors which starts from the 0, 0 point as it seemed so. Maybe consider some more complicated example here, but I'm not going to try to draw a nice picture here. It's tricky because it's much more comprehensive function. So what do we need here? We need to once again, just calculate our partial derivatives and strengthening as into the vector. So towards x, it's a simple exponential functions. Thus, we get, sorry, towards x is just simple polynomial functions as we get x powered y minus 1 multiplied by the power, and although it's y exponential functions, thus we get a logarithm of x and multiply it by x power y. That's how our gradient looks like. It's quite easy, but it's nice to have one single concepts to describe the behavior of our function for all the possible points.