[MUSIC] You shouldn't think that any set should be classified as being either open or closed. For example, let's consider, A set in the coordinate plane such that the first coordinate is strictly greater than zero and the second is non negative. We just need the first quadrant. Now, what we see is that this is neither open nor closed set. Why it's not open? It's not open because if I choose somewhere the point here. This is a point which belongs to the set. But whenever I choose any ball centered here, Doesn't depend on the radius. Radius can be taken as a small number. This ball doesn't lie within the set. It goes beyond the border. So, that means that this set is not open, but, also it's not closed. Because, I can easily construct a conversion sequence which is coming closer. These are the terms they are approaching, this part of the border. So the limit lies on this part of the boundary, but this part of the boundary doesn't belong to this set. So this is an example of a set which is not open, not closed. We talked about convergent sequences. We remember from the single calculus, that there are operations on convergent sequences. For example, given two sequences which are convergent to their respective limits. What can we say about the sum or a difference or a list of sequences. We can multiply all terms or we can convergent sequence by some number. Is it true that the resulting sequence will be convergent? All such facts also are true for n-dimensional space. For example, if we have two, Sequences, the first, M ranges from one to infinity, and the second, And we suppose both are convergent so they have the limits. When M turns to infinity, the limit equals X is 0. And if M turns to infinity, The limit of this sequence equals Y zero. We know how to add up vectors, so what will be the sum of two sequences? We take m terms, and we get the sum by simply adding up the terms. So this is a resulting sequence. And it can be proven that this is a convergent sequence with limit equals the sum Of limits. The same true for the difference. So, I also can add here a minus symbol. And I also can form a sequence by multiplying. Let me choose xn sequence, and let me multiply by some scalar. So I have a sequence made of terms of their Initial sequence XM. The limit will definitely be equal to lambda x0. Now we start exploring functions in rn. We're used to a function when there is just one independent variable. So usually We wrote y, dependent variable, is a function of x, where x belonged to Some set D, which is called the domain, and D is the set of real numbers. Now we need to get you used to a function which depends on many variables now. So, now a point x is from n-dimensional space,. And we'll indicate it using notation. Y dependent variable is a real number, as earlier, here we write f, and we won't be using colons. We'll be writing, which is more convenient, in one line I said x belongs to rne But actually, we always indicate some domain. Domain is the set of point from rn and sometimes we deal with the natural domains. For example, let's consider a function like this, y = this is the square root of 1- x1 squared- x2 squared. A function depends on x1 and x2. Well clearly, D is not the whole coordinate plane that would just a circle. And I'll write an inequality. According to our previous definition, this is a closed ball or rather circle in coordinate plane, centered at zero who's radius is one. And this is a national domain for this particular function. Now, when we draw x from the domain and substitute into the, by the way, the totals of arguments, we have an arguments for each such function. We calculate Y value and Y value is a real number. So when we consider all such Y values, we get the range for the function. For example, if we go back to this example the function, what will be the range of the function. Square root of 1 minus x1 squared minus x2 squared. The range in this particular example will be as can be easily seen enclosed into. So this is the range of this function. Graphical representation of functions of many variables or we explore level curves. What is a level curve? We're considering the case of two independent variables, so n=2. Then a function y is a function of x1 and x2. If we fix Y value in which using x1, x2 from the domain of this particular function. For given value of Y, we get a curve and at any point of this curve, The value of the function is the same which equals y. Let's consider a particular example, for example, y equals the product of x1 times x2. And, I'll be considering domain, Which is letter D, the first quadrant. Although the domain of this function is the whole coordinate plane, so the whole r2, but I am interested in drawing level curves which belong to the first quadrant. Now if I choose a particular value for y, let it be one. I can draw this curve, and this curve, we know, is called hyperbola. Looks like that. If I change y value into something else, let it be two, I also get a hyperbola. And change again, this time I can choose a value smaller than one, let it be one-half. So each time I have some curve, and this curve is called the level curve. Well the name is from problem geography drone. So if I choose any point on this curve the y value will you be the same. And this is especially helpful, because otherwise I would have to draw the graph of this function. But the graph of this function won't be necessarily three dimensional. Because x1, x2 these are coordinates of a point from the coordinate plane. And y is a value and I plot this value along the third dimensional axis of which probably should point at me. Level curves widely used in economics. For example, in the consumer theory, we consider a utility function of a consumer. So when we write u, representing utility function, where x1 is the quantity of the first good consumed and x2 is the quantity of the second good consumed. And when we fix the value of the utility at some level, u bar, we can draw the curve, which will look exactly like this. If this utility function is provided by the same formula x1, x2. These level curves for utility function are called indifference curves in microeconomics. Indifference curves. And clearly, they can have any shape, depending on the utility function itself. And later on, we'll try to explore such indifference curves in more detail. Also microeconomics, we get level curves when we consider production functions. Production functions, if we consider only two factors of conduction like capital and labor, is the function whose value Q denoting the quantity of the produced goods. This is the function of labor denoted by capital L and capital K. And when we fix the quantity produced Q at some level, then we draw a similar level curve. Now it has a different name. It' s called, the level curve for a production function is called isoquant. Once again, we can choose a similar function. So the function becomes Q=LK. And when we draw these isoquants, we get the same picture, a family of hyperbolas. The bigger the Q, the output, the farther away from the origin is such a hyperbola.