 Hello, and welcome to this screencast on Section 9.1, Functions of Several Variables and Three-Dimensional Space, from the textbook Active Calculus Multivariable. This screencast is going to cover functions of several variables and representing functions of two variables. A function f of two independent variables is a rule that assigns to each ordered pair x, y, in some set D, exactly one real number, f of x, y. Up to this point, you've likely studied functions of a single variable in Calculus 1 and 2. While single variable functions take a single real number as input, functions of two variables take an ordered pair as input. Let's look at some examples. So the functions f and h here are functions of two independent variables. We can evaluate these functions at ordered pairs for which they are defined. For example, f evaluated at the ordered pair one negative four is equal to negative two. So f assigns the ordered pair one negative four the real number negative two. Similarly, we see that h of three two is equal to one seventh. Now, there's no reason to restrict ourselves to just two independent variables and defining functions. For example, we can define functions of three independent variables, x, y, and z, as we have here. And in general, we can define a function of however many independent variables that we'd like. As with functions of a single variable, it's important to understand the set of inputs. The domain of a function f is the set of all inputs for which the function is defined. As an example, consider the two variable functions f and h from before. Looking at the rule for f, we see it is defined for all ordered pairs x, y of real numbers. So the domain is going to be all ordered pairs of real numbers, which we can denote by our superscript two, which stands for two copies of the real numbers. On the other hand, the function h is not defined whenever x squared minus y is equal to zero. Therefore, the set of inputs for which h is defined is all ordered pairs such that x squared does not equal y. One of the techniques we use to study functions of a single variable is to create a table of values. We can do the same for functions of two variables, but our table will look slightly different. For example, consider this function f. Here is a table of some of f's values. The leftmost column of the table over here represents values of x, and the top row represents values of y. For example, we see that f evaluated at the ordered pair when x equals one and y equals 1.5 is given by this entry of the table. So when x equals one and y equals 1.5, the function outputs 0.2617. Similarly, f evaluated at the ordered pair when x equals two and y equals two is given by this entry of the table. We can also represent two variable functions using graphs. The graph of a function f is the set of all points that are ordered triples, where the first coordinate is the x value, the second coordinate is the y value, and the third coordinate is the value of the function at the ordered pair x, y. Note that points on this graph have three coordinates, and therefore these graphs are in three dimensions. We let our superscript three denote the set of all ordered triples of real numbers. We also often refer to the graph of the function f of two variables as the surface generated by f. To plot points in three dimensions, we need to set up a coordinate system with three mutually perpendicular axes. These axes are the x-axis, the y-axis, and the z-axis. We call these the coordinate axes. Pictured here are just the positive parts of these axes. And there are essentially two different ways we could set up a 3D coordinate system, as shown here. We call the coordinate system a right hand system. If we point the index finger of our right hand along the positive x-axis and our middle finger along the positive y-axis, then our thumb points in the direction of the positive z-axis. A similar idea holds for the left hand system. Following mathematical conventions, we're going to choose to use a right hand system throughout this course. Now, suppose we want to graph the function f that we looked at before. To do this, we're going to set up a right hand system. And now, in this graph, each ordered pair x, y in the domain of f gets assigned to a z-value, which is the output of our function f. For example, consider the ordered pair where x equals 0.5 and y equals 1. Our function f assigns this ordered pair a z-value. The z-value here is approximately 0.43. In doing this, this gives us a point with three coordinates and three space. The first coordinate is the x-value, the second is the y-value, and the third is the z-value. If we do this for all ordered pairs, x, y, and the domain of f, and we connect all the resulting points, this forms a surface. Pictured here, we've chosen to display what the surface looks like just for some non-negative x and y values. But we can also graph this function for negative x and y values in the domain. And to do that, to give you an idea of what that would look like, we can extend the coordinate axes in the negative direction and plot points from the surface there as well.