Functions of Two Variables. Worked Examples

Functions of two variables. Worked examples In this next example, we will sketch the graphs of three-dimensional functions. So z here as a function of x and y. On this second one, z is also a function of x and y,it just happens not to depend on y. When you graph these, we'd suggest consideringthe x axis and the y axis separately. So what happens if you consider x equals 0 or ifyou consider z equals 0? As you graph these, let's see what you can do. So why don't we start by examining this function, z-squared equals x squared plus ysquared. So x is pointing toward us, y to the right, and z up. A nice way to get startedwith these problems is to just try setting the variables x and y variously equal to zeroand then see what happens when we do so; this should give us a surface instead ofa curve and we'll see what curve we get. For instance, if we set x equal to zero, thenwe just get z is the square root of y squared. So we see that z is the absolute value of y. This means that whatever the surfacelooks like, we know what it would look like if we sliced it in the plane of theblackboard. We know that it just looks like—this is just the graph of y equals 0, zequals 0. So, now if you think about it, what we just said works just as well for xinstead of for y. So if were to graph this in the xz-plane, where we set y equals 0,then we would get. Let's draw the graph of y equals x³ + y² in blue and the graph of y equals 4 – x² inwhite. If we set z equal to 0, there is one solution, which is this point here. But whatwill be interesting is if we set z equal to some positive value such as 2. So if we saidthat z was equal to 2, then we have 2 = x³ + y². Solving this is equivalent to sayingthat x³ + y² = 4.

So, at the height of 2, we will have a circle with radius 2. This is just the equation fora circle with radius 2. And so at height 2, we just have a circle. And actually as youcan see, there's nothing special about 2; at every height, we're just going to have acircle. So this is what's called a cone. Now for B, we can expect something funny to happen when we go over here to bbecause it doesn't depend on y. Let's see if we can see how the fact that z doesn'tdepend on y enters into our picture. So we'll just walk over here and we'll consider zequals x squared.So again, we have our x-axis, x-, y- and z-axes. Now, let's consider what this looks like when we intersect the xz- plane. When yequals 0, the equation does not change. Therefore, z equals x squared. We knowwhat that looks like: it is a parabola. It lies in the xz- plane and goes in and out of theboard. But now if you think about it, what it means to say that this function does notdepend on y is that we have the exact same picture at every value of y. So if we goout here, then we are going to have the same picture. If we go over here, we aregoing to have the same picture. And in fact what you will get is a prism.The graph of a parabola can be made into a rectangular prism by stretching out itsshape. We could call this a prism of a parabola. Now let's see if we can get more insight from these two pictures. So look whathappened in this instance: as you vary y, the picture had to be unchanged. So hereare the function z; it obviously did not depend on y.

The fact that the equation of a cone is z equals x squared plus y squared and thefact that this function is not dependent on y are one in the same fact. Now, if we goover to the prism. Here our function z very much depends on both x and y. But noticethat it depends on x and y only in the sense that z equals the radius r which is equalto x squared plus y squared.Therefore, it is because the dependence of z on x and y can be rewritten as adependence on r that this surface has radial symmetry. As in translation symmetry ofthe prism, we can expect that if the dependence of z on x and y can be rewritten as adependence on r, then we will get radial symmetry.