Visualizing surfaces in three dimensions. Review: Trying to draw a function of two variables
As a first step, we can plot the graph of f, which is a function of two variables. How to vizualize function of 2 variable? z -> [Graph] z=f(x, y) To determine the possible values of x and y, we can construct graph of this equation. As with equations involving a single variable, the possible values of x and y depend on the shape of this graph. We plot a point for each of them in the xy- plane, whose height is the value of a function of these parameters. So we’ll plot- let’s say z equals f(x, y)-and now that will actually become a surface in space. So for each value of x and y, here we have (x, y) in the xy- plane, then we’ll plot in the point in space at position (x y, z) equals f(x, y). If we combine all these points together, they will give us a surface that sits in space. Given function of two variables, how do we represent it geometrically? Let’s take our first example. We are given the function f of (x, y) equal to negative y. However, it does not depend on x, which is not a problem. It is still a valid function of x and y. [Graph] The function z=-y is constant with respect to x. Because the curve is a plane, we can draw its graph by examining the plane in space defined by z being equal to -y. If we want to draw the graph of z = -y, we can use the axes shown on the blackboard. If we look at what happens in the yz-plane, which is the plane of the blackboard, we will see that it will look like a line that goes downward with slope 1 If we change x, nothing happens because x does not appear in this equation. If instead of setting x equal to 0, we set x equal to 1 or minus 1, it still looks exactly the same. Now we have a plane that contains the x-axis and slopes downward with a slope of 1. It’s hard to draw; but you can see immediately what the big problem with graphs will be: They are hard to read.
