 Hello and welcome to this second screencast on section 9.1, functions of several variables and three-dimensional space. This screencast is going to cover traces, contour maps, and level curves. As you probably deduced from our last video, understanding trends and functions of two variables can be challenging, as can sketching their graphs, like the one pictured here. To help us with these tasks, we introduce traces. A trace of a function f of two independent variables x and y is a curve that either fixes the x-coordinate to be some constant c and lets y vary, or it fixes the y-coordinate to be some constant c and lets x vary. For example, consider the function f graphed on the previous slide. If we hold x constant at 1, we can consider f of a function of y alone. This is the trace of f where x equals 1. Similarly, if we hold y constant at 0.5, we can consider f as a function of x only. This is the trace where y equals 0.5. Now, these are just two examples of traces. We can choose other values of x or y to hold constant, and this would give us other traces to consider. These values can be chosen to help us better understand trends in our graph. Pictured here, the red curve shows the trace of f where x equals 1. You can think of this as slicing our graph on the line x equals 1. Such a trace can help us better understand what happens to f as y changes and x is held constant at 1. And pictured here, the red curve shows the trace of f where y equals 0.5. Such a trace can help us better understand what happens to f as x changes and y is held constant at 0.5. Another notion that can better help us understand trends and functions of two variables are level curves, also known as contours. A level curve or contour of a function f of two independent variables is a curve of the form k equal to f of x, y, where k is some constant. Notice a level curve is obtained from holding z values, or the outputs of the function, constant. You might have seen this idea before if you've ever come across a topographical map. These are maps of places in the world often involving mountains, where curves on the map show regions of constant altitude. Consider again the function f. Here are three examples of level curves. The first example fixes z at negative 0.4. This curve will depict all values of f where z is equal to negative 0.4. Similarly, the second and third examples will depict all values of f where z equals 0.1 and z equals 0.3 respectively. Pictured here on the left, we have graphed various level curves of the function f, most of which are labeled by the z value that was held constant. On the right, we have superimposed these on the graph of f. As this shows, looking at various level curves can give us information about the function f. For example, in the first quadrant of the graph, the level curves suggest that the graph of f increases as we approach the point where x equals 0.7 and y equals 1. Since the level curves that we cross as we approach this point have increasing z values no matter which direction we come from. This trend is confirmed by examining the graph of the surface on the right and looking in the first quadrant.