Functions of Two Variables

Functions of two variables Examples: Functions of several variables f (x, y) = x2 + y 2 ⇒ f (1, 2) = 5 etc. f (x, y) = xy 2 ex+y f (x, y, z) = xy log z Ideal gas law: P = kT /V . Dependent and independent variables In z = f (x, y) we say x, y are independent variables and z is a dependent variable. This indicates that x and y are free to take any values and then z depends on these values. For now it will be clear which are which, later we’ll have to take more care. Graphs For the function y = f (x): there is one independent variable and one dependent variable, which means we need 2 dimensions for its graph. Graphing technique: go to x then compute y = f (x) then go up to height y. For z = f (x, y) we have two independent and one dependent variable, so we need 3 dimen­ sions to graph the function. The technique is the same as before. Example: Consider z = f (x, y) = x2 + y 2 . To make the graph: go to (x, y) then compute z = f (x, y) then go up to height z. √ We show the plot of three points: f (0, 0) = 0, f (1, 1) = 2 and f (0, 2) = 2. z� ... . . .. . ... . . . . . . ... . .. . • • • x� • (1,1) • √ 2 y� The figure above shows more than just the graph of three points. Here are the steps we used to draw the graph. Remember, this is just a sketch, it should suggest the shape of the graph and some of its features. 1. First we draw the axes. The z-axis points up, the y-axis is to the right and the x-axis comes out of the page, so it is drawn at the angle shown. This gives a perspective with the eye somewhere in the first octant. 2. The yz-traces are those curves found by setting x = a constant. We start with the trace when x = 0. This is an upward pointing parabola in the yz-plane. √ 3. Next we sketch the trace with z = 3. This is a circle of radius 3 at height z = 3. Note, the traces where z = constant are generally called level curves. This is enough for this graph. Other graphs take other traces. You should expect to do a certain amount of trial and error before your figure looks right.