Question 11.3.1: Consider the function of two variables defined by the equati......
Consider the function of two variables defined by the equation
z = x² + y² (∗)
What are the level curves? Draw both a set of level curves and the graph of the function.
The variable z can only assume values ≥ 0. Each level curve has the equation
x² + y² = c (∗∗)
for some c ≥ 0. We see that these are circles in the xy-plane centred at the origin and with radius \sqrt{c}, as in Fig. 11.3.5.
As for the graph of (∗), all the level curves are circles. For y = 0, we have z= x². This shows that the graph of (∗) cuts the xz-plane (where y = 0) in a parabola. Similarly, for x = 0, we have z = y², which is the graph of a parabola in the yz-plane. In fact, the graph of (∗) is obtained by rotating the parabola z = x² around the z-axis. This surface of revolution is called a paraboloid, with its lowest part shown in Fig. 11.3.6. Five of the level curves in the xy-plane are also indicated.
