91Ó°ÊÓ

In this exercise, we start with analyzing the equation given: \( yz = 1 \). Here, the equation expresses a relationship between the variables \( y \) and \( z \). Their product is always equal to 1. This shows an interesting interaction: as \( y \) changes, \( z \) must adjust to maintain the equation's balance. Since the equation lacks an \( x \) term, the surface does not change with \( x \) values.By rearranging the equation to \( z = \frac{1}{y} \), we see that \( z \) varies concerning \( y \) in a hyperbolic manner. If you input different values for \( y \), you'll see corresponding reciprocal values for \( z \). This kind of relationship hints at a hyperbolic surface. Given that we only analyze the \( yz \) relationship, we're handling a two-variable function. This helps in understanding the core characteristics of the hyperbolic shape before we delve deeper.

Let's plot this relationship on a two-dimensional plane formed by the axes \( y \) and \( z \). The equation \( z = \frac{1}{y} \) implies that as \( y \) moves toward zero from either the positive or the negative side, \( z \) becomes very large in magnitude. This behavior is typical for hyperbolas.In this plot:

• As \( y \) becomes positive and small, \( z \) increases positively and significantly.
• As \( y \) is negative and close to zero, \( z \) decreases negatively, becoming quite large in the negative sense.
• Hyperbolas have two distinct curves, reflecting the two branches of this figure.

Recognizing these branches and their trailing off towards infinity is crucial. They form a signature symmetrical appearance that helps identify the nature of the surface-forming equation.

Moving to three dimensions, we consider how the hyperbolic shape extends in a 3D space. The given equation \( yz = 1 \) lacked an \( x \) variable, indicating that any \( yz \) pair satisfying the equation can be matched with any \( x \) value. Consequently, for every valid \( y, z \) combination, an entire line parallel to the \( x \)-axis can be drawn.This property leads to the formation of a hyperbolic cylinder. Essentially, imagine each 2D curve lying flat on its plane being stretched indefinitely along the \( x \)-axis. This infinite extension results in a collection of such lines, forming what is known as a hyperbolic surface.To visualize, think of each branch of the hyperbola as a tunnel stretching endlessly in the \( x \) direction. This process solidifies the concept of turning two-dimensional equations into three-dimensional structures—integral for understanding spatial surfaces.