Problem 60 - Thomas Calculus

Question

Sketch the surfaces in Exercises \(13-76\) $$ z=x^{2}-y^{2}-1 $$

Step-by-Step Solution

Answer

The surface is a hyperbolic paraboloid, shifted down by 1 unit.

Step 1: Identify the Type of Surface

The equation given is \( z = x^2 - y^2 - 1 \). This equation resembles the standard form of a hyperbolic paraboloid given by \( z = x^2/a^2 - y^2/b^2 \). However, our equation has constants \( a = b = 1 \) and a constant \(-1\) added.

Step 2: Determine the Orientation

In a hyperbolic paraboloid \( z = x^2 - y^2 \), the surface is saddle-shaped, curving upwards along the \( x \)-axis and downwards along the \( y \)-axis. The constant \( -1 \) will shift the entire surface downwards by one unit.

Step 3: Analyze Cross Sections

Set \( x = k \) for various constants \( k \). The cross sections will be parabolas of the form \( z = k^2 - y^2 - 1 \), opening downwards. Set \( y = k \), resulting in parabolas \( z = x^2 - k^2 - 1 \) opening upwards. These cross-sections help visualize the saddle shape.

Step 4: Plotting the Surface

To sketch the surface, plot a few key cross-sections such as when \( x = 0 \) (giving \( z = -y^2 - 1 \)) and when \( y = 0 \) (providing \( z = x^2 - 1 \)). The intersection of these curves and symmetry about the axes verifies the shape. The center is at \((0, 0, -1)\).

Key Concepts

3D Surfaces

In mathematics, three-dimensional (3D) surfaces are fascinating shapes that exist in a three-dimensional space. They are defined using an equation with three variables, often represented as functions connecting variables like \( x \), \( y \), and \( z \) together. A 3D surface can be curved or flat and comes in various forms, such as planes, spheres, or paraboloids. For the hyperbolic paraboloid defined by \( z = x^2 - y^2 - 1 \), understanding how each coordinate impacts the surface is key. Here, the shape relies on the balance of the squares of \( x \) and \( y \), which together define its saddle shape. The constant \", \(-1\), simply modifies the height of the surface, shifting it downwards in the \( z \)-axis by one unit. This creates the platform, or level base, of the overall structure. Recognizing these fundamental principles helps in comprehending not just lines or planes, but entire 3D surfaces.

Cross Sections

Cross sections are slices or intersected views of a 3D surface. By slicing the hyperbolic paraboloid \( z = x^2 - y^2 - 1 \), you can determine its shape along different planes. Depending on whether you hold \( x \) or \( y \) constant, the resulting cross sections form parabola shapes.

• If \( x = k \), the resulting equation is \( z = k^2 - y^2 - 1 \). This depicts parabolas that open downward.
• If \( y = k \), you'll find \( z = x^2 - k^2 - 1 \), which shows parabolas opening upward.

These cross-sectional views are immensely helpful for visualizing what a saddle shape looks like, making it easier to sketch or graph. By understanding how these sections intersect and interact, one can achieve a more profound comprehension of the entire 3D surface and its defining characteristics. This method demystifies the 3D complexity by breaking it down into comprehensible 2D views.

Saddle Shape

The term "saddle shape" describes a specific kind of surface often seen in hyperbolic paraboloids like \( z = x^2 - y^2 - 1 \). This shape resembles a horse saddle due to its distinct curvature properties. The surface curves upward in one direction while curving downward in another, giving it a unique and intriguing geometry. In the context of our equation:

• Along the \( x \)-axis, the curve rises like a bowl (upward parabola).
• Along the \( y \)-axis, it dips like a valley (downward parabola).

The interaction between these opposing curves is what gives rise to the saddle appearance. The lowered center point (at (0, 0, -1)) marks the minimum point of the saddle, while the sides perpetually rise or sink as they extend along the respective axes. This saddle shape is not only visually distinct but plays a significant role in fields such as calculus, architecture, and engineering, where understanding of such complex, interdependent surfaces is essential.