Problem 20 - Essential Calculus Early Transcendentals

Question

Use traces to sketch and identify the surface. \(x=y^{2}-z^{2}\)

Step-by-Step Solution

Answer

The surface is a hyperbolic paraboloid due to its saddle shape.

Step 1: Identify the Surface Equation

The given surface equation is a hyperbolic paraboloid, determined by the form of the equation, which is \[ x = y^2 - z^2 \]. This indicates a saddle shape, as it is a standard form of a hyperbolic paraboloid.

Step 2: Finding the Trace in the xy-plane

To find the trace in the xy-plane, we set \(z = 0\). After substitution, the equation reduces to \[ x = y^2 \]. This is the equation of a parabola opening along the positive x-axis.

Step 3: Finding the Trace in the xz-plane

For the trace in the xz-plane, we set \(y = 0\). This gives us the equation \[ x = -z^2 \]. This is another parabola, but it opens along the negative x-axis, showing part of the saddle form.

Step 4: Finding the Trace in the yz-plane

To find the trace in the yz-plane, set \(x = 0\). The equation transforms to \[ 0 = y^2 - z^2 \] or \[ y^2 = z^2 \]. This simplifies to two lines \( y = z \) and \( y = -z \), illustrating a hyperbola in this plane.

Step 5: Sketch the Surface

Combine the observations from each plane. The xy-plane gives an upward opening parabola, the xz-plane gives a downward opening parabola, and the yz-plane provides intersecting lines at a hyperbola. Together, these traces reflect a saddle-shaped surface, confirming it is indeed a hyperbolic paraboloid.

Key Concepts

Surface Sketching

In surface sketching, our aim is to draw a three-dimensional representation of a mathematical surface given its equation. For the hyperbolic paraboloid described by the equation \( x = y^2 - z^2 \), surface sketching brings together the information from different plane intersections to form a comprehensive visualization.

Initially, we need to understand that a hyperbolic paraboloid has a distinctive saddle shape, meaning it curves upwards in one direction and downwards in a perpendicular direction. By combining traces from the xy-plane, xz-plane, and yz-plane, we can create a sketch that captures the complex shape of this surface. Each trace provides crucial insights into how the surface behaves in different slices and guides us in forming the full 3D sketch.

Traces in Coordinate Planes

Traces in coordinate planes reveal how a surface intersects with specific planes such as the xy-plane, xz-plane, and yz-plane. These intersections transform the 3D equation into familiar 2D shapes that are easier to analyze. Let's break it down:

• **Trace in the xy-plane**: By setting \( z = 0 \), the equation \( x = y^2 \) emerges, showing a parabola that opens along the x-axis.
• **Trace in the xz-plane**: Here, set \( y = 0 \) to get the equation \( x = -z^2 \), another parabola, but this time opening in the opposite direction.
• **Trace in the yz-plane**: Setting \( x = 0 \), we find \( y^2 = z^2 \), resulting in intersecting lines \( y = z \) and \( y = -z \), forming a hyperbola.

Understanding the shapes that arise in each trace helps piece together the overall shape of the hyperbolic paraboloid.

Saddle Shape

The term "saddle shape" is often synonymous with hyperbolic paraboloids due to their characteristic form. Imagine a saddle resting on horseback, arching upwards and downwards simultaneously along perpendicular axes. This form is innate to hyperbolic paraboloids like \( x = y^2 - z^2 \).

Recognizing the saddle shape is crucial when identifying a hyperbolic paraboloid from multiple traces. The xy-plane and xz-plane provide parabolas opening in opposite directions, creating the appearance of this saddle or "swooping" peculiar form. Meanwhile, the yz-plane adds the necessary complexity through hyperbola-like intersections, confirming the 3D nature of this unique shape.

Surface Equation Identification

Identifying a surface equation involves analyzing its form and recognizing standard patterns found in mathematical definitions. The equation \( x = y^2 - z^2 \) exemplifies a hyperbolic paraboloid. Here's how:

• This equation is quadratic in terms of \( y \) and \( z \) and linear in \( x \), a common trait of saddle shapes.
• Notice the difference of squares \( y^2 - z^2 \), which often characterizes hyperbolic paraboloids, since they alternate in directions for curvatures.

With these observations, students can efficiently link this equation to its 3D representation, recognizing the hyperbolic paraboloid by its defining saddle shape properties. Understanding this setup is vital for acknowledging surface behavior from traces.