Problem 40 - Calculus an Applied Approach

Question

Describe the traces of the surface in the given planes. $$ \text { Surface } \quad \text { Planes } $$ $$ y^{2}+z^{2}-x^{2}=1 \quad x y \text { -plane, } x z \text { -plane, } y z \text { -plane } $$

Step-by-Step Solution

Answer

The traces are hyperbolas in the xy-plane: \( y^{2}-x^{2}=1 \), and xz-plane: \( z^{2}-x^{2}=1 \) and a circle in the yz-plane \( y^{2}+z^{2}=1 \)

Step 1: Trace in xy-plane

Set \(z = 0\) in the surface equation: \( y^{2}+0^{2}-x^{2}=1 \). Simplify equation to get: \( y^{2}-x^{2}=1 \).

Step 2: Trace in xz-plane

Set \(y = 0\) in the surface equation: \( 0^{2}+z^{2}-x^{2}=1 \). Simplify equation to get: \( z^{2}-x^{2}=1 \).

Step 3: Trace in yz-plane

Set \(x = 0\) in the surface equation: \( y^{2}+z^{2}-0^{2}=1 \). Simplify equation to get: \( y^{2}+z^{2}=1 \).

Key Concepts

Traces in Coordinate Planes

Understanding the concept of traces in coordinate planes is essential for visualizing complex surfaces in three-dimensional space. A trace is essentially a cross-section of a 3D figure, given by setting one of the coordinates to zero. Consider the surface described by the equation \( y^{2} + z^{2} - x^{2} = 1 \).

When we want to find the trace in the xy-plane, we set \( z = 0 \) because this plane is defined by all points where \( z \) is zero. After substituting \( z = 0 \), the equation simplifies to \( y^{2} - x^{2} = 1 \), which represents a hyperbola in two dimensions. Similarly, for the xz-plane and yz-plane, setting \( y = 0 \) and \( x = 0 \) respectively, gives us additional 2D slices of our original surface. In each case, the resulting curve offers a simplified glimpse at the structure of the 3D object when viewed from a specific axis.

Solving 3D Space Equations

Solving 3D space equations involves setting one variable to zero to study the intersection of the surface with the respective coordinate plane. Let's demonstrate this with our given equation \( y^{2} + z^{2} - x^{2} = 1 \). When we solve for the trace in the xz-plane, we begin by letting \( y = 0 \) because the xz-plane is the 2D plane in space where the y-value is always zero.

Upon eliminating \( y \), our equation becomes \( z^{2} - x^{2} = 1 \). What you're seeing is not only a simpler equation but also a specific visualization of the surface in 3D space regarding the xz-plane. To convey this concept in a classroom or homework setting, one can utilize graphing software or 3D models to show how these traces reflect certain features of the original surface.

Surface Equations

Surface equations like \( y^{2} + z^{2} - x^{2} = 1 \) represent a relationship between three variables that, when graphed, produce a 3D shape or surface. The equation provided is an example of a hyperboloid, a type of quadric surface.

Different surfaces have different standard forms of equations - for spheres, cylinders, cones, etc., and each one provides a distinct set of geometric properties. When we analyze such surfaces by finding their traces, we're essentially taking a 'slice' of the surface by intersecting it with a plane, resulting in a 2D representation that can be easier to study and understand.

3D Calculus

In the realm of 3D calculus, understanding surface equations and their intersections (traces) with different planes is fundamental. This sub-field of calculus deals with problems involving rates of change and accumulation in three dimensions. Traces help us to break down a complex 3D surface into more manageable 2D curves.

For example, determining the area of a complex surface could involve integrating over these simpler 2D traces. Furthermore, should one need to find the volume under a given surface, the technique of stacking infinitesimally thin 'slices' or traces of the surface can provide the solution. The integration of traces is a powerful tool that leverages the concept of limits, which is a cornerstone of calculus.