Problem 21 - Calculus Early Transcendentals: Pearson New International Edition

Question

The graph of an equation in \(x, y\), and \(z\) is symmetric with respect to the \(x y\)-plane if replacing \(z\) by \(-z\) results in an equivalent equation. What condition leads to a graph that is symmetric with respect to each of the following? (a) \(y z\)-plane (b) \(z\)-axis (c) origin

Step-by-Step Solution

Answer

(a) Replace \(x\) with \(-x\), (b) Replace \(x\) and \(y\) with \(-x\) and \(-y\), (c) Replace \(x\), \(y\), \(z\) by \(-x\), \(-y\), \(-z\).

Step 1: Understanding Symmetry in the y-z Plane

To check for symmetry with respect to the \(y z\)-plane, we need to see if replacing \(x\) with \(-x\) gives us an equivalent equation. This occurs because each point \((x, y, z)\) reflected over the \(y z\)-plane becomes \((-x, y, z)\). Thus, the condition required is that the equation remains unchanged when \(x\) is replaced with \(-x\).

Step 2: Understanding Symmetry in the z-axis

An equation is symmetric with respect to the \(z\)-axis if replacing both \(x\) with \(-x\) and \(y\) with \(-y\) results in an equivalent equation. This is because reflecting a point \((x, y, z)\) about the \(z\)-axis gives us \((-x, -y, z)\). Therefore, the condition for this symmetry is that the equation remains unchanged under the transformation \((x, y)\) to \((-x, -y)\).

Step 3: Understanding Symmetry in the Origin

For a graph to be symmetric with respect to the origin, replacing \(x\) with \(-x\), \(y\) with \(-y\), and \(z\) with \(-z\) must produce an equivalent equation. This stems from the fact that reflecting any point \((x, y, z)\) through the origin results in the point \((-x, -y, -z)\). Hence, the required condition is that the equation is unchanged when all three variables are replaced by their negatives.

Key Concepts

xy-plane symmetry

The concept of symmetry in the context of the xy-plane is an intriguing one. Symmetry here implies, essentially, a kind of mirror reflection, which results when you replace the variable \(z\) with \(-z\) in an equation. Consider this akin to looking into a mirror that's lying flat through the xy-plane. If the equation remains unchanged after this transformation, its graph is symmetric with respect to the xy-plane. This means all points with coordinates \((x, y, z)\) have corresponding points \((x, y, -z)\), highlighting their symmetric relationship.

• This condition is common in many physical and mathematical systems, often reflecting real-world scenarios where the system experiences uniform conditions above and below a particular plane.
• Examples of such symmetry can be seen in even functions of \(z\) that consist solely of even powers, like \(z^2\) or constant terms independent of \(z\).

Grasping this idea helps in understanding how systems and equations behave consistently along specific axes.

yz-plane symmetry

To determine symmetry with respect to the yz-plane, focus on what occurs when you replace \(x\) with \(-x\) in the equation. This is like taking each point plotted in three-dimensional space and reflecting it across the yz-plane, creating a mirror image on either side of this plane.

• If the equation remains identical after this swap, it suggests a symmetric property across the yz-plane.
• This symmetry implies that for every point \((x, y, z)\), there is an equal counterpart at \((-x, y, z)\).

Symmetry of this type can often be visualized in shapes and physical objects that do not change their structure when flipped left to right over the yz-plane. Understanding this concept is vital when analyzing equations to describe structures or phenomena in physics where such symmetry can simplify calculations or explain invariant properties.

z-axis symmetry

For z-axis symmetry, the process involves changing both \(x\) to \(-x\) and \(y\) to \(-y\). This results in symmetry around the z-axis, akin to spinning an object around a central vertical line and seeing no change in its appearance.

• Confirmed when every equation remains the same even after substituting both variables \(x\) and \(y\) with their negative counterparts.
• This symmetry results in each point \((x, y, z)\) having a reflected point \((-x, -y, z)\), preserving the same distance and position relative to the z-axis.

Z-axis symmetry is particularly significant in scenarios involving rotational symmetry, where an object or system looks the same even after being rotated. Grasping this understanding aids in modeling and analyzing symmetrical objects or patterns revolving around a central axis, which is common in many engineering and architectural designs.

origin symmetry

Origin symmetry in three-dimensional equations entails a complete negation of all three variables: \(x\), \(y\), and \(z\). Replace them with their negatives, and check if the equation remains consistent.

• This transformation is equivalent to turning an object around, comparing it to its inverted self through the center of its coordinate system.
• An equation maintaining its form under such conditions means that every point \((x, y, z)\) correlates with \((-x, -y, -z)\).

This demonstrates a more extensive type of symmetry, in which all dimensions are evenly represented around a central origin point. Understanding origin symmetry can be essential for specialists in fields like physics or chemistry, where origin-centered operations or transformations play a critical role in the study of molecular structures or forces acting symmetrically on a body.