Problem 40 - Thomas Calculus

Question

Sketch the surfaces in Exercises \(13-76\) $$ \left(y^{2} / 4\right)-\left(x^{2} / 4\right)-z^{2}=1 $$

Step-by-Step Solution

Answer

The surface is a hyperboloid of two sheets, symmetric about the y-axis.

Step 1: Identify the Type of Surface

The given equation \( \left(y^{2} / 4\right)-\left(x^{2} / 4\right)-z^{2}=1 \) resembles the general form of a hyperboloid of two sheets.

Step 2: Rewrite the Equation

To make it clearer, rewrite the equation as \( \frac{y^2}{4} - \frac{x^2}{4} - z^2 = 1 \). Notice that the \( y^2 \) term and \( x^2 \) term both have the same denominator, indicating symmetry in the \( x-y \) plane.

Step 3: Analyze Cross Sections

Set \( z = 0 \), which results in the equation \( \frac{y^2}{4} - \frac{x^2}{4} = 1 \), this is a hyperbola in the \( x-y \) plane. If you set \( x = 0 \), the equation becomes \( \frac{y^2}{4} - z^2 = 1 \), another hyperbola in the \( y-z \) plane.

Step 4: Identify Symmetry

The surface is symmetric about the \( y \)-axis since the equation can be rewritten with \( y \) as having a positive coefficient while \( x \) and \( z \) have negative coefficients. This suggests the surface extends along the \( y \)-axis.

Step 5: Sketch the Surface

Using the information from the cross sections, sketch two opposing bowls (one below and one above the point \( y=0 \)). These open along the \( y \)-axis, forming a hyperboloid of two sheets. The intersection with the \( y \) axis is along the hyperboloid's \( y \) direction.

Key Concepts

Surfaces in Multivariable Calculus

In multivariable calculus, the concept of surfaces can be compared to exploring three-dimensional structures in space. One such surface, highlighted by the equation \( \frac{y^2}{4} - \frac{x^2}{4} - z^2 = 1 \), is the hyperboloid of two sheets. This surface is a fascinating example due to its unique structure of two separate, opposing bowls that open in opposite directions. When we analyze its mathematical representation, we can glean insights about the nature of three-dimensional space.

• The signs and coefficients of each squared term in the equation give clues about the shape's structure and orientation.
• The term with a positive coefficient (in this case \( y^2 \)) indicates the axis along which the surface opens.

Understanding surfaces like the hyperboloid of two sheets helps build a foundation in comprehending how equations can represent complex three-dimensional forms.

Cross Sections of Surfaces

When exploring surfaces, a useful tool is analyzing cross sections, which are basically slices of the surface taken parallel to various planes. For the hyperboloid of two sheets represented by the given equation, examining these cross sections enriches our understanding of its shape.

• Setting \( z = 0 \) transforms the equation into \( \frac{y^2}{4} - \frac{x^2}{4} = 1 \), sketching a hyperbola in the \( x-y \) plane.
• Likewise, setting \( x = 0 \) changes it to \( \frac{y^2}{4} - z^2 = 1 \), which is a hyperbolic curve in the \( y-z \) plane.

Understanding these cross sections helps in visualizing the surface's complexity. It's like examining a sculpture by viewing it from different angles; each cross-sectional view reveals a new aspect of the overall shape.

Symmetry in Surfaces

Symmetry is a fundamental concept in geometry that greatly simplifies the understanding and representation of surfaces. In the equation \( \frac{y^2}{4} - \frac{x^2}{4} - z^2 = 1 \), symmetry plays a key role. The surface shows symmetry with respect to the \( y \)-axis due to the positive coefficient of \( y^2 \).

• Symmetry about the \( y \)-axis implies that the shape of the surface is mirrored on either side of this axis.
• Furthermore, the identical coefficients of \( x^2 \) and \( y^2 \) suggest rotational symmetry along the \( y \)-axis.

This symmetry offers a clearer insight into the structure's design and aids in predicting its behavior in any mathematical or physical context where such shapes might appear.