Problem 8 - Thomas Calculus

Question

In Exercises \(1-12,\) match the equation with the surface it defines. Also, identify each surface by type (paraboloid, ellipsoid, etc.) The surfaces are labeled (a ) \(-(1)\) . $$ z^{2}+x^{2}-y^{2}=1 $$

Step-by-Step Solution

Answer

The surface is a hyperboloid of one sheet.

Step 1: Analyze the Equation

The given equation is \( z^2 + x^2 - y^2 = 1 \). Notice that the equation has squares of the variables on both sides of the equality.

Step 2: Standard Form Identification

This equation can be rearranged to the standard form of \( z^2 + x^2 - y^2 = 1 \), which matches the form \( Ax^2 + By^2 + Cz^2 = D \). Here, \( A = 1, B = -1, C = 1, \) and \( D = 1 \).

Step 3: Compare with Known Quadric Surfaces

The equation resembles the standard form of a hyperboloid of one sheet, where both \( x^2 \) and \( z^2 \) have the same sign, and \( y^2 \) has the opposite sign.

Step 4: Identify the Surface Type

Since the equation is of the form \( x^2 + z^2 - y^2 = 1 \), this indicates a hyperboloid of one sheet. The defining feature is the different signs of the variables and the positive constant on the right side.

Key Concepts

Hyperboloid of One Sheet

A hyperboloid of one sheet is a fascinating three-dimensional surface that looks like a twisted cylinder. The unique aspect of this surface is its 'saddle' shape, where two sections curve outward and one section curves inward.
The general equation for a hyperboloid of one sheet is:

• \( \frac{x^2}{a^2} + \frac{y^2}{b^2} - \frac{z^2}{c^2} = 1 \)

In this form:

• \(a^2, b^2,\) and \(c^2\) are positive constants that determine the dimensions along the x, y, and z-axes, respectively.
• Notice the model includes both addition and subtraction. This is the hallmark nature of hyperboloids.

In simple terms, the positive signs indicate expansion along two axes, while the negative sign suggests contraction along the other axis.

Standard Form Identification

Identifying the standard form of an equation is crucial in recognizing the type of quadric surface it represents. For the given equation \( z^2 + x^2 - y^2 = 1 \), matching it to the standard form is fairly straightforward. The standard form is:

• \( Ax^2 + By^2 + Cz^2 = D \)

Here, each coefficient before the variables equals 1 or -1, and the equation is set to a constant 1 on the right side.

• \(A = 1\)
• \(B = -1\)
• \(C = 1\)
• \(D = 1\)

The hyperboloid's standard equation form helps us easily confirm its structure and orientation in space.

Equation Analysis

Analyzing the equation \( z^2 + x^2 - y^2 = 1 \) is like deciphering a code to understand the shape and nature of a surface. The first thing to notice is the combination of terms, with both \( z^2 \) and \( x^2 \) taking similar signs, while \( y^2 \) signs differ. This pattern confirms that it's a hyperboloid of one sheet.
To further break it down:

• The positive term \( z^2 + x^2 \) indicates axes that stretch outwards in their respective directions.
• The negative term \( -y^2 \) suggests an axis that's pulled inward or contracted.

This signature arrangement of positive and negative terms for squared variables reveals the hyperboloid’s symmetrical alignment and provides insight into the surface's actual appearance. The constant on the right side (1) governs the scaling of this geometric surface.