Problem 33 - Thomas Calculus

Question

Sketch the surfaces in Exercises \(13-44.\) ASSORTED $$z=1+y^{2}-x^{2}$$

Step-by-Step Solution

Answer

The surface is a hyperbolic paraboloid, resembling a saddle shape.

Step 1: Analyze the given equation

The equation given is \( z = 1 + y^2 - x^2 \). This is a quadratic equation involving variables \(x\) and \(y\), indicating the equation is related to a surface.

Step 2: Identify the type of surface

The given equation resembles the standard hyperbolic paraboloid form, which can be expressed as \( z = Ax^2 + By^2 + C \). Here, \(A = -1\), \(B = 1\), and \(C = 1\), signaling a saddle surface shape, typical for a hyperbolic paraboloid.

Step 3: Determine the intercepts

To find intercepts, set one variable to zero and solve for \(z\). - When \(x = 0\), \( z = 1 + y^2 \), a parabola opening upwards in the \(zy\)-plane.- When \(y = 0\), \( z = 1 - x^2 \), a parabola opening downwards in the \(zx\)-plane.- At \( z = 0 \), \( y^2 = x^2 - 1 \). Positive values of \(y^2\) appear outside \(x = \pm 1\).

Step 4: Sketch the surface based on observations

Using the intercepts and understanding the standard form of a hyperbolic paraboloid, start by drawing the parabolic sections:- Parabolas intersecting at the origin: opening upward along \(y\)-axis and downward along \(x\)-axis.- Extend the curves and connect to form the saddle-like shape characterized by a minimal point (saddle point) at the origin (\(0,0,1\)).

Step 5: Label and finish the sketch

Label the axes \(x\), \(y\), and \(z\). Clearly mark the parabolic sections along respective planes. Indicate the saddle point and note the nature of the surface being a hyperbolic paraboloid with the center at \((0,0,1)\). This can be a rough representation of how the surface behaves across the coordinate planes.

Key Concepts

Hyperbolic Paraboloid

A hyperbolic paraboloid is a fascinating and unique 3D surface in multivariable calculus, often referred to as a saddle surface.
Its curvature properties are distinct in that it curves upward in one direction and downward in another. This unique combination of curves gives the hyperbolic paraboloid its saddle-like shape. The standard equation for a hyperbolic paraboloid can be expressed as \( z = Ax^2 + By^2 + C \), where the coefficients \( A \) and \( B \) determine the direction of the opening of the parabolas on the coordinate planes.

• If \( A \) is positive and \( B \) is negative, the parabola opens upwards along the \( y \)-axis and downwards along the \( x \)-axis, or vice versa depending on their signs.
• These opposite curvatures are the defining feature of hyperbolic paraboloids.

In the given exercise, the equation \( z = 1 + y^2 - x^2 \) shows a hyperbolic paraboloid where:

• It opens upwards along the \( y \)-axis and downwards along the \( x \)-axis, forming a saddle.
• The saddle point or the lowest point in the z-direction in this case is at \((0, 0, 1)\).

Surface Sketching

Sketching the surface of an equation like a hyperbolic paraboloid can help in visualizing how the surface behaves in space.
Understanding the parabolic cross-sections relevant to the equation is the key to accurate sketching. When tackling \( z = 1 + y^2 - x^2 \), first focus on the cross-sectional curves:

• In the \( zy \)-plane with \( x = 0 \), it becomes a parabola \( z = 1 + y^2 \), curving upwards.
• In the \( zx \)-plane with \( y = 0 \), it becomes \( z = 1 - x^2 \), curving downwards.
• These curves intersect at the surface's center, showcasing the saddle point.

Visualizing these intersections helps in outlining the saddle shape.
Additionally, plotting intercept points can guide where the surface crosses the planes and how these curves combine to form the 3D saddle shape.
Remember to label axes and note crucial points like the saddle point for clarity.

Quadratic Surfaces

Quadratic surfaces are 3-dimensional surfaces represented by second-degree polynomial equations in three variables. These surfaces can take various forms such as ellipsoids, paraboloids, and hyperboloids.
The hyperbolic paraboloid discussed in this exercise is a type of quadratic surface.
Quadratic surfaces are broadly categorized by their general equation: \[ Ax^2 + By^2 + Cz^2 + 2Dxy + 2Exz + 2Fyz + Gx + Hy + Iz + J = 0 \]Certain coefficients will determine the specific type of quadratic surface.

• When a surface has mixed positive and negative coefficients among \( A, B, C \), it often results in a hyperbolic surface, like the one in our exercise.
• Understanding these coefficients and their arrangements can help identify the surface type quickly.

By recognizing these patterns, students can practice and improve their ability to identify and sketch various quadratic surfaces efficiently.
This is crucial in multivariable calculus, as these surfaces appear frequently across different topics and applications.