Problem 47 - Thomas Calculus

Question

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

Step-by-Step Solution

Answer

The surface is a hyperbolic paraboloid with parabolas opening in opposite directions.

Step 1: Identify the Surface Type

The given equation \( z = 1 + y^2 - x^2 \) is a quadratic equation in three variables. This is a standard form of a hyperbolic paraboloid, recognizable by the presence of both \( y^2 \) (positive) and \( -x^2 \) (negative) terms.

Step 2: Identify Cross Sections

To understand what the surface looks like, identify cross-sections of the surface by setting \( x \) or \( y \) to a constant value. For example, when \( y = k \), \[ z = 1 + k^2 - x^2 \]This represents a family of parabolas opening downward along the \( x \)-axis.Similarly, when \( x = h \), \[ z = 1 + y^2 - h^2 \]This represents a family of parabolas opening upward along the \( y \)-axis.

Step 3: Identify the Axis of Symmetry

Notice that the surface is symmetric with respect to the \( z \)-axis at \( x=0 \) and \( y=0 \). This symmetry helps in sketching the surface accurately.

Step 4: Identify Intersections with Coordinate Planes

Check where the surface intersects the coordinate planes:- In the plane \( z=0 \):Set \( 1 + y^2 - x^2 = 0 \), which implies \( y^2 = x^2 - 1 \). This involves imaginary solutions for most values, meaning no real intersection when \( z = 0 \) unless \( x = \pm 1 \) and \( y = 0 \) (intersection in x-direction edges).- In the plane \( xy \):Since \( x = 0 \) or \( y = 0 \), curves are formed but don't intersect z=0, merely showing direction.

Step 5: Sketch the Surface

Using all information, sketch a saddle-shaped surface. The graph should show parabolas opening in opposite directions. The general shape will be a saddle where the surface curves upward along the \( y \)-axis and downward along the \( x \)-axis.

Key Concepts

Quadratic Surfaces

Quadratic surfaces are three-dimensional surfaces defined by a second-degree polynomial equation in three variables, typically expressed as \( ax^2 + by^2 + cz^2 + dxy + exz + fyz + gx + hy + iz + j = 0 \). These surfaces can have varied and interesting shapes, making them a crucial part of studies in multivariable calculus and geometry. Some common types of quadratic surfaces include ellipsoids, hyperboloids, and paraboloids.

A hyperbolic paraboloid, as seen in the equation \( z = 1 + y^2 - x^2 \), is one type of quadratic surface. It features a combination of positive and negative squared terms, leading to a distinctive saddle shape. Unlike an ellipsoid which is closed and smooth, the hyperbolic paraboloid is open and has regions curving in opposite directions. Recognizing these surfaces requires identifying the characteristic squared terms in their equations.

Cross Sections

Cross sections are incredibly useful for understanding the shape of three-dimensional surfaces. A cross section is essentially a two-dimensional slice of a surface. By examining cross sections, one can gain insights into how the surface behaves in different directions.

• When we set \( y = k \) in our hyperbolic paraboloid equation \( z = 1 + k^2 - x^2 \), we find a set of parabolas that open downwards. These are called vertical cross sections along the \( y \)-axis.
• Similarly, when \( x = h \), the equation becomes \( z = 1 + y^2 - h^2 \), producing parabolas opening upwards along the \( x \)-axis. These portray the surface's behavior in the opposite direction.

By analyzing these cross sections, one can better visualize the overall 3D shape of the surface, helping sketch it more accurately.

Axis of Symmetry

The axis of symmetry is a critical feature in understanding the symmetry of a surface. For the hyperbolic paraboloid \( z = 1 + y^2 - x^2 \), symmetry plays a major role in determining its shape.

• This surface is symmetric about the \( z \)-axis at \( x = 0 \) and \( y = 0 \). In simpler terms, if you were to take the surface and rotate or flip it around these axes, it would look the same.

The axis of symmetry simplifies the process of graphing these surfaces, as it helps to predict how different regions relate to each other. If a surface is symmetric along a certain axis, then features found on one side of the axis will mirror those on the other, reducing the complexity involved in sketching.

Intersections with Coordinate Planes

Understanding where a surface intersects the coordinate planes can provide a clearer picture of its behavior in space. These intersections provide points and lines that can be crucial for sketching the surface.

• For the given surface \( z = 1 + y^2 - x^2 \), checking intersections with the \( xz \) and \( yz \) planes does not yield straightforward lines or parabolas due to imaginary solutions. Instead, the plane \( z = 0 \) shows intersections of \( x = \pm 1 \) and \( y = 0 \), highlighting edges where the surface touches or crosses this plane.
• In the \( xy \) plane, when \( z = 0 \), no simple intersections occur but directional parabolas form.

These intersections help define the boundaries and height limits, essential components in conveying the overall layout and structure of the surface in a graph.