Problem 65 - Thomas Calculus

Question

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

Step-by-Step Solution

Answer

The surface is a downward-opening paraboloid with vertex at the origin.

Step 1: Identify the Type of Surface

The given equation \(z = -(x^2 + y^2)\) is a variation of the standard form of the equation of a paraboloid. Specifically, it is a downward-opening paraboloid because it is expressed in the form \(z = -r^2\), where \(r^2 = x^2 + y^2\).

Step 2: Recognize the Characteristics of the Paraboloid

This surface is symmetric around the z-axis. As \(x\) and \(y\) increase or decrease, the value of \(z\) becomes more negative, indicating that the paraboloid opens downward. The vertex of this paraboloid is at the origin \((0, 0, 0)\).

Step 3: Sketch the Surface

To sketch the paraboloid, note the shape formed: a bowl opening downwards. The cross-section in planes parallel to the xy-plane are circles. Begin by plotting the vertex at the origin. Then plot points for various values of \((x,y)\) such as \((1,0,-1), (0,1,-1), (1,1,-2)\), and so on. Draw the curves to form a shape extending downward symmetrically.

Key Concepts

3D Surfaces

Generally, 3D surfaces are representations of equations or functions in a three-dimensional space. They involve x, y, and z coordinates. In our equation, the surface is defined as a paraboloid, which is a common type of 3D surface shaped like a bowl. This bowl can open upwards or downwards. The equation we have, \( z=-(x^2+y^2) \), forms a downward-opening paraboloid.

• Such surfaces are defined by algebraic equations involving all three axes.
• They help us visualize mathematical equations in real space.

A paraboloid is especially useful in understanding intersection points, angles, and distances on a 3D surface. This is pivotal in fields like engineering and computer graphics, where understanding space dimensions is crucial.

Sketching Techniques

When sketching 3D surfaces like a paraboloid, consider viewing the problem from different angles. Starting with recognizing the vertex is crucial. For \( z=-(x^2+y^2) \), the vertex is at (0,0,0), which means the bowl starts here before extending downwards. Next, you identify the symmetry. Since this is symmetric around the z-axis, drawing cross-sections or slices can simplify the visualization. Use points such as \((1,0,-1), (0,1,-1)\) as reference to discover how far the surface dips.

• Plot key points that lie on different cross-sections of the surface.
• Use circles to depict horizontal slices of the paraboloid.

By connecting these plotted points with smooth lines, you get the overall shape of the surface, which aids in understanding the 3D object better.

Coordinate Geometry

Coordinate geometry is a math discipline that combines algebra with geometric concepts through the coordinate plane. For surfaces, it describes how we can express three-dimensional figures with equations involving coordinates \((x, y, z)\). In our example, the paraboloid \(z=-(x^2+y^2)\) uses coordinate geometry to show how a change in x or y affects the z value.

• The x and y together define a plane, while z gives height or depth.
• By experiencing shifts in \((x, y)\), you visually grasp how a 3D object forms.

This type of geometry is crucial because it helps solve problems related to distances and intersections in careers like architecture or physics where 3D object visualization is necessary.

Surface Symmetry

Surface symmetry in geometry refers to a balanced and proportionate similarity found across different segments of a surface. For paraboloids, this symmetry revolves mainly around the z-axis. Understanding symmetry is key in predicting the behavior of surfaces without extensive calculations. In the paraboloid \(z=-(x^2+y^2)\), symmetry means every point \( (x, y, z) \) has an equivalent point \((-x, -y, z)\) on the opposite side.

• It implies surface uniformity, saving time in calculations.
• Recognizing symmetry simplifies sketching and understanding the surface.

This characteristic helps when solving real-world problems since you can infer properties of one part based on another without separate analysis, impacting fields like aerodynamics and design.