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Chapter 10: Problem 40

Sketch the appropriate traces, and then sketch and identify the surface. $$x+y^{2}+z^{2}=2$$

Short Answer

Expert verified
The given equation represents a three-dimensional parabolic surface, or a paraboloid, that opens to the positive x direction. It can be envisioned as a series of increasing circular cross sections (traces) along the x-axis that forms a parabolic shape.

Step by step solution

01

Identify the type of surface

The given equation is in the form \(x = y^{2} + z^{2}\), which means it represents a paraboloid. Paraboloids open along the plane of the variable that isn't squared, which in this case is the x-plane. So this will be a paraboloid opening to the positive x direction.
02

Find the traces

The traces are the curves obtained by cutting the surface with planes parallel to the coordinate planes. In this case:- For the x=0 trace, set x=0 in the equation, which gives \(y^{2}+z^{2}=2\), a circle of radius \(\sqrt{2}\) in the yz-plane.- For the y=0 trace, set y=0 in the equation, which gives \(x+z^{2}=2\), a parabola opening to the positive x direction in the xz-plane.- For the z=0 trace, set z=0 in the equation. This gives \(x+y^{2}=2\), another parabola opening to the positive x direction in the xy-plane.
03

Sketch the traces

Now that we have the equations for the traces, we can sketch them.- The x=0 trace is a circle of radius \(\sqrt{2}\) in the yz-plane. Since we're dealing with three dimensions, this circle can be thought of as a cylinder stretching along the x-axis.- The y=0 and z=0 traces are parabolas opening to the positive x direction. Note that these parabolas have the same shape due to symmetry.
04

Combine the sketches to form the surface

Taking these traces together, we can see that as x increases, the radius of the 'circular cross sections' (the yz slices) increase. This creates a parabolic shape opening towards the positive x-axis, illustrating that the original function represents a paraboloid opening in the positive x direction.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Traces
Traces, in the context of three-dimensional surfaces, are crucial for visualizing complex geometrical shapes. They represent the intersection of a surface with planes that are parallel to the coordinate planes.

To understand this better, imagine slicing a 3D object with perfectly flat, parallel sheets. Each of these intersections, or traces, provides valuable insights into the structure of the object.

For the given equation, when we examine the traces, we observe:
  • The x=0 trace: Setting x=0 yields the equation \(y^2 + z^2 = 2\), which describes a circle with radius \(\sqrt{2}\) in the yz-plane.
  • The y=0 trace: Here, setting y=0 changes the equation to \(x + z^2 = 2\), identifying it as a parabola in the xz-plane.
  • The z=0 trace: With z=0, the equation becomes \(x + y^2 = 2\), forming another parabola in the xy-plane.
Combining these traces can illustrate the entire surface they are part of, aiding in the understanding of the geometric shape represented by a function.
Basics of 3D Sketching
3D sketching is an essential skill for visualizing mathematical and geometric shapes in three-dimensional space. The challenge lies in representing three-dimensional objects on two-dimensional media, like paper or a computer screen.

To sketch the paraboloid from the exercise, you start by plotting each trace separately in their respective planes.
  • The circle traced on the yz-plane suggests a circular cross-section of the paraboloid as it cuts through.
  • The parabolas traced in both the xz-plane and xy-plane suggest that these curves open toward the positive x-axis, confirming the direction in which the paraboloid opens.
By connecting these traces smoothly, you can visualize the curved surface of the paraboloid. The technique of combining slices from different vantage points helps us form a complete picture of complex 3D objects.
Exploring Conic Sections
Conic sections are the curves obtained by intersecting a plane with a cone. Key conic sections include circles, ellipses, parabolas, and hyperbolas.

In three-dimensional space, these curves appear naturally when analyzing traces of surfaces like the paraboloid.
  • The circle \(y^2 + z^2 = 2\) represents a conic section, indicating a symmetrical, round shape.
  • The parabola \(x + y^2 = 2\) signifies a curve that opens in one direction, reflecting the shape's gradual outward spreading.
  • Each conic section provides a different perspective on the overall structure and behavior of the 3D shape.
By understanding conic sections, we gain insights into the geometric properties and symmetries of diverse shapes, like the smooth, inward-folding form of a paraboloid. This understanding enhances both visualization and problem-solving abilities in a three-dimensional context.

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Most popular questions from this chapter

Among all sets of nonnegative numbers \(p_{1}, p_{2}, \ldots, p_{n}\) that sum to \(1,\) find the choice of \(p_{1}, p_{2}, \ldots, p_{n}\) that minimizes \(\sum_{k=1}^{n} p_{k}^{2}\)

Use the Cauchy-Schwartz Inequality in \(n\) dimensions to show that \(\sum_{k=1}^{n}\left|a_{k}\right| \leq\left(\sum_{k=1}^{n}\left|a_{k}\right|^{2 / 3}\right)^{1 / 2}\left(\sum_{k=1}^{n}\left|a_{k}\right|^{4 / 3}\right)^{1 / 2}\)

Use geometry to identify the cross product (do not compute!). $$\mathbf{j} \times(\mathbf{j} \times \mathbf{k})$$

$$\text { Show that }\|\mathbf{a} \times \mathbf{b}\|^{2}=\|\mathbf{a}\|^{2}\|\mathbf{b}\|^{2}-(\mathbf{a} \cdot \mathbf{b})^{2}$$

You are asked to work with vectors of dimension higher than three. Use rules analogous to those introduced for two and three dimensions. $$\|\mathbf{a}\| \text { for } \mathbf{a}=\langle 1,0,-3,-2,4,1\rangle$$
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