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Chapter 2: Problem 27

Sketch or describe the surfaces in \(\mathbb{R}^{3}\) of the equations presented. $$4 x^{2}+y^{2}=16$$

Short Answer

Expert verified
An elliptical cylinder extending along the z-axis with elliptical cross-sections in the xy-plane.

Step by step solution

01

Identify the Type of Surface

The equation given is \(4x^2 + y^2 = 16\). This equation resembles the equation of an ellipse when viewed in a plane. This particular equation does not have a variable \(z\), indicating that the surface is a cylinder in \(\mathbb{R}^3\) that extends infinitely along the \(z\)-axis.
02

Rewrite in Standard Form

To understand the form of the ellipse, rewrite the equation as \(\frac{x^2}{4} + \frac{y^2}{16} = 1\). This is the standard form equation of an ellipse centered at the origin with semi-major axis along the \(y\)-axis, with length 4, and semi-minor axis along the \(x\)-axis, with length 2.
03

Describe the Cylinder

The equation \(4x^2 + y^2 = 16\) represents an elliptical cylinder in \(\mathbb{R}^3\). It is an infinite cylinder whose cross-sectional ellipse lies in the \(xy\)-plane. The axis of the cylinder is parallel to the \(z\)-axis, meaning the cylinder extends infinitely upwards and downwards along this axis.
04

Visualize and Sketch

To sketch the surface, first draw the ellipse \(\frac{x^2}{4} + \frac{y^2}{16} = 1\) in the \(xy\)-plane. This ellipse is centered at the origin with a width of 4 along the \(x\)-axis and a height of 8 along the \(y\)-axis. Extend this ellipse infinitely in the positive and negative \(z\) direction to represent the full three-dimensional elliptical cylinder.

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Key Concepts

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

Ellipse
An ellipse is a geometric shape that looks like a stretched or compressed circle. It has two main lengths called axes: the semi-major axis and the semi-minor axis. The semi-major axis is the longest diameter of the ellipse and the semi-minor axis is the shortest diameter. In our given problem, the ellipse is described by the equation \( \frac{x^2}{4} + \frac{y^2}{16} = 1 \).

Here are some important characteristics of an ellipse:
  • The equation of an ellipse is typically given in the form \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \).
  • \(a\) and \(b\) are the lengths of the semi-major and semi-minor axes, respectively.
  • If \(a > b\), the ellipse is elongated along the \(x\)-axis; if \(b > a\), it is elongated along the \(y\)-axis.
  • The center of the ellipse is the point from which the axes extend, commonly at the origin \((0,0)\) in 2D space.
  • In our exercise, the ellipse is centered at the origin with a semi-major axis of 4 along the \(y\)-axis and a semi-minor axis of 2 along the \(x\)-axis.
These properties help us understand the shape and orientation of the ellipse on the plane.
Cylinder
A cylinder in three-dimensional space (\(\mathbb{R}^3\)) is generated when a curve is extended along a line. For an elliptical cylinder, as in our example, the base of the cylinder is shaped like an ellipse.

The equation \(4x^2 + y^2 = 16\) represents this type of cylinder. The cylinder is described by an ellipse in the \(xy\)-plane that extends along the \(z\)-axis. Here's how we can look at cylinders:
  • A cylinder takes its shape by extending a two-dimensional cross-section along an additional dimension, in this case, the \(z\)-dimension.
  • Since the problem's equation lacks the \(z\)-variable, it implies that the ellipse is extended infinitely parallel to the \(z\)-axis, forming the cylinder.
  • This elliptic cross-section provides the base of the cylinder which gives it its name: elliptical cylinder.
  • Every cross-section parallel to the \(xy\)-plane produces the same ellipse, meaning the structure does not change along the \(z\)-direction.
Understanding the properties of the elliptical cylinder helps to visualize how it extends infinitely in 3D space.
Equation in three dimensions
Equations in three-dimensional space are used to define surfaces or solids. The expansion to 3D allows us to add another axis, typically the \(z\)-axis, helping to describe objects in terms of length, width, and height.

In the case of the elliptical cylinder given by \(4x^2 + y^2 = 16\), the following aspects are highlighted:
  • The absence of a \(z\)-term indicates that the surface extends parallel to the \(z\)-axis. The cylinder is unbounded in the \(z\)-dimension.
  • Equations in 3D can be visualized by interpreting how surfaces behave in space, meaning, every point \((x, y, z)\) satisfies the equation.
  • While equations in 2D define flat shapes, equations in 3D add a spatial component, converting these shapes into solids or surface shells.
  • Similarly, modifying or adding values of \(z\) can shift, tilt, or scale sections of the geometry in 3D space.
Recognizing an equation's 3D nature allows for better comprehension of how objects are formed and interact in physical environments.

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Most popular questions from this chapter

Show that \(f: \mathbb{R}^{n} \rightarrow \mathbb{R}^{m}\) is continuous at all points if and only if the inverse image of every open set is open.

Find \((\partial / \partial s)(f \circ T)(1,0),\) where \(f(u, v)=\cos u \sin v\) and \(T: \mathbb{R}^{2} \rightarrow \mathbb{R}^{2}\) is defined by \(T(s, t)=\left(\cos \left(t^{2} s\right), \log \sqrt{1+s^{2}}\right)\)

Consider the spiral given by \(\mathbf{c}(t)=\left(e^{t} \cos (t), e^{t} \sin (t)\right)\) Show that the angle between \(c\) and \(c^{\prime}\) is constant.

Let \(h: \mathbb{R}^{3} \rightarrow \mathbb{R}^{5}\) and \(g: \mathbb{R}^{2} \rightarrow \mathbb{R}^{3}\) be given by \(h(x, y, z)=\left(x y z, e^{x z}, x \sin (y), \frac{-9}{x}, 17\right)\) and \(g(u, v)=\left(v^{2}+2 u, \pi, 2 \sqrt{u}\right) .\) Find \(\mathbf{D}(h \circ g)(1,1)\)

The position vector for a particle moving on a helix is \(\mathbf{c}(t)=\left(\cos (t), \sin (t), t^{2}\right).\) (a) Find the speed of the particle at time \(t_{0}=4 \pi\) (b) Is \(\mathbf{c}^{\prime}(t)\) ever orthogonal to \(\mathbf{c}(t) ?\) (c) Find a parametrization for the tangent line to \(\mathbf{c}(t)\) at \(t_{0}=4 \pi\) (d) Where will this line intersect the \(x y\) plane?
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