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Chapter 17: Problem 1

Give a parametric description for a cylinder with radius \(a\) and height \(h,\) including the intervals for the parameters.

Short Answer

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Question: Give a parametric description of a cylinder with radius a and height h. Answer: The parametric description of a cylinder with radius a and height h can be given as follows: x(u, v) = a*cos(u) y(u, v) = a*sin(u) z(u, v) = v with the parameter intervals 0 鈮� u 鈮� 2蟺 and 0 鈮� v 鈮� h.

Step by step solution

01

Parameterize the circular base of the cylinder

First, let's parameterize the circular base of the cylinder in the \(xy\)-plane with a radius of \(a.\) We can use the following equations: \(x(u) = a\cos(u)\) \(y(u) = a\sin(u)\) where \(0 \le u \le 2\pi.\)
02

Introduce the height parameter

Now, let's introduce the height parameter, \(v.\) Notice that the height of the cylinder stretches from 0 to \(h\) along the \(z\)-axis. We can use the following equation: \(z(v) = v\) where \(0 \le v \le h.\)
03

Combine the parameterized equations

To parameterize the entire cylinder, we need to combine the parameterized equations for the circular base and the height: \(x(u, v) = a\cos(u)\) \(y(u, v) = a\sin(u)\) \(z(u, v) = v\) These equations describe the points on the surface of the cylinder with radius \(a\) and height \(h.\) Finally, let's include the intervals for the parameters: \(0 \le u \le 2\pi\) \(0 \le v \le h\)

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Key Concepts

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

Cylinder Parameterization
To understand cylinder parameterization, we first need to know that it involves expressing a cylindrical surface in terms of two parameters. These parameters are often taken to be two independent variables that help us describe any point on the cylinder. A cylinder, in this context, is conceptualized with a circular base and a set height, often aligned with one of the three coordinate axes, such as the z-axis.

The parameterization process involves determining equations that detail the cylinder's surface considering a specified radius and height. Typically, for a cylinder aligned along the z-axis, one parameter describes the circular nature of the base (often using trigonometric functions for the x and y coordinates), while the other details the vertical stretch along the axis. Together, these give us a full representation of the cylinder's surface, enabling calculations and graphic representations with ease.
Circular Base
The circular base of a cylinder is essentially a circle, commonly found in the xy-plane when the cylinder is oriented vertically. To parameterize this base, we use trigonometric functions, particularly sine and cosine these functions make it easy to describe circular shapes in terms of angles.

Here's how it works:
  • The parameter \(u\), ranging from 0 to \(2\pi\), represents the angle around the circle.
  • To convert this angle to x and y coordinates, we use the equations \(x(u) = a\cos(u)\) and \(y(u) = a\sin(u)\), where \(a\) is the radius.
This parameterization allows us to pinpoint any location on the circular base simply by varying \(u\), seamlessly drawing out the entire circle through its basal parameter.
Height Parameter
Adding depth to our understanding of cylinders, the height parameter controls the extent of the cylinder along the vertical axis. This component is crucial when shifting from a simple circle to a full cylindrical shape.

For a cylinder standing on the xy-plane and stretching along the z-axis, the height parameter, often labeled \(v\), indicates how far along the z direction a point is located.
  • This parameter has a range starting from 0 at the base and reaching up to \(h\) at the top of the cylinder.
  • The equation \(z(v) = v\) defines this relationship, meaning that as \(v\) varies, it literally moves the parametrized point up the z-axis from 0 to the maximum height \(h\).
Combining the height parameter with the circular base equations provides a comprehensive method to define every point on the cylinder's curved surface.

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Most popular questions from this chapter

Fourier's Law of heat transfer (or heat conduction ) states that the heat flow vector \(\mathbf{F}\) at a point is proportional to the negative gradient of the temperature; that is, \(\mathbf{F}=-k \nabla T,\) which means that heat energy flows from hot regions to cold regions. The constant \(k>0\) is called the conductivity, which has metric units of \(J /(m-s-K)\) A temperature function for a region \(D\) is given. Find the net outward heat flux \(\iint_{S} \mathbf{F} \cdot \mathbf{n} d S=-k \iint_{S} \nabla T \cdot \mathbf{n} d S\) across the boundary S of \(D\) In some cases, it may be easier to use the Divergence Theorem and evaluate a triple integral. Assume \(k=1 .\) \(T(x, y, z)=100+x^{2}+y^{2}+z^{2} ; D\) is the unit sphere centered at the origin.

Gauss' Law for gravitation The gravitational force due to a point mass \(M\) at the origin is proportional to \(\mathbf{F}=G M \mathbf{r} /|\mathbf{r}|^{3},\) where \(\mathbf{r}=\langle x, y, z\rangle\) and \(G\) is the gravitational constant. a. Show that the flux of the force field across a sphere of radius \(a\) centered at the origin is \(\iint_{S} \mathbf{F} \cdot \mathbf{n} d S=4 \pi G M\) b. Let \(S\) be the boundary of the region between two spheres centered at the origin of radius \(a\) and \(b,\) respectively, with \(a

Miscellaneous surface integrals Evaluate the following integrals using the method of your choice. Assume normal vectors point either outward or upward. $$\begin{aligned} &\iint_{S} \nabla \ln |\mathbf{r}| \cdot \mathbf{n} d S, \text { where } S \text { is the hemisphere } x^{2}+y^{2}+z^{2}=a^{2}\\\ &\text { for } z \geq 0, \text { and where } \mathbf{r}=\langle x, y, z\rangle \end{aligned}$$

Green's First Identity Prove Green's First Identity for twice differentiable scalar-valued functions \(u\) and \(v\) defined on a region \(D:\) $$\iiint_{D}\left(u \nabla^{2} v+\nabla u \cdot \nabla v\right) d V=\iint_{S} u \nabla v \cdot \mathbf{n} d S$$where \(\nabla^{2} v=\nabla \cdot \nabla v .\) You may apply Gauss' Formula in Exercise 48 to \(\mathbf{F}=\nabla v\) or apply the Divergence Theorem to \(\mathbf{F}=u \nabla v\)

Choosing a more convenient surface The goal is to evaluate \(A=\iint_{S}(\nabla \times \mathbf{F}) \cdot \mathbf{n} d S,\) where \(\mathbf{F}=\langle y z,-x z, x y\rangle\) and \(S\) is the surface of the upper half of the ellipsoid \(x^{2}+y^{2}+8 z^{2}=1\) \((z \geq 0)\) a. Evaluate a surface integral over a more convenient surface to find the value of \(A\) b. Evaluate \(A\) using a line integral.
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