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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 ≤ u ≤ 2π and 0 ≤ v ≤ 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

One of Maxwell's equations for electromagnetic waves is \(\nabla \times \mathbf{B}=C \frac{\partial \mathbf{E}}{\partial t},\) where \(\mathbf{E}\) is the electric field, \(\mathbf{B}\) is the magnetic field, and \(C\) is a constant. a. Show that the fields \(\mathbf{E}(z, t)=A \sin (k z-\omega t) \mathbf{i}\) and \(\mathbf{B}(z, t)=A \sin (k z-\omega t) \mathbf{j}\) satisfy the equation for constants \(A, k,\) and \(\omega,\) provided \(\omega=k / C\). b. Make a rough sketch showing the directions of \(\mathbf{E}\) and \(\mathbf{B}\).

Suppose a solid object in \(\mathbb{R}^{3}\) has a temperature distribution given by \(T(x, y, z) .\) The heat flow vector field in the object is \(\mathbf{F}=-k \nabla T,\) where the conductivity \(k>0\) is a property of the material. Note that the heat flow vector points in the direction opposite to that of the gradient, which is the direction of greatest temperature decrease. The divergence of the heat flow vector is \(\nabla \cdot \mathbf{F}=-k \nabla \cdot \nabla T=-k \nabla^{2} T(\text {the Laplacian of } T) .\) Compute the heat flow vector field and its divergence for the following temperature distributions. $$T(x, y, z)=100 e^{-\sqrt{x^{2}+y^{2}+z^{2}}}$$

Suppose a solid object in \(\mathbb{R}^{3}\) has a temperature distribution given by \(T(x, y, z) .\) The heat flow vector field in the object is \(\mathbf{F}=-k \nabla T,\) where the conductivity \(k>0\) is a property of the material. Note that the heat flow vector points in the direction opposite to that of the gradient, which is the direction of greatest temperature decrease. The divergence of the heat flow vector is \(\nabla \cdot \mathbf{F}=-k \nabla \cdot \nabla T=-k \nabla^{2} T(\text {the Laplacian of } T) .\) Compute the heat flow vector field and its divergence for the following temperature distributions. $$T(x, y, z)=100 e^{-x^{2}+y^{2}+z^{2}}$$

Let \(f\) be differentiable and positive on the interval \([a, b] .\) Let \(S\) be the surface generated when the graph of \(f\) on \([a, b]\) is revolved about the \(x\) -axis. Use Theorem 17.14 to show that the area of \(S\) (as given in Section 6.6 ) is $$ \int_{a}^{b} 2 \pi f(x) \sqrt{1+f^{\prime}(x)^{2}} d x $$

Mass and center of mass Let \(S\) be a surface that represents a thin shell with density \(\rho .\) The moments about the coordinate planes (see Section 16.6 ) are \(M_{y z}=\iint_{S} x \rho(x, y, z) d S, M_{x z}=\iint_{S} y \rho(x, y, z) d S\) and \(M_{x y}=\iint_{S} z \rho(x, y, z) d S .\) The coordinates of the center of mass of the shell are \(\bar{x}=\frac{M_{y z}}{m}, \bar{y}=\frac{M_{x z}}{m},\) and \(\bar{z}=\frac{M_{x y}}{m},\) where \(m\) is the mass of the shell. Find the mass and center of mass of the following shells. Use symmetry whenever possible. The cylinder \(x^{2}+y^{2}=a^{2}, 0 \leq z \leq 2,\) with density \(\rho(x, y, z)=1+z\)

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