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Chapter 17: Problem 12

Parametric descriptions Give a parametric description of the form \(\mathbf{r}(u, v)=\langle x(u, v), y(u, v), z(u, v)\rangle\) for the following surfaces. The descriptions are not unique. Specify the required rectangle in the uv- plane The cone \(z^{2}=4\left(x^{2}+y^{2}\right),\) for \(0 \leq z \leq 4\)

Short Answer

Expert verified
Question: Provide the parametric description of the cone given by the equation \(z^2 = 4(x^2 + y^2)\) and specify the required rectangle in the uv-plane. Answer: The parametric description of the cone is \(\mathbf{r}(u, v) = \langle u\cos(v), u\sin(v), 2u \rangle\), and the required rectangle in the uv-plane is \(0 \leq u \leq 2, 0 \leq v \leq 2\pi\).

Step by step solution

01

Convert given equation into cylindrical coordinates

In cylindrical coordinates, we have the following relation: $x = r * \cos(\theta), y = r * \sin(\theta), z = z $ Now, plug these into the equation of the cone: \(z^2 = 4(x^2 + y^2)\). After substituting, we get: \(z^2 = 4(r^2\cos^2(\theta) + r^2\sin^2(\theta)) = 4r^2(\cos^2(\theta) + \sin^2(\theta)) = 4r^2\)
02

Parametric transformation

Now, we can write the parametric representation of the cone as follows: $x(u, v) = u\cos(v), y(u, v) = u\sin(v), z(u, v) = 2u $ Here, we used v to represent the angle \(\theta\) and u for the radius r. From the equation \(z^2 = 4r^2\), we can see that \(z = 2r\) and therefore we have substituted the parametric expression of z as \(z(u, v) = 2u\).
03

Specify the required uv-plane rectangle

To find the domain and the range of u and v, we make use of the given range for z \((0 \leq z \leq 4)\). Recall that \(z(u, v) = 2u\). If \(0 \leq z \leq 4\), then we have \(0 \leq 2u \leq 4\). Therefore, for the domain of u, we have \(0 \leq u \leq 2\). Next, we must find the range for v. Since v corresponds to the angle \(\theta\) in cylindrical coordinates, it completes one full rotation along the z-axis. Therefore, we can set the range for v as \(0 \leq v \leq 2\pi\). Now, we can write the required rectangle in uv-plane: $0 \leq u \leq 2, 0 \leq v \leq 2\pi$ The parametric description of the cone is \(\mathbf{r}(u, v) = \langle u\cos(v), u\sin(v), 2u \rangle\), and the required rectangle in the uv-plane is \(0 \leq u \leq 2, 0 \leq v \leq 2\pi\).

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Key Concepts

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

Parametric Equations
In mathematics, parametric equations define a group of quantities as explicit functions of one or more independent variables, known as parameters. Parametric representations are commonly used to express the coordinates of the points that make up a geometric object such as a curve or surface, in a coordinate system.

For example, in the context of the exercise, the cone surface is described parametrically by the set of equations \(x(u, v) = u\cos(v),\) \(y(u, v) = u\sin(v),\) and \(z(u, v) = 2u\). Here, \(u\) and \(v\) are the parameters: \(u\) can be thought of as the distance from the origin (on the surface of the cone), while \(v\) represents the angle around the cone's central axis. This parametric form is powerful as it can easily describe complex curves and surfaces in a simplified manner.

Advantages of Parametric Equations

  • Ease of plotting curves and surfaces.
  • Potential simplification of complicated geometric objects.
  • Flexibility to manipulate the shape and size by altering parameters.
Cylindrical Coordinates
Cylindrical coordinates are a way of describing the location of points in three-dimensional space using the three coordinates \(\rho, \phi, z\). It's an extension of the two-dimensional polar coordinate system with an added height component \(z\). The coordinate \(\rho\) indicates the radial distance from the axis of symmetry (typically the z-axis), \(\phi\) is the angle around the axis (analogous to the angle \(\theta\) in polar coordinates), and \(z\) is the height.

In the step-by-step solution, we use cylindrical coordinates to transform the equation of a cone into a more manageable form. This transformation simplifies solving the problem since the symmetry of the cone naturally aligns with the cylindrical coordinate system. Converting Cartesian coordinates to cylindrical coordinates can also be particularly useful when dealing with problems that have rotational symmetry, such as the case of the cone in the exercise.

Transforming Equations to Cylindrical Coordinates

  • \(x = \rho\cos(\phi)\)
  • \(y = \rho\sin(\phi)\)
  • \(z = z\)
Conic Sections
Conic sections are the curves obtained by intersecting a plane with a double-napped right circular cone. Depending on the angle of the plane with respect to the cone's axis, the intersection can be a circle, an ellipse, a parabola, or a hyperbola.

In the context of the given textbook exercise, we are dealing with a specific type of conic section, namely a circular cone. The cone's equation given by \(z^2 = 4(x^2 + y^2)\) represents a perfect funnel shape, and it's clear that for every point \(x, y, z\) on the cone, \(z\) is directly proportional to the distance from the origin in the \(xy\)-plane. This property is what allows us to set up the parametric equations described in the first section.

Role of Conic Sections in Parametric Descriptions

  • Conic sections provide geometric shapes that can be efficiently described using parametric equations.
  • They are used in various fields, from astronomy to physics, to describe orbits and other spatial phenomena.
  • Their simple geometric properties allow for straightforward translation into other coordinate systems such as cylindrical or spherical coordinates.

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Most popular questions from this chapter

Find the upward flux of the field \(\mathbf{F}=\langle x, y, z\rangle\) across the plane \(\frac{x}{a}+\frac{y}{b}+\frac{z}{c}=1\) in the first octant where \(a, b,\) and \(c\) are positive real numbers. Show that the flux equals \(c\) times the area of the base of the region. Interpret the result physically.

Consider the rotational velocity field \(\mathbf{v}=\langle-2 y, 2 z, 0\rangle\) a. If a paddle wheel is placed in the \(x y\) -plane with its axis normal to this plane, what is its angular speed? b. If a paddle wheel is placed in the \(x z\) -plane with its axis normal to this plane, what is its angular speed? c. If a paddle wheel is placed in the \(y z\) -plane with its axis normal to this plane, what is its angular speed?

Streamlines and equipotential lines Assume that on \(\mathbb{R}^{2}\), the vector field \(\mathbf{F}=\langle f, g\rangle\) has a potential function \(\varphi\) such that \(f=\varphi_{x}\) and \(g=\varphi_{y},\) and it has a stream function \(\psi\) such that \(f=\psi_{y}\) and \(g=-\psi_{x}\). Show that the equipotential curves (level curves of \(\varphi\) ) and the streamlines (level curves of \(\psi\) ) are everywhere orthogonal.

A beautiful flux integral Consider the potential function \(\varphi(x, y, z)=G(\rho),\) where \(G\) is any twice differentiable function and \(\rho=\sqrt{x^{2}+y^{2}+z^{2}} ;\) therefore, \(G\) depends only on the distance from the origin. a. Show that the gradient vector field associated with \(\varphi\) is \(\mathbf{F}=\nabla \varphi=G^{\prime}(\rho) \frac{\mathbf{r}}{\rho},\) where \(\mathbf{r}=\langle x, y, z\rangle\) and \(\rho=|\mathbf{r}|\)b. Let \(S\) be the sphere of radius \(a\) centered at the origin and let \(D\) be the region enclosed by \(S\). Show that the flux of \(\mathbf{F}\) across \(S\) is \(\iint_{S} \mathbf{F} \cdot \mathbf{n} d S=4 \pi a^{2} G^{\prime}(a)\) c. Show that \(\nabla \cdot \mathbf{F}=\nabla \cdot \nabla \varphi=\frac{2 G^{\prime}(\rho)}{\rho}+G^{\prime \prime}(\rho)\) d. Use part (c) to show that the flux across \(S\) (as given in part (b)) is also obtained by the volume integral \(\iiint_{D} \nabla \cdot \mathbf{F} d V\) (Hint: Use spherical coordinates and integrate by parts.)

Prove that for a real number \(p\) with \(\mathbf{r}=\langle x, y, z\rangle, \nabla \cdot \nabla\left(\frac{1}{|\mathbf{r}|^{p}}\right)=\frac{p(p-1)}{|\mathbf{r}|^{p+2}}\).
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