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Chapter 14: Problem 2

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

Short Answer

Expert verified
Answer: The parametric description of a cone with radius a and height h is given by: \((x, y, z) = (a(1-t)\cos(s), a(1-t)\sin(s), ht)\) The intervals for the parameters are \(0 \leq t \leq 1\) and \(0 \leq s \leq 2\pi\).

Step by step solution

01

Write the parameterized equation for the cone

We need to find a parametric equation that describes the cone, in terms of coordinates \((x, y, z)\). For a cone with radius \(a\) and height \(h\), we can use the parameters \(t\) and \(s\) to define the coordinates as follows: \((x, y, z) = (a(1-t)\cos(s), a(1-t)\sin(s), ht)\)
02

Determine the intervals for the parameters

Now that we have the parameterized equation, we need to define the intervals for the parameters, \(t\) and \(s\). To describe the entire cone, we need to take the following intervals for the parameters: For \(t\): Since \(t\) defines the height of the cone, the interval for \(t\) should be \(0 \leq t \leq 1\). For \(s\): Since \(s\) defines the circular base of the cone, the interval for \(s\) should be \(0 \leq s \leq 2\pi\).
03

Final Solution

The parametric description of a cone with radius \(a\) and height \(h\) is given by: \((x, y, z) = (a(1-t)\cos(s), a(1-t)\sin(s), ht)\) with the intervals for the parameters being \(0 \leq t \leq 1\) and \(0 \leq s \leq 2\pi\).

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Most popular questions from this chapter

The Navier-Stokes equation is the fundamental equation of fluid dynamics that models the flow in everything from bathtubs to oceans. In one of its many forms (incompressible, viscous flow), the equation is $$\rho\left(\frac{\partial \mathbf{V}}{\partial t}+(\mathbf{V} \cdot \nabla) \mathbf{V}\right)=-\nabla p+\mu(\nabla \cdot \nabla) \mathbf{V}.$$ In this notation, \(\mathbf{V}=\langle u, v, w\rangle\) is the three-dimensional velocity field, \(p\) is the (scalar) pressure, \(\rho\) is the constant density of the fluid, and \(\mu\) is the constant viscosity. Write out the three component equations of this vector equation. (See Exercise 40 for an interpretation of the operations.)

Suppose an object with mass \(m\) moves in a region \(R\) in a conservative force field given by \(\mathbf{F}=-\nabla \varphi\) where \(\varphi\) is a potential function in a region \(R .\) The motion of the object is governed by Newton's Second Law of Motion, \(\mathbf{F}=m \mathbf{a}\) where a is the acceleration. Suppose the object moves from point \(A\) to point \(B\) in \(R\). a. Show that the equation of motion is \(m \frac{d \mathbf{v}}{d t}=-\nabla \varphi\) b. Show that \(\frac{d \mathbf{v}}{d t} \cdot \mathbf{v}=\frac{1}{2} \frac{d}{d t}(\mathbf{v} \cdot \mathbf{v})\) c. Take the dot product of both sides of the equation in part (a) with \(\mathbf{v}(t)=\mathbf{r}^{\prime}(t)\) and integrate along a curve between \(A\) and \(B\). Use part (b) and the fact that \(\mathbf{F}\) is conservative to show that the total energy (kinetic plus potential) \(\frac{1}{2} m|\mathbf{v}|^{2}+\varphi\) is the same at \(A\) and \(B\). Conclude that because \(A\) and \(B\) are arbitrary, energy is conserved in \(R\)

Use the procedure in Exercise 57 to construct potential functions for the following fields. $$\mathbf{F}=\langle x, y\rangle$$

Find the work required to move an object in the following force fields along a line segment between the given points. Check to see whether the force is conservative. $$\mathbf{F}=\langle x, y\rangle \text { from } A(1,1) \text { to } B(3,-6)$$

Prove that for a real number \(p,\) with \(\mathbf{r}=\langle x, y, z\rangle, \nabla \cdot \frac{\langle x, y, z\rangle}{|\mathbf{r}|^{p}}=\frac{3-p}{|\mathbf{r}|^{p}}.\)
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