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Chapter 14: Problem 4

How do you determine whether a vector field in \(\mathbb{R}^{3}\) is conservative?

Short Answer

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Question: Determine whether the given vector field \(\mathbf{F}(x, y, z) = P(x, y, z) \mathbf{i} + Q(x, y, z) \mathbf{j} + R(x, y, z) \mathbf{k}\) is conservative.

Step by step solution

01

Understanding the curl of a vector field

The curl of a vector field gives information about the rotation or spinning of that field. Mathematically, it is computed using the following formula for a given vector field \(\mathbf{F}(x, y, z) = P(x, y, z) \mathbf{i} + Q(x, y, z) \mathbf{j} + R(x, y, z) \mathbf{k}\): $$ \mathrm{curl}\ \mathbf{F}(x, y, z) = \nabla \times \mathbf{F} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\ P & Q & R \end{vmatrix} $$ To find the curl, we need to compute the determinant of this matrix.
02

Computing the determinant

We compute the determinant as follows: $$ \nabla \times \mathbf{F} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\ P & Q & R \end{vmatrix} = \Bigg(\frac{\partial R}{\partial y} - \frac{\partial Q}{\partial z}\Bigg)\mathbf{i} - \Bigg(\frac{\partial R}{\partial x} - \frac{\partial P}{\partial z}\Bigg)\mathbf{j} + \Bigg(\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}\Bigg)\mathbf{k} $$
03

Check if curl is zero

A vector field is conservative if its curl is equal to the zero vector. We can check whether the components of the curl we calculated above are equal to zero: $$ \frac{\partial R}{\partial y} - \frac{\partial Q}{\partial z} = 0 \\ \frac{\partial R}{\partial x} - \frac{\partial P}{\partial z} = 0 \\ \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} = 0 $$ If all these conditions are satisfied, then the curl of the vector field is zero, and we can conclude that the vector field is conservative. Otherwise, it's not conservative. In summary, to determine whether a vector field in \(\mathbb{R}^3\) is conservative, we need to compute its curl and verify if it's equal to the zero vector.

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Most popular questions from this chapter

\(A\) scalar-valued function \(\varphi\) is harmonic on a region \(D\) if \(\nabla^{2} \varphi=\nabla \cdot \nabla \varphi=0\) at all points of \(D\) Show that if \(\varphi\) is harmonic on a region \(D\) enclosed by a surface \(S\) then $$\iint_{S} \nabla \varphi \cdot \mathbf{n} d S=0$$

The cone \(z^{2}=x^{2}+y^{2},\) for \(z \geq 0,\) cuts the sphere \(x^{2}+y^{2}+z^{2}=16\) along a curve \(C\) a. Find the surface area of the sphere below \(C,\) for \(z \geq 0\). b. Find the surface area of the sphere above \(C\). c. Find the surface area of the cone below \(C,\) for \(z \geq 0\).

Let \(\mathbf{F}=\langle z, 0,0\rangle\) and let \(\mathbf{n}\) be a unit vector aligned with the axis of a paddle wheel located on the \(x\) -axis (see figure). a. If the paddle wheel is oriented with \(\mathbf{n}=\langle 1,0,0\rangle,\) in what direction (if any) does the wheel spin? b. If the paddle wheel is oriented with \(\mathbf{n}=\langle 0,1,0\rangle,\) in what direction (if any) does the wheel spin? c. If the paddle wheel is oriented with \(\mathbf{n}=\langle 0,0,1\rangle,\) in what direction (if any) does the wheel spin?

The rotation of a three-dimensional velocity field \(\mathbf{V}=\langle u, v, w\rangle\) is measured by the vorticity \(\omega=\nabla \times \mathbf{V} .\) If \(\omega=\mathbf{0}\) at all points in the domain, the flow is irrotational. a. Which of the following velocity fields is irrotational: \(\mathbf{V}=\langle 2,-3 y, 5 z\rangle\) or \(\mathbf{V}=\langle y, x-z,-y\rangle ?\) b. Recall that for a two-dimensional source-free flow \(\mathbf{V}=(u, v, 0),\) a stream function \(\psi(x, y)\) may be defined such that \(u=\psi_{y}\) and \(v=-\psi_{x} .\) For such a two-dimensional flow, let \(\zeta=\mathbf{k} \cdot \nabla \times \mathbf{V}\) be the \(\mathbf{k}\) -component of the vorticity. Show that \(\nabla^{2} \psi=\nabla \cdot \nabla \psi=-\zeta\). c. Consider the stream function \(\psi(x, y)=\sin x \sin y\) on the square region \(R=\\{(x, y): 0 \leq x \leq \pi, 0 \leq y \leq \pi\\}\). Find the velocity components \(u\) and \(v\); then sketch the velocity field. d. For the stream function in part (c), find the vorticity function \(\zeta\) as defined in part (b). Plot several level curves of the vorticity function. Where on \(R\) is it a maximum? A minimum?

Prove the following identities. Assume that \(\varphi\) is \(a\) differentiable scalar-valued function and \(\mathbf{F}\) and \(\mathbf{G}\) are differentiable vector fields, all defined on a region of \(\mathbb{R}^{3}\). $$\nabla \times(\nabla \times \mathbf{F})=\nabla(\nabla \cdot \mathbf{F})-(\nabla \cdot \nabla) \mathbf{F}$$
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