/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 50 Find a vector field \(\mathbf{F}... [FREE SOLUTION] | 91影视

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Chapter 14: Problem 50

Find a vector field \(\mathbf{F}\) with the given curl. In each case, is the vector field you found unique? $$\operatorname{curl} \mathbf{F}=\langle 0,1,0\rangle$$

Short Answer

Expert verified
Answer: A possible vector field \(\mathbf{F}\) for the given curl is given by: $$\mathbf{F} = \left\langle \frac{\partial g(x,z)}{\partial x}, 1, \frac{\partial g(x,z)}{\partial z} \right\rangle,$$ where \(g(x,z)\) is a function of \(x\) and \(z\). The vector field found is not unique, as the function \(g(x,z)\) can vary, leading to multiple different vector fields \(\mathbf{F}\) for the given curl.

Step by step solution

01

Define the given curl

We are given the curl of a vector field \(\mathbf{F}\): $$\operatorname{curl} \mathbf{F}=\langle 0,1,0\rangle$$
02

Integrate each component of the curl

In order to find a potential function \(\phi\) that is associated with the vector field \(\mathbf{F}\), we need to integrate each component of the given curl with respect to the corresponding variables: $$\phi_x = \int (\operatorname{curl}\mathbf{F})_i dx \implies \phi_x = \int 0 dx = 0$$ $$\phi_y = \int (\operatorname{curl}\mathbf{F})_j dy \implies \phi_y = \int 1 dy = y + g(x,z)$$ $$\phi_z = \int (\operatorname{curl}\mathbf{F})_k dz \implies \phi_z = \int 0 dz = 0$$
03

Combine components to find the potential function

We can now combine the components of \(\phi\) to find the associated potential function \(\phi(x,y,z)\): $$\phi(x,y,z) = y + g(x,z)$$
04

Compute the gradient of the potential function

In order to find the vector field \(\mathbf{F}\), we need to compute the gradient of the potential function \(\phi(x,y,z)\). This is given by: $$\mathbf{F} = \nabla \phi = \left\langle \frac{\partial \phi}{\partial x}, \frac{\partial \phi}{\partial y}, \frac{\partial \phi}{\partial z} \right\rangle$$ Now compute the gradients: $$\frac{\partial \phi}{\partial x} = \frac{\partial}{\partial x}(y + g(x,z)) = \frac{\partial g(x,z)}{\partial x}$$ $$\frac{\partial \phi}{\partial y} = \frac{\partial}{\partial y}(y + g(x,z)) = 1$$ $$\frac{\partial \phi}{\partial z} = \frac{\partial}{\partial z}(y + g(x,z)) = \frac{\partial g(x,z)}{\partial z}$$
05

Finalize the vector field

By substituting the gradients of \(\phi\) back into the formula for the vector field \(\mathbf{F}\), we get: $$\mathbf{F} = \left\langle \frac{\partial g(x,z)}{\partial x}, 1, \frac{\partial g(x,z)}{\partial z} \right\rangle$$
06

Discuss the uniqueness of the found vector field

In our solution, the vector field \(\mathbf{F}\) depends on the function \(g(x,z)\). As the function \(g(x,z)\) can vary (even by constants), it means that there can be multiple different vector fields \(\mathbf{F}\) for the given curl. Therefore, the vector field we found is not unique.

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Most popular questions from this chapter

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.

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

a. Prove that the rotation field \(\mathbf{F}=\frac{\langle-y, x\rangle}{|\mathbf{r}|^{p}},\) where \(\mathbf{r}=\langle x, y\rangle\) is not conservative for \(p \neq 2\) b. For \(p=2,\) show that \(\mathbf{F}\) is conservative on any region not containing the origin. c. Find a potential function for \(\mathbf{F}\) when \(p=2\)

Let \(S\) be a surface that represents a thin shell with density \(\rho .\) The moments about the coordinate planes (see Section 13.6 ) 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 \(\bar{x}=\frac{M_{y z}}{m}, \bar{y}=\frac{M_{x z}}{m}, \bar{z}=\frac{M_{x y}}{m},\) where \(m\) is the mass of the shell. Find the mass and center of mass of the following shells. Use symmetry whenever possible. The constant-density half cylinder \(x^{2}+z^{2}=a^{2},-h / 2 \leq y \leq h / 2, z \geq 0\)

Let \(S\) be the paraboloid \(z=a\left(1-x^{2}-y^{2}\right),\) for \(z \geq 0,\) where \(a>0\) is a real number. Let \(\mathbf{F}=\langle x-y, y+z, z-x\rangle .\) For what value(s) of \(a\) (if any) does \(\iint_{S}(\nabla \times \mathbf{F}) \cdot \mathbf{n} d S\) have its maximum value?
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