Step-by-step explanation:
Given:
![\textbf{F} = (2xy + z^3)\hat{\textbf{i}} + x^3\hat{\textbf{j}} + 3xz^2\hat{\textbf{k}}](https://tex.z-dn.net/?f=%5Ctextbf%7BF%7D%20%3D%20%282xy%20%2B%20z%5E3%29%5Chat%7B%5Ctextbf%7Bi%7D%7D%20%2B%20x%5E3%5Chat%7B%5Ctextbf%7Bj%7D%7D%20%2B%203xz%5E2%5Chat%7B%5Ctextbf%7Bk%7D%7D)
This field will have a scalar potential
if it satisfies the condition
. While the first x- and y- components of
are satisfied, the z-component doesn't.
![(\nabla \times \textbf{F})_z = \left(\dfrac{\partial F_y}{\partial x} - \dfrac{\partial F_x}{\partial y} \right)](https://tex.z-dn.net/?f=%28%5Cnabla%20%5Ctimes%20%5Ctextbf%7BF%7D%29_z%20%3D%20%5Cleft%28%5Cdfrac%7B%5Cpartial%20F_y%7D%7B%5Cpartial%20x%7D%20-%20%5Cdfrac%7B%5Cpartial%20F_x%7D%7B%5Cpartial%20y%7D%20%5Cright%29)
![\:\:\:\:\:\:\:\:\: = 3x^2 - 2x \ne 0](https://tex.z-dn.net/?f=%5C%3A%5C%3A%5C%3A%5C%3A%5C%3A%5C%3A%5C%3A%5C%3A%5C%3A%20%3D%203x%5E2%20-%202x%20%5Cne%200)
Therefore the field is nonconservative so it has no scalar potential. We can still calculate the work done by defining the position vector
as
![\vec{\textbf{r}} = x \hat{\textbf{i}} + y \hat{\textbf{j}} + z \hat{\textbf{k}}](https://tex.z-dn.net/?f=%5Cvec%7B%5Ctextbf%7Br%7D%7D%20%3D%20x%20%5Chat%7B%5Ctextbf%7Bi%7D%7D%20%2B%20y%20%5Chat%7B%5Ctextbf%7Bj%7D%7D%20%2B%20z%20%5Chat%7B%5Ctextbf%7Bk%7D%7D)
and its differential is
![\textbf{d} \vec{\textbf{r}} = dx \hat{\textbf{i}} + dy \hat{\textbf{j}} + dz \hat{\textbf{k}}](https://tex.z-dn.net/?f=%5Ctextbf%7Bd%7D%20%5Cvec%7B%5Ctextbf%7Br%7D%7D%20%3D%20dx%20%5Chat%7B%5Ctextbf%7Bi%7D%7D%20%2B%20dy%20%5Chat%7B%5Ctextbf%7Bj%7D%7D%20%2B%20dz%20%5Chat%7B%5Ctextbf%7Bk%7D%7D)
The work done then is given by
![\displaystyle \oint_c \vec{\textbf{F}} • \textbf{d} \vec{\textbf{r}} = \int ((2xy + z^3)\hat{\textbf{i}} + x^3\hat{\textbf{j}} + 3xz^2\hat{\textbf{k}}) • (dx \hat{\textbf{i}} + dy \hat{\textbf{j}} + dz \hat{\textbf{k}})](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Coint_c%20%5Cvec%7B%5Ctextbf%7BF%7D%7D%20%E2%80%A2%20%5Ctextbf%7Bd%7D%20%5Cvec%7B%5Ctextbf%7Br%7D%7D%20%3D%20%5Cint%20%28%282xy%20%2B%20z%5E3%29%5Chat%7B%5Ctextbf%7Bi%7D%7D%20%2B%20x%5E3%5Chat%7B%5Ctextbf%7Bj%7D%7D%20%2B%203xz%5E2%5Chat%7B%5Ctextbf%7Bk%7D%7D%29%20%E2%80%A2%20%28dx%20%5Chat%7B%5Ctextbf%7Bi%7D%7D%20%2B%20dy%20%5Chat%7B%5Ctextbf%7Bj%7D%7D%20%2B%20dz%20%5Chat%7B%5Ctextbf%7Bk%7D%7D%29)
![\displaystyle = (x^2y + xz^3) + x^3y + xz^3|_{(1, -2, 1)}^{(3, 1, 4)}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%3D%20%28x%5E2y%20%2B%20xz%5E3%29%20%2B%20x%5E3y%20%2B%20xz%5E3%7C_%7B%281%2C%20-2%2C%201%29%7D%5E%7B%283%2C%201%2C%204%29%7D)
![= 422](https://tex.z-dn.net/?f=%3D%20422)
Answer:
D
Step-by-step explanation:
Since we don't know how many people we don't know how many servings would be consumed. therefore this is an indirect variation
Correct Answer: A. Scatter plot A
Explanation: The line y=x slopes upward from the lower left corner of the graph to the upper right corner, at a 45-degree angle. The scatter plot whose points most closely match that trend is scatter plot A.
Answer:
Step-by-step explanation:
First equation:
15 - x = - 3
-x = - 3 - 15
-x = -18 (now negative signs on b.s cut eachother)
[x = 18]
Second equation :
15 - x = 3
-x = 3 - 15
- x = -12 ( now negative sign on b.s cut eachother)
[x = 12]
so x = 18 and x = 12 are not same
Answer:
idk
Step-by-step explanation:
i wish i did