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:
marka girişi yok mu güzel ama çok iyi Dersler ve doğru şekilde a-na-rim olur diye ben
Answer:
87 sq in
Step-by-step explanation:
Surface area of the net
![= 3(8) + 3(6) + (2) \frac{1}{2} (6)(5) + (5)(3) \\ \\ = 24 + 18 + 30 + 15 \\ \\ = 87 \: {in}^{2}](https://tex.z-dn.net/?f=%20%3D%203%288%29%20%2B%203%286%29%20%2B%20%282%29%20%5Cfrac%7B1%7D%7B2%7D%20%286%29%285%29%20%2B%20%285%29%283%29%20%5C%5C%20%20%5C%5C%20%20%3D%2024%20%2B%2018%20%2B%2030%20%2B%2015%20%5C%5C%20%20%5C%5C%20%20%3D%2087%20%5C%3A%20%20%7Bin%7D%5E%7B2%7D%20)
Answer:
add the absolute value for both, boom ur answer
Step-by-step explanation: