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Vesnalui [34]
3 years ago
15

F=(2xy +z³)i + x³j + 3xz²k find a scalar potential and work done in moving an object in the field from (1,-2,1) to (3,1,4)​

Mathematics
1 answer:
Alex73 [517]3 years ago
5 0

Step-by-step explanation:

Given:

\textbf{F} = (2xy + z^3)\hat{\textbf{i}} + x^3\hat{\textbf{j}} + 3xz^2\hat{\textbf{k}}

This field will have a scalar potential \varphi if it satisfies the condition \nabla \times \textbf{F}=0. While the first x- and y- components of \nabla \times \textbf{F} 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)

\:\:\:\:\:\:\:\:\: = 3x^2 - 2x \ne 0

Therefore the field is nonconservative so it has no scalar potential. We can still calculate the work done by defining the position vector \vec{\textbf{r}} as

\vec{\textbf{r}} = x \hat{\textbf{i}} + y \hat{\textbf{j}} + z \hat{\textbf{k}}

and its differential is

\textbf{d} \vec{\textbf{r}} = dx \hat{\textbf{i}} + dy \hat{\textbf{j}} + dz \hat{\textbf{k}}

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}})

\displaystyle = (x^2y + xz^3) + x^3y + xz^3|_{(1, -2, 1)}^{(3, 1, 4)}

= 422

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Explanation:  

Please follow the diagram in attachment.  

As we know median from vertex C to hypotenuse is CM  

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3 years ago
What are the solutions to the equation x - 7/x = 6
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Step-by-step explanation:

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To find Sn, we need to draw out equations for each a₇ and a₁₉

In an arithmetic progression,

Sn = a + (n-1)d

Where Sn = sum of the A.P

a = first term

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a₇ = 32

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32 = a + 6d ........equation (i)

a₁₉ = 140

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