 # A particle moves along line segments from the origin to the points(3,0,0),(3,3,1),(0,3,1), and back to the origin under the influence of the force field F(x,y,z)=z^2i+3xyj+4y^2k. Use Stokes' Theorem to find the work done. glasskerfu 2021-02-09 Answered
A particle moves along line segments from the origin to the points(3,0,0),(3,3,1),(0,3,1), and back to the origin under the influence of the force field $F\left(x,y,z\right)={z}^{2}i+3xyj+4{y}^{2}k$.
Use Stokes' Theorem to find the work done.
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Step 1
From stokes theorem $\oint F.dr=\oint \oint \mathrm{\nabla }×FdS$.
Thus, calculate the curl of $F={z}^{2}i+3xyj+4{y}^{2}k$:
$\mathrm{\nabla }×F=\left[\begin{array}{ccc}i& j& k\\ \frac{\partial }{\partial x}& \frac{\partial }{\partial y}& \frac{\partial }{\partial z}\\ {z}^{2}& 3xy& 4{y}^{2}\end{array}\right]$
$=8yi+2zj+3yk$
Step 2
Thus, $\oint F.dr=\oint \oint \left(8ydydz+2zdxdz+3ydxdy\right)$. Now as per the given information $0\le x\le 3,0\le y\le 3\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}0\le z\le 1$.
Therefore the integral becomes:
$\oint F.dr={\int }_{z=0}^{1}{\int }_{y=0}^{3}8ydydz+{\int }_{z=0}^{1}{\int }_{x=0}^{3}2zdxdz+{\int }_{x=0}^{3}{\int }_{y=0}^{3}3ydxdy$
$=\frac{1}{2}\left(8×{3}^{2}+2×3+3×3×{3}^{2}\right)$
$=\frac{159}{2}$
$=79.5$ unit
Step 3
Thus work done is 79.5 unit.