Question

Find the work done by the force field f(x, y, z) = x − y2, y − z2, z − x2 on a particle that moves along the line segment from (0, 0, 1) to (3, 1, 0).

Answer 1

The path can be parameterized by $\mathbf r(t)=\langle0,0,1\rangle(1-t)+\langle3,1,0\rangle t=\langle3t,t,1-t\rangle$ with $0\le t\le1$. So, the work done by $\mathbf F(x,y,z)=\langle x-y^2,y-z^2,z-x^2\rangle$ is given by the line integral

$\displaystyle\int_C\mathbf F\cdot\mathrm d\mathbf r$

where $C$ is the line segment. Equivalently,

$\displaystyle\int_{t=0}^{t=1}\langle3t-t^2,t-(1-t)^2,1-t-9t^2\rangle\cdot\langle3,1,-1\rangle\,\mathrm dt$
$=\displaystyle\int_0^1(5t^2+13t-2)\,\mathrm dt$
$=\dfrac{37}6$