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 (2, 1, 0).

Answer 1

Parameterize the line segment (call it
`C`) by


`\mathbf r(t)=\langle0,0,1\rangle(1-t)+\langle2,1,0\rangle t=\langle2t,t,1-t\rangle`

where
`0\le t\le1`. Then the work done by
`\mathbf f(x,y,z)` along the line segment is


`\displaystyle\int_C\mathbf f(x,y,z)\cdot\mathrm d\mathbf r=\int_(t=0)^(t=1)\mathbf f(x(t),y(t),z(t))\cdot\langle2,1,-1\rangle\,\mathrm dt`

`=\displaystyle\int_0^1\langle2t-t^2,t-(1-t)^2,(1-t)-(2t)^2\rangle\cdot\langle2,1,-1\rangle\,\mathrm dt`

`=\displaystyle\int_0^1(t^2+8t-2)\,\mathrm dt=\frac73`