To find the work done using Stoke's theorem,given $4 $ points in space and $\mathbf{F}(x,y,z) = z^2~\mathbf{i}+ 2xy~\mathbf{j}+ 4y^2~\mathbf{k}$

Question

A particle moves along line segments from the origin to the points $(1,0,0)$,$(1,2,1)$,$(0,2,1)$ and back to the origin under the influence of force field $$\mathbf{F}(x,y,z) = z^2~\mathbf{i}+ 2xy~\mathbf{j}+ 4y^2~\mathbf{k}$$

Find the work done

I know how to calculate the work done using line integral by considering each line segment but since it's a tedious process, I'm supposed to use Stoke's theorem in this case but I don't know how to get a suitable surface enclosed by the boundary $C$ here.Do I have to get an equation for plane through $4$ points?How?

Answer 1

Plotting the four points (using, for instance, geogebra) you should be able to easily see that they define a parallelogram in space. In particular, they define the parallelogram $$\mathbf r(s,t) = s\mathbf u_1 + t\mathbf u_2,\quad s,t\in [0,1]$$ where $\mathbf u_1 = \mathbf i$ and $\mathbf u_2 = 2\mathbf j + \mathbf k$ are two of the sides of the parallelogram.

So then you can set up your surface integral in terms of $s$ and $t$.