I am trying to solve the following problem:
I shall calculate the line integral in the given vector-field F. You can ignore the text under the Field.
I started to parametrisize the curve itself, one parametrisation for each way (I,II,III) - so I was asking myself if this is necessary or if there's a smarter way to do that.
After I got some paramaters z(t) and y(t), I ran into the following problem and this is the main reason I am calling out for help:
The given vectorfield has an x component and a component which is dependend on x. I am not sure what to with this. I would have argued that we can ignore the x component of the vector-field, because no work is done there and each dot product will return 0. Nonetheless there is still an x variable in the z component of the field and I do not have any parameters for x, so basically my dot product would include an x.
My calculations just for way II: $$\vec{s}=(2+(0-2)t,0+(2-0)t)\\ z(t)=2-2t , y(t)=2t$$
If the line segment II goes from $(0,2,0)$ to $(0,0,2)$, then the parameterization should be $s=(0,2−2t,2t), 0\leq t\leq 1.$ In the integral, you wouldn't have an $x$, since $x=0.$ I'm interpreting the diagram as lying in the $yz$-plane, so $x=0$ and only the second component of $F$ is non-zero.