I am given the following exercise:
Calculate the work for the force if $$\overrightarrow{F}=\overrightarrow{i}(x^2-y)+\overrightarrow{j}(y^2-z)+\overrightarrow{k}(z^2-x)$$
where the path of integration begins from $(0,0,0)$ and reaches at the point $A(1,1,1)$ along the line $OA$..
According,to my notes:
$$x=y=z=t, 0 \leq t \leq 1$$ $$\overrightarrow{R}=t(\hat{i}+\hat{j}+\hat{j})$$ $$\frac{d\overrightarrow{R}}{ds}=\overrightarrow{i}+\overrightarrow{j}+\overrightarrow{k}$$ $$\overrightarrow \cdot \frac{d\overrightarrow{R}}{ds}=3(s^2-s)$$ $$W=\int_0^1 \overrightarrow{F} \frac{d\overrightarrow{R}}{ds}ds= \dots = \frac{-1}{2}$$
But..do we have to take at the beggining the variable $t$ and then $s$ , is there a difference?or could I take just one of them?
There's a mixup. You should indeed use either $t$ or $s$ consistently and not mix them.