Question

Let F(r)=(2y,-z,x);calculate integer F(r) x dr?
let F(r)=(2y,-z,x);calculate integer F(r) x dr along the curve x=cost,y=sint,z=2cost from t=0 to t= pi/2....

Answer 1

I'm going to assume a lot of things since this is typed strangely.
I'm assuming you want the line integral of the vector field F along the given curve.\[\int_{a}^{b}\overrightarrow{F}(r)\cdot d\overrightarrow{r}=\int_{a}^{b}\overrightarrow{F}(\overrightarrow{r}(t))\cdot \overrightarrow{r'}(t)\]so we need the parameterization of our vector function.\[\overrightarrow{F}(\overrightarrow{r}(t))=<2y(t),-z(t),x(t)>=<2\sin t,-2\cos t,\cos t>\]and we need the derivative of the curve\[\overrightarrow{r'}(t)=<-\sin t,\cos t,-2\sin t>\]out integrand will be the dot-product if these two vector quantities.\[\int_{a}^{b}\overrightarrow{F}(\overrightarrow{r}(t))\cdot \overrightarrow{r'}(t)dt\]\[=\int_{0}^{\pi/2}<2\sin t,-2\cos t,\cos t>\cdot<-\sin t, \cos t,-2\sin t>dt\]So now just take the dot-product and integrate.