Question

# z = x Let be the curve of intersection of the cylinder x ^ 2 + y ^ 2 = 1 and the plane , oriented positively when viewed from above . Let S be the inside of this curve , oriented with upward -pointing normal . Use Stokes ' Theorem to evaluate int S curl F* dS if F = yi + zj + 2xk.

z = x Let be the curve of intersection of the cylinder $$\displaystyle{x}^{{2}}+{y}^{{2}}={1}$$ and the plane , oriented positively when viewed from above . Let S be the inside of this curve , oriented with upward -pointing normal . Use Stokes ' Theorem to evaluate $$\displaystyle\int{S}{c}{u}{r}{l}{F}\cdot{d}{S}{\quad\text{if}\quad}{F}={y}{i}+{z}{j}+{2}{x}{k}$$.

2021-03-09

Step 1
Stock's Theorem:-
$$\displaystyle\int_{{C}}{F}.{d}{r}\int\int{c}{u}{r}{l}{\left({F}\right)}{d}{s}$$
Given that a curve say D (as name not mention in problem) of intersection of the cylinder $$\displaystyle{x}^{{2}}+{y}^{{2}}={1}$$ and the plane z=x and $$F=y\ i+z\ j+ 2x\ k$$
Also, S be inside of this curve, oriented with upward-pointing normal, parameterize of this curve is
$$\displaystyle{r}={<}{x},{y},{z}\ge{<}{x},{y},{x}{>}$$ as z = z
$$\displaystyle{r}={\cos{{0}}}{i}+{\sin{{0}}}{j}+{\cos{{0}}}{k}$$
since, we know $$\displaystyle{x}={1}.{\cos{{0}}}={\cos{{0}}},{y}={1}.{\sin{{0}}}={\sin{{0}}},{z}={x}={\cos{{0}}}$$ as radius =1
Here, $$\displaystyle{0}\le{0}\le{2}\pi{\left({a}{s}{x}^{{2}}+{y}^{{2}}={1}^{{2}}\right)}$$
then,
$$\displaystyle{r}={<}{\cos{{0}}},{\sin{{0}}},{\cos{{0}}}{>}$$
$$\displaystyle\frac{{{d}{r}}}{{{d}{0}}}={r}'={<}-{\sin{{0}}},{\cos{{0}}},-{\sin{{0}}}{>}$$
$$F(x,y,z)=yi+zj+2xk$$
$$\displaystyle{F}{\left({r}{\left({0}\right)}\right)}={\sin{{0}}}{i}+{\cos{{0}}}{j}+{2}{\cos{{0}}}{k}$$
Step 2
Now,
$$\displaystyle\int_{{D}}{F}{d}{r}={\int_{{0}}^{{{2}\pi}}}{F}{\left({r}{\left({0}\right)}\right)}\frac{{{d}{r}}}{{{d}{0}}}{d}{0}$$
$$\displaystyle\int_{{D}}{F}{d}{r}={\int_{{0}}^{{{2}\pi}}}{<}{\sin{{0}}},{\cos{{0}}},{2}{\cos{{0}}}{>}{<}-{\sin{{0}}},{\cos{{0}}},-{\sin{{0}}}{>}{d}{0}$$
$$\displaystyle\int_{{D}}{F}{d}{r}={\int_{{0}}^{{{2}\pi}}}{\left(-{{\sin}^{{2}}{0}}+{{\cos}^{{2}}{0}}-{2}{\sin{{0}}}{\cos{{0}}}\right)}{d}{0}$$
Note: $$\displaystyle-{{\cos}^{{2}}{0}}-{{\sin}^{{2}}{0}}={\cos{{20}}},{\sin{{20}}}={2}{\sin{{0}}}{\cos{{0}}}$$
$$\displaystyle\int_{{D}}{F}{d}{r}={\int_{{0}}^{{{2}\pi}}}{\left({\cos{{20}}}-{\sin{{20}}}\right)}{d}{0}={{\left[\frac{{{\sin{{20}}}}}{{2}}+\frac{{{\cos{{20}}}}}{{2}}\right]}_{{0}}^{{{2}\pi}}}$$
Note: $$\displaystyle-{\sin{{\left({2}\pi\right)}}}={\sin{{\left({0}\right)}}}={0},{\cos{{\left({2}\pi\right)}}}-{\cos{{\left({0}\right)}}}={1}-{1}={0}$$
$$\displaystyle\int_{{D}}{F}{d}{r}={0}$$
By, using Stock's Theorem, we have
$$\displaystyle\int\int{c}{u}{r}{l}{\left({F}\right)}{d}{s}=\int_{{D}}{F}.{d}{r}={0}$$
$$\displaystyle\int\int{c}{u}{r}{l}{\left({F}\right)}{d}{s}={0}$$