I am trying to verify Stokes' Theorem for a hemisphere with radius 3. I have only worked and found examples where the unit sphere has been used and I'm not sure how to factor in the value of the radius. This is the full question I am trying to answer:
Verify Stokes' Theorem for the hemisphere $D: x^2 + y^2 + z^2 = 9, z\geq0$ its bounding circle $C: x^2 + y^2 = 9, z=0$ and the vector field $\overrightarrow{A} = y\overrightarrow{i} - x\overrightarrow{j}$.
(I'm sure this is a really stupid question so please forgive me, I'm really struggling and nothing I've read or watched has helped me get to the solution.)
Suppose for the moment that $C$ is the unit circle in the plane $z=0$ ($x^2+y^2=1$), and $D$ is the hemisphere with radius $1$ ($x^2+y^2+z^2=1$, $z\ge0$). We parameterize $C$ by
$$\vec r(t)=\cos t\,\vec\imath+\sin t\,\vec\jmath$$
with $0\le t\le2\pi$, and $D$ by
$$\vec s(u,v)=\cos u\sin v\,\vec\imath+\sin u\sin v\,\vec\jmath+\cos v\,\vec k$$
with $0\le u\le2\pi$ and $0\le v\le\pi$.
For the hemisphere with radius $3$ ($x^2+y^2+z^2=9$, $z\ge0$), we make the following adjustments:
$$\vec r(t)=\color{red}3\cos t\,\vec\imath+\color{red}3\sin t\,\vec\jmath$$
$$\vec s(u,v)=\color{red}3\cos u\sin v\,\vec\imath+\color{red}3\sin u\sin v\,\vec\jmath+\color{red}3\cos v\,\vec k$$