Question

Evaluate the surfacuse stokes's theorem to evaluate f · dr

Answer 1

Since you're asked to use Stoke's theorem, I'm interpreting the question to be asking to compute the line integral

$\displaystyle\int_{\mathcal C}\mathbf f(x,y,z)\cdot\mathrm d\mathbf r$

which by Stoke's theorem is equivalent to the surface integral

$\displaystyle\iint_{\mathcal S}\nabla\times\mathbf f(x,y,z)\cdot\mathrm d\mathbf S$

where $\mathcal S$ is the positively-oriented surface with boundary $\mathcal C$.

Given $\mathbf f(x,y,z)=2y\,\mathbf i+3z\,\mathbf j+x\,\mathbf k$, we get curl

$\nabla\times\mathbf f=-3\,\mathbf i-\mathbf j-2\,\mathbf k$

We parameterize the surface $\mathcal S$ by

$\mathbf s(u,v)=5(1-u)(1-v)\,\mathbf i+5u(1-v)\,\mathbf j+5v\,\mathbf k$

with $0\le u\le1$ and $0\le v\le1$. Then we take the partial derivatives of $\mathbf s$ and check their cross product:

$\mathbf s_u\times\mathbf s_v=25(1-v)(\mathbf i+\mathbf j+\mathbf k)$

Now,

$\mathrm d\mathbf S=(\mathbf s_u\times\mathbf s_v)\,\mathrm du\,\mathrm dv$

so the surface integral reduces to

$\displaystyle-150\int_{v=0}^{v=1}\int_{u=0}^{u=1}(1-v)\,\mathrm du\,\mathrm dv=-75$