 Home

Codex Wiki

List of universities

University of Oxford

Course of the University of Oxford

Undergraduate course of the University of Oxford

Mathematics course of the University of Oxford

Past exam of the mathematics course of the University of Oxford

2021

Prelims

Mathematics V

{c}

The <curl> is $\nabla\times\mathbf F=(x,y,-2z)$. For the upward orientation of the paraboloid, the <vector surface area element> is $(2x,2y,1)\,dx\,dy$. Therefore <Stokes theorem> gives
$$
\begin{aligned}
\oint_C\mathbf F\cdot d\mathbf r
&=\iint_{x,y\geq0,\ x^2+y^2\leq4}\{2x^2+2y^2-2(4-x^2-y^2)\}\,dx\,dy\\
&=\int_0^{\pi/2}\int_0^2(4r^2-8)r\,dr\,d\theta
=\boxed{0}.
\end{aligned}
$$
The upward normal gives the prescribed anticlockwise boundary orientation viewed from above.

Solution

Past exam of the mathematics course of the University of Oxford / 2021 / Prelims / Mathematics V / 3 / a / i / Solution

Past exam of the mathematics course of the University of Oxford / 2021 / Prelims / Mathematics V / 3 / a / i / Solution

 Ancestors (13)

 Discussion (0)

 Articles by others on the same topic (0)

Past exam of the mathematics course of the University of Oxford / 2021 / Prelims / Mathematics V / 3 / a / i / Solution

 Ancestors (13)

 Discussion (0)  Subscribe (1)

 Articles by others on the same topic (0)

 Discussion (0)