Question

Find the work done by the vector field F~ = x 2y~i + 1 3 x 3~j + xy~k along the curve of intersection C of the paraboloid z = y 2 − x 2 and the cylinder x 2 + y 2 = 1, oriented counterclockwise as viewed from the above.

Answer 1

Let $x=\cos t$ and $y=\sin t$. Then $C$ can be parameterized by

$\vec r(t)=\cos t\,\vec\imath+\sin t\,\vec\jmath+(\sin^2t-\cos^2t)\,\vec k$

with $0\le t\le2\pi$, and its derivative is

$\dfrac{\mathrm d\vec r}{\mathrm dt}=-\sin t\,\vec\imath+\cos t\,\vec\jmath+4\sin t\cos t\,\vec k$

Now,

$\vec F(x,y,z)=x^2y\,\vec\imath+\dfrac{x^3}3\,\vec\jmath+xy\,\vec k$

$\implies\vec F(\vec r(t))=\cos^2t\sin t\,\vec\imath+\dfrac{\cos^3t}3\,\vec\jmath+\cos t\sin t\,\vec k$

Then the work done by $\vec F$ along $C$ is

$\displaystyle\int_C\vec F(x,y,z)\cdot\mathrm d\vec r=\int_0^{2\pi}\vec F(\vec r(t))\cdot\frac{\mathrm d\vec r}{\mathrm dt}\,\mathrm dt=\int_0^{2\pi}\left(3\cos^2t\sin^2t+\frac{\cos^4t}3\right)\,\mathrm dt=\boxed{\pi}$