Tilted disks Let S be the disk enclosed by

QUESTION:

Tilted disks Let S be the disk enclosed by the curve \(C: \mathbf{r}(t)=\langle\cos \varphi \cos t, \sin t, \sin \varphi \cos t\rangle\), for \(0 \leq t \leq 2 \pi\). Where \(0 \leq \varphi \leq \pi / 2\) is a fixed angle.

Consider the vector field F = a x r. where \(\mathbf{a}=\left\langle a_{1}, a_{2}, a_{3}\right\rangle\), 'Lil is a constant nonzero vector and \(\mathbf{r}=\langle x, y, z\rangle\). Show that the circulation is a maximum when a points in the direction of the normal to S.

Text Transcription:

C: r(t) = langle cos varphi cos t, sin t, sin varphi cos t rangle

0 leq t leq 2pi

0 leq varphi pi/2

a = langle a_1, a_2, a_3 rangle

r = langle x, y, z rangle