- #1
Gameowner
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Homework Statement
Question Attached
Homework Equations
The Attempt at a Solution
So here I'm attempting b), I know \nabla \times \vec F\ is the curl, which in this case is defined by the matrix
\left[ \begin {array}{ccc} x&y&z\\ \noalign{\medskip}{\frac {d}{{\it <br /> dx}}}&{\frac {d}{{\it dy}}}&{\frac {d}{{\it dz}}}\\ \noalign{\medskip}<br /> -z&-x&{y}^{2}\end {array} \right]
which gives me the vector \nabla \times \vec F\ = 2*yi+j-k
Since my given vector is a function of theta and z, I apply the change of coordinates x=rcos(theta), y=rsin(theta), z=r which effectively changes the vector from
r(theta,z)=(sqrt(z)*cos(theta) , sqrt(z)/2*sin(theta) , z)
into
r(theta,r)=(sqrt(r)*cos(theta) , sqrt(r)/2*sin(theta) , r)
The equation of the integral being \int\int_S \nabla \times \vec F\cdot \hat n\, dS
so the \hat n\ is given by \hat n = \vec R_\theta \times \vec R_r for downward pointing normal.
Rtheta
\left[ \begin {array}{c} -\sqrt {r}\sin \left( \theta \right) <br /> \\ \noalign{\medskip}1/2\,\sqrt {r}\cos \left( \theta \right) <br /> \\ \noalign{\medskip}0\end {array} \right]
Rr
\left[ \begin {array}{c} 1/2\,{\frac {\cos \left( \theta \right) }{<br /> \sqrt {r}}}\\ \noalign{\medskip}1/4\,{\frac {\sin \left( \theta<br /> \right) }{\sqrt {r}}}\\ \noalign{\medskip}1\end {array} \right]
Thus taking the cross product and then yielding the normal
n=
\left[ \begin {array}{c} 1/2\,\sqrt {r}\cos \left( \theta \right) <br /> \\ \noalign{\medskip}\sqrt {r}\sin \left( \theta \right) <br /> \\ \noalign{\medskip}-1/4\end {array} \right]Then I substitute the cylindrical coordinates x=rcos(theta), y=rsin(theta), z=r into my vector F to get.
\left[ \begin {array}{c} 2\, \left( r \right) \sin \left( \theta<br /> \right) \\ \noalign{\medskip}1\\ \noalign{\medskip}-1\end {array}<br /> \right]
Thus finally, yielding the integral for the flux as.
\int _{0}^{2\,\pi}\!\int _{0}^{h}\!{r}^{ 1.5}\sin \left( \theta<br /> \right) \cos \left( \theta \right) +{r}^{ 0.5}\sin \left( \theta<br /> \right) +1/4{dr}\,{d\theta}
Just want to know if everything I've done so far is correct(most of it were wild/educated guesses), and the domain for z is confusing me with the h>0, when I'm evaluating the integral, what would I put as h?
Thank you.
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