- #1

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Say we have a vector function ##\vec{F} (\vec{r})=\hat{\phi}##.

a. Calculate ##\oint_C \vec{F} \cdot d\vec{\ell}##, where C is the circle of radius R in the xy plane centered at the origin

b. Calculate ##\int_H \nabla \times \vec{F} \cdot d\vec{a}##, where H is the hemisphere above the xy plane with boundary curve C.

c. Calculate ##\int_D \nabla \times \vec{F} \cdot d\vec{a}##, where D is the disk in the xy plane bounded by C.

(Verify Stokes's theorem in both cases.)

Solution:

First off, isn't it unnecessary to say verify Stokes's theorem in both cases? If we calculate a line integral, and then a surface integral bounded by the same curve, Stokes's theorem states that they're the same, yes?

Moving on...

a. We have [tex]\vec{F} \cdot d\vec{\ell}=(0,0,1)(dr, d\theta, rsin\theta \ d\phi)=rsin\theta \ d\phi[/tex]

So [tex]\oint_C \vec{F} \cdot d\vec{\ell} = Rsin\theta \int_0^{2 \pi} d\phi= 2 \pi R sin \theta [/tex]

Is this correct?

b. We have ##F_r=F_{\theta}=0## and ##F_{\phi}=1##, so for the curl I find [tex]\nabla \times \vec{F}=(\frac{1}{r}tan\theta, 0, -\frac{1}{r})[/tex]

I think this is right so far. But in calculating ##\int_H \nabla \times \vec{F} \cdot d\vec{a}##, I'm not sure what to use as ##d\vec{a}##. Any help? Thanks in advance.