- #1
Reshma
- 749
- 6
The vector area is given by the integral [itex]\vec a = \int_s d\vec a[/itex] for a surface s.
I have three problems to prove based on this:
1] Find the vector area of a hemispherical bowl of radius R.
I solved a bit of this.
The area element for a hemisphere is given by [tex]d\vec a = R^2\sin \theta d\theta d\phi \hat r[/tex].
So,[tex]\vec a = R^2\int_{0}^{\frac{\pi}{2}}\sin \theta d\theta \int_{0}^{2\pi} d\phi \hat r[/tex]
How am I supposed to integrate over the unit vector [itex]\hat r[/itex]?
2] Show [tex]\vec a = 0[/tex] for any closed surface.
3] Show [tex] \vec a[/tex] is the same for all surfaces sharing the same boundary.
I have three problems to prove based on this:
1] Find the vector area of a hemispherical bowl of radius R.
I solved a bit of this.
The area element for a hemisphere is given by [tex]d\vec a = R^2\sin \theta d\theta d\phi \hat r[/tex].
So,[tex]\vec a = R^2\int_{0}^{\frac{\pi}{2}}\sin \theta d\theta \int_{0}^{2\pi} d\phi \hat r[/tex]
How am I supposed to integrate over the unit vector [itex]\hat r[/itex]?
2] Show [tex]\vec a = 0[/tex] for any closed surface.
3] Show [tex] \vec a[/tex] is the same for all surfaces sharing the same boundary.