- #1
S. Moger
- 53
- 2
Homework Statement
Compute [itex]\int_S \vec{F} \cdot d\vec{S}[/itex]
[itex]\vec{F} = (xz, yz, z^3/a)[/itex]
S: Sphere of radius [itex]a[/itex] centered at the origin.
Homework Equations
[itex]x = a \sin(\theta) \cos(\varphi)[/itex]
[itex]y = a \sin(\theta) \sin(\varphi)[/itex]
[itex]z = a \cos(\theta) [/itex]
Phi : 0->2 pi, Theta : 0->pi/2 .
The Attempt at a Solution
[itex]\vec{F} = a^2 \cos(\theta) \cdot \{ \sin(\theta) \cos(\varphi), \sin(\theta) \sin(\varphi), \cos(\theta)^2 \}[/itex]
[itex]d\vec{S} = \frac{ \partial{\vec{r} }} {\partial{\theta} } \times \frac{ \partial{\vec{r}}}{\partial{\varphi}} d\theta d\varphi = a^2 \sin(\theta) \cdot \{ \sin(\theta) \cos(\varphi), \sin(\theta) \sin(\varphi), \cos(\theta) \}[/itex]
[itex]\int_S \vec{F} \cdot d\vec{S}[/itex] = [itex]\int_\varphi d\varphi \int_\theta ... d\theta = 2 \pi a^4 \int_\theta ... d\theta = 9 \pi a^4 / 10[/itex]
While the correct answer is [itex]\frac{4}{5} \pi a^4[/itex] .I'm relatively sure this isn't a book-keeping issue, I double checked the computations manually and with mathematica. Maybe I'm missing something (or maybe there's an easier way to "see" the answer).