Problem integrating over a sphere.

[S. Moger]

Homework Statement

Compute \int_S \vec{F} \cdot d\vec{S}

\vec{F} = (xz, yz, z^3/a)

S: Sphere of radius a centered at the origin.

Homework Equations

x = a \sin(\theta) \cos(\varphi)
y = a \sin(\theta) \sin(\varphi)
z = a \cos(\theta)

Phi : 0->2 pi, Theta : 0->pi/2 .

The Attempt at a Solution

\vec{F} = a^2 \cos(\theta) \cdot \{ \sin(\theta) \cos(\varphi), \sin(\theta) \sin(\varphi), \cos(\theta)^2 \}

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) \}

\int_S \vec{F} \cdot d\vec{S} = \int_\varphi d\varphi \int_\theta ... d\theta = 2 \pi a^4 \int_\theta ... d\theta = 9 \pi a^4 / 10

While the correct answer is \frac{4}{5} \pi a^4 .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).

 

[vela]

What's the integrand you came up with?

 

[mfb]

Stokes' theorem might lead to an easier integral (didn't test it).

$$\vec{F} = a^2 \cos(\theta) \cdot \{ \sin(\theta) \cos(\varphi), \sin(\theta) \sin(\varphi), \cos(\theta)^2 \}$$

This is still in cartesian coordinates? Then it shouldn't have that prefactor.

 

[S. Moger]

I got this:
\int_S \vec{F} \cdot d\vec{S}= a^4 \int \int \sin(\theta)^3 \cos(\theta) + \sin(\theta) \cos(\theta)^4 d\varphi d\theta

The phi variable "disappears" ( = 1) because of the trigonometric identity, so I separate its integral from the expression involving theta.

The expression for the F-vector should be valid as it stands (in cartesian coordinates). I inserted the expressions for x y z in theta, phi and a, and extracted z.

 

[vela]

When you integrate the $\sin^3\theta \cos\theta$ term, it'll vanish because the integrand is odd around $\theta = \pi/2$. Is that what you found? The final answer pops out from the second integral.

 

[mfb]

Ah right, you have the z everywhere.

 

[S. Moger]

When you integrate the $\sin^3\theta \cos\theta$ term, it'll vanish because the integrand is odd around $\theta = \pi/2$. Is that what you found? The final answer pops out from the second integral.

If I integrate it from 0 to pi/2 I should get 1/4. (It's just \frac{\sin^4{\theta}}{4}, with theta=0 vanishing, if I'm not mistaken). But makes me think about the range (integration limits, not sure about the word for it). But if I spanned -pi/2 to pi/2 I would get zero in any case

 

[vela]

Oh, I didn't notice you had written down the wrong limits. $\theta$ does indeed go from 0 to $\pi$, not $\pi/2$.

 

[S. Moger]

Yes, that's it! Thank you very much.