1. ## surface integrals

evaluate double int over S (e^z) dS
where S is the surface of the sphere x^2 + y^2 + z^2 = a^2

I know dS = a^2(sintheta)(d.theta)(d.phi) using the parametrization

r = (a.sintheta.cosphi,a.sintheta.sinphi,acosphi)

so from there do I just calculate

int int e^(acosphi) a^2(sintheta)(d.theta)(d.phi)

?

many thanks

2. Originally Posted by James0502
evaluate double int over S (e^z) dS
where S is the surface of the sphere x^2 + y^2 + z^2 = a^2

I know dS = a^2(sintheta)(d.theta)(d.phi) using the parametrization

r = (a.sintheta.cosphi,a.sintheta.sinphi,acosphi)

so from there do I just calculate

int int e^(acosphi) a^2(sintheta)(d.theta)(d.phi)

?

many thanks
The surface can be parametrized as $\displaystyle \bold{g} : [0,2\pi]\times [0,\pi] \to \mathbb{R}^3$ as $\displaystyle \bold{g}(\theta,\phi) = (a^2\cos \theta \sin \phi, a^2 \sin \theta \sin \phi, a^2 \cos \phi)$.

The function that you have is $\displaystyle f(x,y,z) = e^z$.

Therefore, $\displaystyle \iint_S f dS = \int_0^{\pi} \int_0^{2\pi} f(\bold{g}(\theta,\phi)) \cdot \left| \frac{\partial \bold{g}}{\partial \theta} \times \frac{\partial \bold{g}}{\partial \phi} \right| d\theta ~ d\phi$

Compute that for your surface integral.