I have two problems on surface integrals.(adsbygoogle = window.adsbygoogle || []).push({});

1] I have a constant vector [itex]\vec v = v_0\hat k[/itex]. I have to evaluate the flux of this vector field through a curved hemispherical surface defined by [itex]x^2 + y^2 + z^2 = r^2[/itex], for z>0. The question says use Stoke's theorem.

Stoke's theorem suggests:

[tex]\int_s \left(\vec \nabla \times \vec v\right) \cdot d\vec a = \int_p \vec v \cdot d\vec l[/tex]

But the curl of this vector comes out to be zero :yuck:. Am I going right? How is the surface integral evaluated?

2] I have a vector field [itex]\vec A = y\hat i + z\hat j + x\hat k[/itex]. I have to find the value of the surface integral:

[tex]\int_s \left(\vec \nabla \times \vec A\right) \cdot d\vec a[/tex]

The surface S here is a paraboloid defined by:

[tex]z = 1 - x^2 - y^2[/tex]

I evaluated the curl and it comes out to be:

[tex]\vec \nabla \times \vec A = -1\left(\hat i + \hat j + \hat k\right)[/tex]

I need help here on the procedure to evaluate the surface integral.

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Homework Help: 2 Surface integral problems

**Physics Forums | Science Articles, Homework Help, Discussion**