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Help with vector fields and integration

A vector field
F= 2XZ(i) + YZ(j) + Z^2(k)

for the volume enclosed by the hemisphere

X^2 + Y^2 + Z^2 = a^2 for Z greater than or equal to zero

Show that part of the surface integral over the disc in the Z=0 plane is zero, because the unit normal to the plane is (n)=-(k)


I have put unit vectors in brackets. I can see qualitatively how this works, but I can't seem to prove it. I'm not sure whether to apply it to the vector field of the volume enclosed. When I apply it to the vector field I just get F=0. Which would obviously not dot product with anything. But this seems a bit to simple for an answer.

Help would be really appreciated!
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ztibor
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Original post by Bcx
A vector field
F= 2XZ(i) + YZ(j) + Z^2(k)

for the volume enclosed by the hemisphere

X^2 + Y^2 + Z^2 = a^2 for Z greater than or equal to zero

Show that part of the surface integral over the disc in the Z=0 plane is zero, because the unit normal to the plane is (n)=-(k)


I have put unit vectors in brackets. I can see qualitatively how this works, but I can't seem to prove it. I'm not sure whether to apply it to the vector field of the volume enclosed. When I apply it to the vector field I just get F=0. Which would obviously not dot product with anything. But this seems a bit to simple for an answer.

Help would be really appreciated!


You should to apply to the hemisphere. Parametrize it with aa and ϕ\phi at z=0.

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