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I understand most of the work involved with these types of questions, but there is one point in an example I'm following that I don't understand.

1. The problem statement, all variables and given/known data

Evaluate:

[tex]I = \int{(z^2)}dS[/tex] over the positive quadrant of a sphere, where (x,y > 0).

2. Relevant equations

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

[tex]\underline{\hat{n}} = \frac{\nabla f}{|\underline{\nabla}f|}[/tex]

3. The attempt at a solution

Project onto the xy plane:

[tex]dS = \frac{dxdy}{\underline{\hat{n}} \cdot \underline{\hat{k}}}[/tex]

At any point on the surface:

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

Therfore:

[tex]\int{(z^2)}dS = \int{\int{(1-x^2-y^2)}}\frac{dxdy}{\underline{\hat{n}} \cdot \underline{\hat{k}}}[/tex]

[tex]\underline{\hat{n}} = \frac{\nabla f}{|\underline{\nabla}f|}[/tex]

[tex]= x \underline{\hat{i}}+y\underline{\hat{j}}+\left (\sqrt{1-x^2-y^2} \right)\underline{\hat{k}}[/tex]

I don't understand how to get to the next step:

[tex]\underline{\hat{n}} \cdot \underline{\hat{k}} = \sqrt{1-x^2-y^2}[/tex]

Its probably something really simple, knowing me.

I've tried the dot product, but couldn't get the answer from that.

Would anyone be gracious enought to end my torment?

I hope my Tex is okay. Its changed since I was last here, though it could be my reinstallation of ProText. Everything is in bold and the gaps between equations are much bigger.

Thanks.

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# Homework Help: Surface Integral: dot product of two unit vectors

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