Surface integral.

    

Example 8.27   Integrate the function g(x,y,z)=xyz over the upper unit hemisphere, i.e. the surface defined by $x^2+y^2+z^2=1, \; z geq 0$.

 

  

Figure 7: The upper unit hemisphere.

 

• $f(x,y,z)= x^2+y^2+z^2-1=0 \; \Longrightarrow \; \overrightarrow{\nabla} f = (2x,2y,2z)$;
• $\begin{vmatrix}\overrightarrow{\nabla} f \end{vmatrix}= \sqrt{4x^2+4y^2+4z^2}= \sqrt{4}=2$;
• $\overrightarrow{p} = \overrightarrow{k}\; \Longrightarrow \; \overrightarrow{\nabla} f \cdot \overrightarrow{p} = 2z$;
• We integrate:
  $$\begin{align*}I &= \iint_{\mathcal{S}} g(x,y,z) d \sigma = \iint_{x^2+y^2=1} xyz... ...\ \quad &= \frac 14 \int_0^{2 \pi } \sin 2 \theta \; d \theta =0 . \end{align*}$$

Example 8.28