I am taking Vector Calculus and we just learned the Divergence Theorem.
The task is the evaluate $$∬_S (x+y) dS$$ where S is the upper half of the unit sphere with the unit disk in the xy plane closing the surface on the bottom.
We are NOT given the value of F but are supposed to calculate it after calculating the unit normal vector using : $$∬_S F \cdot \hat n dS = ∬_S (x+y) dS $$
I calculated $ \hat n= <x,y,z>$ for the upper half of the unit sphere which gives $F(x,y,z)=<1,1,0>$
but $\hat n = <0,0,-1> $ for the unit disk portion of the surface which gives $F(x,y,z) = <0,0,-x-y>$ when calculating F from: $$ F \cdot \hat n = x+y$$
Can I apply the Divergence theorem AS I'M REQUIRED TO DO when the surface gives me 2 different $\hat n$ each of which gives a different $F(x,y,z)$?
BTW I recognize that both $F(x,y,z)$ that I found are incompressible which tells me there is zero flux across the surfaces but I'm supposed to use the Divergence Theorem to get the answer.
(The problem before this one was to calculate $∬_S (2x+2y+z^2) dS$ where S was the entire unit sphere - which made sense with the Divergence Theorem since there was only one $\hat n$ resulting in only one $F(x,y,z)$.)
I did try to ask the professor via email but his response suggested that he didn't completely read my email since he just said I could use work from the previous part in the problem...