I have this stokes theorem problem but my book doesn't clearly explain to me how the normal vector comes about. Here is the problem:
$$F = [y,z,x]$$ and S is this parabaloid:
So I guess z when expressed as a function of x and y = $1 - (x^2 + y^2)$. Why is this?
So parametrically it seems like the circle can be expressed as: $r(s) = [cos(s), sin(s), 0]$ because z = 0 and it's a circle. So $r'(s) = [-sin(s), cos(s), 0]$ So $$F(r(s)) = [\sin{s}, 0, \cos{s}]$$
Is that right os far?
But then I see the normal vector is $[2x, 2y, 1]$. Why is this?
PRoblem is here:
Why is this defined as the normal vector?