1. The problem statement, all variables and given/known data The vector ﬁeld F is deﬁned in 3-D Cartesian space as F = y(z^2−a^2)i + x(a^2− z^2)j, where i and j are unit vectors in the x and y directions respectively, and a is a real constant. Evaluate the integral Integral:(∇ ×F)·dS, where S is the open surface of the hemisphere x^2+ y^2+ z^2= a^2, z ≥ 0 : (i) by direct integration over the surface S (ii) using Stokes’ theorem 2. Relevant equations Stokes Theorem 3. The attempt at a solution I've calculated the integral over the surface and the line integral around the boundary curve, and both answers are 2*pi*a^4. However, my problem is that when i try to use stokes theorem to switch the integral from the surface of the hemisphere to the surface of the circle on the bottom of the hemisphere, via stokes theorem, my answer is half the correct answer. (∇ ×F).dS = 2a^2 (as z = 0 on circle). 2a^2 = 2x^2+2y^2. The integral of 2x^2+2y^2 over the surface of the circle is 2*pi*integral(r^3 dr) between 0 and a, which just comes out as pi*a^4. I'm not sure where i'm going wrong here, maybe the integrand should be a^2 r dr?? But the field varies over the surface of the circle so i don't think this is right??