Evaluating Surface Integral with Stokes Theorem

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To evaluate the surface integral using Stokes' Theorem, the vector field F is defined as F=(x^2+y^2, x, 2xyz), and the surface S is a hemisphere of radius a in the upper half-space. The boundary curve dr that bounds this surface is the circle defined by x² + y² = a² in the xy-plane. To compute the integral, one can convert to spherical polar coordinates, differentiating to find dr, but determining the limits of integration can be challenging. Visualizing the problem through a diagram can aid in understanding the geometry involved. A clear representation of the surface and its boundary is essential for successful evaluation.
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use the stokes theorem to evaluate the surface integral [curl F dot dS] where
F=(x^2+y^2; x; 2xyz)
and S is an open surface x^2+y^2+z^2=a^2 for z>=0. So i guess its a hemisphere of radius a lying on x-y plane.
I don't see however how to take F dot dr. What is this closed curve dr bounding this hemisphere? I guess we can take spherical polar coordinates, but still once i have x,y,z in terms of r,phi,theta i still don't know "dr" (just differentiate ?) and what will the limits of integration be? Can someone show me?
 
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The boundary of the surface is just the circle x² + y² = a² in the xy plane. Did you try drawing a picture?
 

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