Answer: To compute the two-dimensional divergence of the vector field F = (3y, 4x), we need to apply the divergence operator to F:
div F = ∂Fx/∂x + ∂Fy/∂y
= ∂(3y)/∂x + ∂(4x)/∂y
= 0 + 0
Therefore, the divergence of F is zero, which means that F is a divergence-free or source-free vector field.
To evaluate the two integrals in Green's theorem, we need to parameterize the boundary of the region R, which consists of two curves: y = 9 - x² and y = 0.
Let's first compute the line integrals of F along each curve.
Along y = 9 - x², we have:
∫ F · dr = ∫ (3y, 4x) · (dx, dy)
= ∫ 3(9-x²) dx + 4x dy
= ∫ 27 dx - 3x² dx + 4xy dy
= 27x - x³ + 2xy |y=0^9-x²
= 27x - x³ + 18x(9-x²)
= -x^3 + 171x
Along y = 0, we have:
∫ F · dr = ∫ (3y, 4x) · (dx, dy)
= ∫ 4x dy
= 0
Next, we need to compute the double integral of the curl of F over the region R:
∬ curl F · dA = ∬ (∂Fy/∂x - ∂Fx/∂y) dA
= ∬ (-4) dA
= -4 ∬ dA over R
The region R is bounded by y = 9 - x² and y = 0, and its projection onto the x-axis is the interval [-3, 3]. Therefore, we can write:
∬ dA over R = ∫_{-3}^3 ∫_0^{9-x²} dy dx
= ∫_{-3}^3 (9-x²) dx
= 54
Finally, we can apply Green's theorem:
∫ F · dr = ∬ curl F · dA
or
(-x^3 + 171x) - 0 = -4(54)
-4(54) = -216
Therefore, the two integrals are consistent with each other, and the vector field F is source-free.