Evaluate ∫ F ∙ dr, where F(x, y, z) = -y^2i + xj + 3z^2k and C is the curve of the intersection of the plane y + z = 2 and the cylinder x^2 + y^2 = 9. (Orient C to be counterclockwise when viewed from above.)
SOLUTION The curve C (an ellipse) is shown in the figure. Although ∫ F ∙ dr could be evaluated directly, it's easier to use Stokes' Theorem. We first compute
curl F =
| i j k |
| ∂/∂x ∂/∂y ∂/∂z |
| x 3z^2 | = ( )k
Although there are many surfaces with boundary C, the most convenient choice is the elliptical region S in the plane y + z = 2 that is bounded by C. If we orient S upwards, then C has the induced positive orientation. The projection D of S on the xy-plane is the disk x^2 + y^2 ≤ 9 and so with z = g(x, y) = 2 - y, we have
∫ F ∙ dr = ∫∫ curl F ∙ dS = ∫∫ dA
= ∫₀²ᵡ ∫₀³ (1 + )r dr dθ
= ∫₀²ᵡ [ r^2/2 + ]₀³ dθ
= ∫₀²ᵡ ( 9/2 + )dθ
= 9/2(2π) + 0 = 9π