I am trying to compare an example I have seen in different places with slightly different setups done with differential forms and double integration to just have a feel for how differential forms can help even at a very basic level.
Differential forms:
Calculating the circulation of the vector field $\vec F=2xzi - z \hat j +x^2 \hat k$ along the intersection of the surfaces $\{y^2 + z^2=6\}$ and $\{5x-9y+3z=11\}$ is resolved by turning the field into a one form
$$\alpha_{\vec F}=2xz \mathrm dx - z\mathrm dy+x^2\mathrm dz$$
and applying the differential operator
$$\alpha_{\vec F}=2z \mathrm dx\wedge\mathrm dx + 2x \mathrm dz\wedge\mathrm dx- \mathrm dz\wedge\mathrm dy+2x\mathrm dx\wedge\mathrm dz= \mathrm dy\wedge\mathrm dz$$
This is equivalent to the curl being $\vec i,$ and $\int_{\partial D} \alpha_{\vec F}=\int_{D}\mathrm \alpha_{\vec F}=\int_D \mathrm dy\wedge\mathrm dz$ is the area of the projected intersection of the cylinder with the plane on the $y,z$ plane, or $6\pi.$
Double integrals:
The curl of the vector field is exactly $\nabla x \vec F =1 \hat i.$ Therefore, and given the strict orientation of the curl with the $x$ axis, and that the projection of the ellipse at the intersection of the plane and cone above on the $y,z$ plane "sees" the entire boundary, and it is orthogonal to the curl, the area of integration will be the disk of radius $\sqrt 6$ on the $y,z$ plane, and flux of the curl field through the disk is easily calculated as
$$ \oint \alpha_{\vec F}=\underset{S}{\int\int}1\hat i \mathrm dS=\int_{\theta=0}^{2\pi}\int_{r=0}^{\sqrt 6} 1 r\mathrm dr \mathrm d \theta=6\pi$$
Both systems seem quite straightforward. Would there be any difference in using differential forms?