Wikipedia says:
Let $$ \omega=f_{z}\, \mathrm dx \wedge \mathrm dy + f_{x}\, \mathrm dy \wedge \mathrm dz + f_{y}\, \mathrm dz \wedge \mathrm dx $$ be a 2-form on a surface with parametrization $$\mathbf{x} (s,t)=( x(s,t), y(s,t), z(s,t))\!$$
defined on some domain $D.$
Then, the surface integral of the two-form on the surface $S$ is given by
$$ \int_{S} \omega = \int_D \left[ f_{z} ( \mathbf{x} (s,t)) \frac{\partial(x,y)}{\partial(s,t)} + f_{x} ( \mathbf{x} (s,t))\frac{\partial(y,z)}{\partial(s,t)} + f_{y} ( \mathbf{x} (s,t))\frac{\partial(z,x)}{\partial(s,t)} \right]\, \mathrm ds\, \mathrm dt$$
Now my question is:
Is this the same as $$\int_{D} \omega(\mathbf{x}(s,t))(\frac{\partial \mathbf{x}}{\partial s}(s,t),\frac{\partial \mathbf{x}}{\partial t}(s,t)) ds dt ?$$
If anything is unclear, please let me know.
Yes, @RealAnalysis, it's precisely that. Have you learned about pullback of differential forms? That's what either formula gives you.