Differential of a function vs differential forms

[Damidami]

Hi,
I understand the concept of the differential of a (differentiable) function at a point as a linear transformation that "best" approximates the increment of the function there. So for example the differential of a function f : D \subseteq \mathbb{R}^2 \to \mathbb{R} could maybe be df = 8 dx + 9 dy at a given point, say (2,3)

But I wanted to know how this concept of differential is related to the integral concept of differential form, so for example 8dx + 9dy can be thoght as a differential form and be integrated over a curve. But the differential form usually varies from point to point so it could be more like (3x+y^2)dx + (3xy) dy, so I'm not sure if it also is been thought as a linear map, or should I think of these both concepts as completely different and unrelated things?

Thanks.

 

[Bacle2]

I don't know if this is what you're looking for, but the differential of a differentiable function is a differentiable form, i.e., if f is a differentiable function, then df is a 1-form.