Let me preface by saying I am a physics major. So I am coming at differential forms from the perspective of physics, i.e. work, flows, em fields, etc. My question is this. My understanding is that a basic 1-form dx, dy, or dz takes a vector v = (v1,v2,v3) and gives back the corresponding component of that vector, i.e. the size of that vector's projection onto the corresponding coordinate axis. The complete 1-form, then, such as phi = Adx + Bdy + Cdz, is the sum of multiples of these projection sizes, where the coefficients are functions of x,y,z. My confusion is regarding which vector space these projections "occur" in: R^3, the tangent space or the dual space? In any case, how is it possible to project onto the x, y, or z axes within the tangent or dual spaces? Wouldn't the "axes" in the dual space be the "dx-axis", etc.? Or am I just confusing myself unnecessarily by assuming that the 1-form and the projection (or the input vector and the projection) have to be in the same space? Thanks for your help.