Understanding about representing regular spatial curve using differential form

• A spatial curve is represented as a vector-valued 0-form \(\gamma(s)\).

• The tangent vector of the curve is not the 1-form \(d\gamma\) but the application of this 1-form to the basis \(\frac{\pdiff }{\pdiff s}\). N.B. For a vector-valued 1-form, its application to a vector is not a scalar but a vector.

• When we represent the tangent vectors of a curve using differential form, it is obvious that if \(d\gamma\) is zero at a point \(s_0\), then \(d\gamma(\frac{\pdiff }{\pdiff s})\vert_{s_0}\) is always a zero vector. This means the tangent vector at \(s_0\) cannot be well defined via a differential form, even though a tangent vector geometrically exists. This is why the definition of a regular curve requires \(d\gamma\neq 0\) everywhere on the curve.