Many text books have varying definitions of one-forms. Some texts state that one-forms are simply covariant vectors. However, Goldstein's text (page 290) on classical mechanics gives an example of a one-form in euclidean spherical coordinates that is simply not a vector. It appears to be an ad hoc scaling factor. According to the text, Tensors, Differential Forms, and Variational Principles by Lovelock and Rund, a one-form is simply a specific p-form, and Equation 1.1 on page 131 makes the one-form analogous to the formula for work equalling the force "dotted" into the infinitesimal distance.
Is there a standard definition for a one-form? If so, what is it? Can it be visualized geometrically?