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I understand, or think I do, vector spaces, tangent spaces, reciprocal bases and how to generate them. I also understand how one forms and vectors act linearly upon each other to make a scalar and of the helpful illustrations of how to think of a one forms geometrically. But I have trouble in understanding how we get from a given vector to its associated one form.

One step at a time.

In a book on EM theory some years ago I came accross a diagram showing oblique 2D axes and a vector radiating from the origin. I believe the terminology now not to be the accepted one for co and contra but this can be sorted easily. The coordinates of the vector arrow obtained by dropping perpendiculars to the x and Y axes were called co-variant components. The coordinates obtained by reading the intersection from the vector arrow of the lines parallel to the other axes were called contra-variant components.

This I suppose is just repersenting the vector in a different way.

If we generate the reciprocal axes relative to the original axes and drop perpendiculars to these from the arrowhead of the original vector my thoughts are that this is just the same object referred to a different set of axes and called a different name. I am told that a one form is also called a covariant vector or covector but cannot see how a one form is described by either of these diagrammatic repesentations. I KNOW I am going wrong but WHERE.

If I have not defined the scenario precisely enough I will try to rectify this if you need me to.

Thanks to the SR people for their previous help but it still did not sink in. Perhaps some input from new angles will help.

Thanks Matheinste.