This defines a one- form since it maps into real numbers and represents the rate of change of along a curve with tangent .

In three dimensions one thinks of a gradient as a vector [ normal to surfaces of constant ] but is a one- form and specifies a vector only if there is a metric.

Now how do the components of transform?

But we also have by the chain rule:

which means that

so

and since we have

This is a useful result, that the basis one- form is just .