For a scalar field 
and a world
line of some particle 
, we have
Figure 3.2: World line of particle with four- velocity 
.
If is the tangent to the curve [ the four- velocity of the
particle, see Figure 3.2 ] then:
so
since 
[see section 2.2].
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 
.