Gradients

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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  .