One-form

A 1-form is a type of differential form that can be integrated over a curve in a manifold. In coordinates, a 1-form on \(\mathbb{R}^n\) is an expression of the form:

\[\omega = f_1(x)\,dx^1 + f_2(x)\,dx^2 + \cdots + f_n(x)\,dx^n\]

where each \(f_i(x)\) is a smooth function.

Properties

1-forms are linear functionals on tangent vectors.
They can be pulled back by smooth maps.
The exterior derivative of a 0-form (function) is a 1-form.

Example

Let \(\omega = y\,dx + x\,dy\) on \(\mathbb{R}^2\). For a curve \(\gamma(t) = (t, t^2) \), \( t \in [0,1]\):

\[\int_\gamma \omega = \int_0^1 \left( t^2 \cdot 1 + t \cdot 2t \right) dt = \int_0^1 (t^2 + 2t^2) dt = \int_0^1 3t^2 dt = 1\]

Applications

Line integrals in vector calculus
Fundamental Theorem of Calculus for line integrals
Physics: work done by a force field