Overview
Differential forms is a mathetical structure that gives meaning to objects such as
{% dx %}
so that expressions such as
{% df = \frac{\partial f}{\partial x} dx + \frac{\partial f}{\partial y} dy %}
have a precise mathematical meaning.
One Forms
Given a vector space, {% E %}, one can define the set of one forms to be the set of linear functions on the given vector space into {% \mathbb{R} %} (or possibly some other field)
{% \omega : E \rightarrow \mathbb{R} %}
{% \omega(a \vec{x} + b \vec{y}) = a\omega(\vec{x}) + b\omega(\vec{y}) %}
{% \theta^i(e_i) = \delta^i _j %}
Cartesian Notation
In standard Cartesian space, there are three axes, labeled x,y, and z. Any point is space can be written as a sum of three vectors :
{% (x,y,z) = x \vec{e}_x + y \vec{e}_y + z \vec{e}_z = \sum_i x^i \vec{e}_i %}
- vector {% (1,0,0) %} is often labeled {% e_1 %}. The corresponding one-form {% \theta^1 %} is label {% dx %}
- vector {% (0,1,0) %} is often labeled {% e_2 %} The corresponding one-form {% \theta^2 %} is label {% dy %}
- vector {% (0,0,1) %} is often labeled {% e_3 %} The corresponding one-form {% \theta^3 %} is label {% dz %}
K Forms
A form of degree {% k %} is an alternating tensor over a vector space {% V %}.
{% T(v_1, ... ,v_p, ... v_q,..., v_n) = - T(v_1, ... ,v_q, ... v_p,..., v_n) %}
The set of all alternating tensors of degree {% k %} is denoted {% \Lambda ^k %}.
One of the primary use of forms is in the calculation of n-dimensional volumes. In particular, given n vectors, a n-form will assign a signed volume spanned by the vectors in accordance with the axioms of the Matrix Determinant
Wedge Product
The wedge product is used to create new forms from old. In particular, if {% \omega_1 %} is an n-form and {% \omega_2 %} is an m-form, then {% \omega_1 \wedge \omega_2 %} is an (n+m)-form
- {% \omega_1 \wedge \omega_2 = - \omega_2 \wedge \omega_1 %}
{% dx \wedge dy = -dy \wedge dx %}
This then implies
{% dx \wedge dx = 0 %}