Differential Forms
Overview
One Forms
Given a
vector space, {% E %},
one can defines the set of one forms to be the set of linear functions on the given vector space into
{% \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.
- 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 %}
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 %}
For example, we have
{% dx \wedge dy = -dy \wedge dx %}
