Overview
Exterior differentiation gives meaning to the expression
{% d\omega %}
where {% \omega %}
is a
differential form.
Differntial forms are used in many contexts, but in particular, is used in integration.
Differential of a Function (0-form)
Given a function {% f(x^1,x^2, ..., x^n) %}, (this is a 0-form), the differential is defined to be
{% df = \frac{\partial f}{\partial x_1} dx^1 + \frac{\partial f}{\partial x_2} dx^2 + ... \frac{\partial f}{\partial x_n} dx^n = \sum \frac{\partial f}{\partial x_i} dx^i %}
Extending this definition a bit, if the function {% f %} is vector valued (multi-variable), the differential is defined to be the
total derivative
of {% f %}.
Differential of an m-form
Given a form
{% \omega = f dx^{i1} \wedge dx^{i2} ... dx^{im} %}
{% d \omega = \sum_j \frac{\partial f}{\partial x^{j}} dx^j \wedge dx^{i1} \wedge dx^{i2} ... dx^{im} %}
Linearity
{% d(c_1 \psi_1 + c_2 \psi_2) = c_1 d\psi_1 + c_2 d\psi_2 %}