Why did the word "exterior" get chosen for the idea of "exterior derivative"? What are the intuitive and historical reasons for choosing the word "exterior" for the concept of an exterior derivative of a form? 
The reasoning I've heard about it is the following: let p(t) be a continuous parametric curve, then if you fix t_0, the tangent line to the curve p(t) at t_0 lies "exterior" of the curve p(t), since it is an approximation of p(t) itself.
 A: I) The term  exterior multiplication ("äussere Multiplication") is due to Grassmann, who introduced the term in his book (written in 1844)
Die Wissenschaft der extensiven Grösse oder die Ausdehnungslehre, eine neue Mathematische Disciplin"
As you can check in the table of contents of the book (on page 276), paragraphs §§34,35  are called  Grundgesetze der äusseren Multiplication (Basic laws of exterior multiplication).
Here is the scan of this  book  by Google .
II) The terminology exterior differential ("différentielle extérieure") was introduced in the 1930's by articles of Elie Cartan, inspired by Grassmann.
Here is a secondary reference from an Analysis course by Chatterji and another by Chern and Chevalley, in their analysis of Elie Cartan's mathematical contributions (cf. in particular pages 229 and 230 )
A: I always presumed it is due to its relationship with the exterior product $\wedge$.  (The latter's name seems natural as a complement to the interior or inner product $\cdot$; one is anti-symmetric while the other is symmetric, etc.)
