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.



Why did the word "exterior" get chosen for the idea of "exterior derivative"?
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](http://ia341214.us.archive.org//load_djvu_applet.php?file=3/items/dielinealeausde00grasgoog/dielinealeausde00grasgoog.djvu) 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](http://books.google.fr/books?id=GtaDSFURUNAC&pg=PA449&lpg=PA449&dq=cartan+elie+produit+exterieur&source=bl&ots=5LCsbUx44Q&sig=QaslOOOe4OvcSZZwjfWcfFRjNoo&hl=fr&ei=P299TPioFYre4Abk4u2nDA&sa=X&oi=book_result&ct=result&resnum=2&ved=0CBkQ6AEwAQ#v=onepage&q&f=false) is a secondary reference from an Analysis course by Chatterji and [another](http://www.ams.org/journals/bull/1952-58-02/S0002-9904-1952-09588-4/S0002-9904-1952-09588-4.pdf) by Chern and Chevalley, in their analysis of Elie Cartan's mathematical contributions (cf. in particular pages 229 and 230 )


