# Etymology of “exterior” in “exterior calculus”

What is the origin of the term "exterior" in "exterior calculus"? How does this term relate to "interior products" and "inner products", if it does at all?

• Looks like an E&LU quesuion. – Mr. Black Mar 8 '17 at 15:24

I think it's well known to have been introduced by Grassmann. He explains the word choice in Die lineale Ausdehnungslehre (1844, pp. x-xi):

I have shown how one can understand as product of two segments the parallelogram (...); this product is distinguished in that factors can only be interchanged with a sign change, while at the same time the product of two parallel segments vanishes. This concept stood in contrast to another (...) Namely when I projected one segment perpendicularly onto another, then the arithmetic product of this projection with the segment projected upon represented likewise a product of the two, insofar as the distributive relation with addition also held. But that product was of a whole other kind than the first, as its factors were interchanged without sign change, and the product of two perpendicular segments vanished. I named the former product exterior, the latter interior, reflecting that the former was nonzero only when involving independent directions, the latter only when involving a shared i.e. partly common one.

and again in the second edition (1862, p. 51):

I chose the name exterior multiplication to emphasize that the product is only nonzero when one factor lies entirely outside the span of the other. Exterior multiplication stands in contrast with the interior one (chap. 4).

• So, just to fill in a step you’ve omitted, that’s maybe not obvious to all readers: Grassmann’s original contrast is between the interior and exterior products of vectors in $\mathbb{R}^3$ — aka the dot-product and cross-product. The former of these was later generalised to “inner products”, etc; the later was generalised to “exterior algebra”, etc. – Peter LeFanu Lumsdaine Mar 8 '17 at 10:16
• So it seems that first the pair "inner product" and "outer product" was conceived, but these two terms separated later. The term "interior product" for the contraction with a vector field in differential geometry is then most likely a later innovation because the "exterior product" had no sibling anymore. – shuhalo Mar 8 '17 at 16:53
• @PeterLeFanuLumsdaine : Yes, if by "later" you mean "on a later page of the same book". (Just leaf through, e.g. exterior at the 1862 link above, and interior at ibid., p. 107.) – Francois Ziegler Mar 8 '17 at 22:29