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I have been doing a self-study of Differential Forms and Exterior Calculus using the book "Applied Exterior Calculus" by Dominic Edelen (Dover Publication).

I ran across an operator that apparently represents the inner product of a differential form on a vector. Quoted below is equation 3-4.3 of the Edelen book: enter image description here

The operator symbol that I have not seen before is the backward L symbol between the $V$ and $\omega$ to the left of the first equal sign. Clearly it is defining an inner product as noted by other expressions shown.

As I have never seen this before and have not found any other reference to this symbol I am curious as to its origin or even if it has a name.

Therefore, who invented this, what is it called, and is it now obsolete or just so rarely used as I have not found it in other sources. I have a number of other texts on differential forms as it is not used in any of them as far as I have found.

K7PEH
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    https://en.wikipedia.org/wiki/Interior_product – rubikscube09 Mar 09 '22 at 19:51
  • @rubikscube09 -- I am familiar with that Wiki article and that describes a different symbol then the one that I posted. Hardly the same although I can understand they both are inner product operators. No subscripts on the symbol that I found in the book. – K7PEH Mar 09 '22 at 20:47
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    Clarification that might explain some of the variance in use of terminology. The equation quoted where I was highlighting the use of that unknown to me symbol was actually the operation of vector and 1-form (i.e. omega was defined as a 1-form earlier in the text). So, this is probably obvious in the notation for the membership of the result being a 0-form but it is something that helps explain the use of "inner product" in this section of the book (of course I am guessing at the rationale for the usage). – K7PEH Mar 10 '22 at 00:12

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This is the interior product, often written $\iota_V\omega$. This is an example of a contraction. It is defined more generally for any $k$-form $\omega$, resulting in a $(k-1)$-form: $$\iota_V\omega (v_1,\dots,v_{k-1}) = \omega(V,v_1,\dots,v_{k-1}).$$

Ted Shifrin
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  • As I mentioned in a comment above, I have seen the use of iota with subscript before to represent differential form inner product (aka contraction). But, I had never seen the symbol (mirror reflection of L) used before except in this one book that I cited above. It was that particular symbol about what I was curious. My question was not about inner product or contraction but the symbol itself. I don't even know if it has a name -- I have searched math symbols for LaTeX but could not find it. – K7PEH Mar 09 '22 at 21:15
  • It is not a standard (or even semi-standard) LaTeX character. Some mathematicians who use it call it “hook.” – Ted Shifrin Mar 09 '22 at 21:21
  • Calling it inner product is plain wrong, by the way. – Ted Shifrin Mar 09 '22 at 21:28
  • Ted -- yes, it is a matter of getting used to new terminology which is made more confusing by the variety offerred by various authors. My case in point is this operation has been called: inner product, interior-product, interior multiplication, pull-down (which may be slightly different from what I have found), contraction, and even trace (as is more common in Tensor algebra). – K7PEH Mar 09 '22 at 23:04