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Below I have posted an excerpt of Lee's Book "Introduction to Smooth Manifolds" (page 371). I don't know what the symbol means that looks like a lower-right corner, and I cannot find it via the index, nor does any alternative source for the definition of the divergence operator have that symbol. If anybody could give a hint what it denotes I can maybe find the passage in the book where it is explained, that would be a HUGE help!!!

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up vote 4 down vote accepted

It is the interior product (Wikipedia page, MathWorld page). It's defined on page 334-336 of Lee's book:

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Oy. Does Lee use $\Lambda^k(V)$ to mean $(\Lambda^k(V))^{\ast}$? – Qiaochu Yuan Apr 24 '12 at 23:08
I plead guilty. In ISM, I used $\Lambda^k(V)$ to denote the space of alternating covariant $k$-tensors on $V$, which most people denote by $\Lambda^k(V^*)$. I also used various other non-standard notations for tensor spaces. My notations were internally consistent and seemed to make sense at the time, but I soon decided they were mistaken, and I've regretted them ever since. In the second edition (scheduled to be released around the end of July), I've switched to more standard notations. – Jack Lee Apr 26 '12 at 0:28

Let $X$ be a vector field and $\omega$ a $k$-form. Then $\iota_X\omega = X ~\lrcorner~ \omega$ is the $(k-1)$-form defined by $$(\iota_X\omega)(V_1, \dots, V_{k-1}) = \omega(X,V_1, \dots, V_{k-1})$$ for any vector fields $V_1, \dots, V_{k-1}$. This is called the interior product of $X$ with $\omega$.

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