Here we have additional LaTex conversions:

Here's Green's theorem

{$$ \int_C P\,du + Q\, dv = \int\!\!\!\int_A \left({\partial Q\over \partial u} - {\partial P\over \partial v}\right) {1\over \sqrt{EG-F^2}}\,dA, $$}
 
\int_C P\,du + Q\, dv = \int\!\!\!\int_A \left({\partial Q\over \partial u} - {\partial P\over \partial v}\right) {1\over \sqrt{EG-F^2}}\,dA,

where dA is the element of area of the region R enclosed by the curve C .

Here's from TeXbook (Chapter 17):

{$$ \pi(n) = \sum_{m=2}^n \left\lfloor  \left(\sum_{k=1}^{m-1}\bigl\lfloor(m/k\bigr)\big/\lceil m/k\rceil\big\rfloor \right)^{-1} \right\rfloor $$}
 
\pi(n) = \sum_{m=2}^n \left\lfloor \left(\sum_{k=1}^{m-1}\bigl\lfloor(m/k\bigr)\big/\lceil m/k\rceil\big\rfloor \right)^{-1} \right\rfloor

And from Chapter 18:

{$$ \lim_{n\to\infty} x_n {\rm\ exists} \iff \limsup_{n\to\infty} x_n = \liminf_{n\to\infty} x_n  $$}
 
\lim_{n\to\infty} x_n {\rm\ exists} \iff \limsup_{n\to\infty} x_n = \liminf_{n\to\infty} x_n

Further from Chpater 18:

{$$ {n\choose k} \equiv {\lfloor n/p\rfloor \choose \lfloor k/p\rfloor} {{n\bmod p} \choose {k\bmod p}} \pmod p, $$}
 
{n\choose k} \equiv {\lfloor n/p\rfloor \choose \lfloor k/p\rfloor} {{n\bmod p} \choose {k\bmod p}} \pmod p,