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,