Happy Face Math

Image credit Charlie Smith

Some pointers/explanations in case you find these cryptic:

A happy face to the power of -1 is the inverse of happy face.
Happy face squared or cubed are pretty easy.
"sup" in mathematics is the supremum of a set.
Then we have a partial derivative of Happy Face.
Next, sine of Happy Face = Happy Face sign by the road.

In the second column, Re( :) ) is the real part of Happy Face, which is Happy Face without the eyes (without i's).
Im of Happy Face is the imaginary part of Happy Face = the eyes (the i's).
[Note, i is the imaginary unit used with imaginary numbers.]

Triangle followed by X is the mathematical operation curl (of a vector).
Next, the upside-down triangle is the gradient of Happy Face or "grad" of Happy Face.
Lastly, Log or logarithm of Happy Face