TITLE: A simple explanation of differential calculus and its link to geometry?
QUESTION [5 upvotes]: The wikipedia articles on differential calculus and differential geometry are quite long and not so straightforward for a layman like me. Is there a master of math vulgarization out there that could summarize in couple strong sentences what differential calculus is and how it is linked to geometry? 
I have read this nice post and this one too which give good examples and references but does not really explain what "differential" means, this post is good too but focuses on equations.

REPLY [2 votes]: Euclidean geometry provides equations between lengths, areas and (trigonometric functions of) angles in figures containing straight lines and circles, including tangents to circles. Differential geometry in the plane provides equations containing in addition to the above lengths of arbitrary smooth curves, angles between tangents of such curves, and finally allows to talk about the curvature of smooth curves.
A typical theorem of plane differential geometry is the following: The total turning angle of the tangent vector to any simple closed curve in the plane (meaning: smooth and no self intersections) is $\pm2\pi$ (and not $2k\pi$ for some $k\ne\pm1$).