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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.

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Perhaps looking at the history of calculus and the history of differential geometry would be helpful. Finding the answer to "why was it invented?" would be a good place to start. –  Antonio Vargas Mar 26 '12 at 16:13
    
Thanks Antonio, research on "why it was invented" took me to this simple explanation:Newton needed calculus for two reasons: finding the "slope of a point" (instead of two points) as well as the area under a curve. The first reason led to "differential calculus", the second one to "integral calculus".The "slope of a point" represents the velocity at a certain time (an instant) and is called instantaneous velocity, it is the line tangent to a position versus time graph. With this calculus one is always searching for a ratio. –  Arthur Mamou-Mani Mar 26 '12 at 17:05

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Here is my understanding : first, differential calculus is a toolbox that allows one to actually compute things in differential geometry.

Differential geometry is a way to describe geometric objects with more precision that for instance topology : in differential geometry, we can make sense of such notions as distance and curvature, which we cannot (or in the case of distance, not so precisely) in usual topology. More importantly, since every bit of our space is linked to a piece of some euclidean space, we can use the tools of differential calculus to describe precisely what "speed" or "acceleration" means; first locally, through maps, then globally, by gluing maps together appropriately.

Now very often, the "geometric objects" we study are in fact phase spaces (spaces of all possible positions and speed of a certain number of physical objects), that are seen with a geometric eye, because it allows to use the tools described earlier to predict the evolution of a physical system.

Concerning the word "differential", see it as a pointer to mean that we study the evolution of the really small differences through functions. For instance, the derivative of a single-variable funtion $f:\mathbf{R}\rightarrow\mathbf{R}$ tells us that a very small difference in the variable $x$ in the source will become $f'(x)$ times this difference in the range, if our difference is "infinitesimally small". In other words, to relate to what I wrote earlier : the "small difference" may be seen as the "speed of variation". The "differential" stuff tels us not only how the points ar afected by functions, but also how the "speeds of variation" are modified.

I hope this helped. I don't know what background you have, but if you are say a 2nd year student in university, I would recommend "Introduction to Smooth Manifolds" by John M. Lee. I have a very geometric eye, and this book appealed to me.

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Thank you very much Samuel T. Is there a way to visualize geometrically some of the concepts you have mentionned above such as "speeds of variation", "derivative of single variable function" and "maps glued together"? I have bought the book by John Lee. Looking forward to it! –  Arthur Mamou-Mani Mar 27 '12 at 12:19
    
For the "speed of variation", imagine the graph of your function. Then the "speed of variation" is the steepness of this graph. The derivativ of a single variable function is exactly the same, when the source of your function is the real line. For gluing maps togecther, imagine precisely what the words mean : imagine you have several incomplete maps of the earth, made of very elastic fabric. Then glue them together by the obvious rule that a given point on one map is glued to itself on the other map. Then if you have enough maps, eventually you will get the entire sphere. –  Samuel T Apr 11 '12 at 14:18
    
I think there are some excellent answers here. I'm curious which book of Lee's you're referring to -- he has quite a few advanced textbooks and it might be very difficult to get through even a page if you don't have much previous experience with calculus. –  snarski Apr 15 '12 at 5:17
    
@snarski The book of Lee's is "Introduction to Smooth Manifolds". It requires some background in calculus and topology, say second year university. The "interpretation" comes from what I heard my professors say, as well as my own way of understanding the subject. –  Samuel T Apr 15 '12 at 23:45

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 $k\ne\pm1$).

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Thanks Christian, does differential geometry always happen in a plane? –  Arthur Mamou-Mani Mar 27 '12 at 12:21
    
@arthurmani: Beware! But I thought I'd begin here. –  Christian Blatter Mar 27 '12 at 14:30

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