TITLE: How do ideas in differential geometry expand upon ideas from introductory calculus
QUESTION [5 upvotes]: I just went through first year in mathematics and used Stewart's book for calculus. I am trying to self study differential manifold and I find many concepts such as chart, atlas very similar to that of a function that I am now used to. Furthermore the notion of a vector field seems to be much more nuanced.
Can someone elaborate on how some simple ideas from introductory calculus are expanded upon through the study of differential manifold? 
For example, what is the equivalent of a manifold in introductory calculus? Are charts the same concept as functions? Are vector fields differential operators same as $\partial$?

REPLY [9 votes]: The fact is that the objects of study in differential geometry (smooth manifolds) build on and generalize the objects of study in elementary geometry (lines, circles, spheres, cylinders, etc.) much more than they generalize the objects of calculus.  It's the tools of calculus that are directly generalized to create the fundamental tools of differential geometry. Differentiation of functions gets generalized to create such concepts as tangent vectors to manifolds, velocity vectors to curves, etc. Integration of functions gets generalized to line integrals of differential forms, volumes of manifolds, average values of functions.  These are just the tip of the iceberg. From there, the path leads into sophisticated advanced concepts like de Rham cohomology, Lie groups, connections, and symplectic structures, none of which would make any sense without the foundational tools provided by calculus.